New York Journal of Mathematics

New York J. Math. 15 (2009) 353–392.

The L_{∞}-deformation complex of diagrams of algebras

Yael Frégier, Martin Markl and Donald Yau

Abstract. The deformation complex of an algebra over a colored PROP P is defined in terms of a minimal (or, more generally, cofibrant) model of P. It is shown that it carries the structure of an L_{∞}-algebra which induces a graded Lie bracket on cohomology.

As an example, the L_{∞}-algebra structure on the deformation complex of an associative algebra morphism g is constructed. Another example is the deformation complex of a Lie algebra morphism. The last example is the diagram describing two mutually inverse morphisms of vector spaces. Its L_{∞}-deformation complex has nontrivial l_{0}-term.

Explicit formulas for the L_{∞}-operations in the above examples are given. A typical deformation complex of a diagram of algebras is a fully-fledged L_{∞}-algebra with nontrivial higher operations.

Contents

1. Introduction 354 2. Preliminaries on colored PROPs 355 3. Minimal models and cohomology 362 4. L_{∞}-structure on C_{P}^{∗}(T; T) and deformations 364 5. Deformation complex of an associative algebra morphism 368 6. The higher brackets in C_{As_{B→W}}^{∗}(T; T) 372 7. Deformation complex of a Lie algebra morphism 376 8. The higher brackets in C_{Lie_{B→W}}^{∗} (T; T) 382 9. Deformations of diagrams 386 References 389

Received July 16, 2009. Mathematics Subject Classification. 14D15, 20G10. Key words and phrases. Deformation, colored PROP, diagram of algebras, strongly

homotopy Lie algebra.

The first author worked in the frame of grant F1R-MTH-PUL-08GEOQ of Professor Schlichenmaier. The second author was supported by the grant GA ČR 201/08/0397 and by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.

353^{ISSN 1076-9803/09}

1. Introduction

In this paper, we construct the deformation complex (C_{P}^{∗}(T; T), δ_{P}) of an algebra T over a colored PROP P and observe that it has the structure of an L_{∞}-algebra. The cochain complex (C_{P}^{∗}(T; T), δ_{P}) is so named because its L_{∞}-structure governs the deformations of T in the form of the Quantum Master Equation (13) (Section 4.3).

The existence of an L_{∞}-structure on the deformation complex of an algebra over an operad was proved in 2002 by van der Laan [51]. Van der Laan’s construction was later generalized, in [39], to algebras over properads. The present paper will, however, be based on the approach of the 2004 preprint [36].

Considering colored PROPs is necessary if one is to study L_{∞}-deforma-tions of, say, morphisms or more general diagrams of algebras over a PROP, module-algebras, modules over an associative algebra, and Yetter–Drinfel’d and Hopf modules over a bialgebra. For example, there is a 2-colored PROP As_{B→W} whose algebras are of the form f: U → V, in which U and V are associative algebras and f is a morphism of associative algebras (Example 2.10). Likewise, there is a 2-colored PROP ModAlg whose algebras are of the form (H, A), in which H is a bialgebra and A is an H-module-algebra (Example 2.12). Other examples of colored PROP algebras are given at the end of Section 9.

Let us sketch the construction of the deformation complex (C_{P}^{∗}(T; T), δ_{P}), with details given in Section 3. First we take a minimal model (Definition 3.4) α: (F(E), ∂) → P of the colored PROP P, which should be thought of as a resolution of P. Given a P-algebra ρ: P → End_{T}, we define

C_{P}^{∗}(T; T) = Der(F(E), E),

in which E = End_{T} is considered an F(E)-module via the morphism β = ρα, and Der(F(E), E) denotes the vector space of derivations F(E) → E. The latter has a natural differential δ that sends θ ∈ Der(F(E), E) to θ∂. The reason why we require the minimality of resolutions whenever possible is explained in the first paragraph of Section 3. As we will see in Section 9, the construction sketched out above applies to more general cofibrant resolutions as well.

L_{∞}-deformation complex 355

The L_{∞}-operations on C_{P}^{∗}(T; T) are constructed using graph substitutions (Section 4.4). The usefulness of this very explicit construction of the L_{∞}- operations on C_{P}^{∗}(T; T) is first illustrated with the example of associative algebra morphisms. For a morphism g: U → V of associative algebras, considered as an algebra over the 2-colored PROP As_{B→W}, we are able to write down explicitly all the L_{∞}-operations l_{k} on the deformation complex of g (Theorem 5.5 for k = 1, Theorem 6.2 for k = 2, and Theorem 6.4 for k ≥ 3). As expected, the underlying cochain complex of the deformation complex of g is isomorphic to the Gerstenhaber–Schack^{1} cochain complex [14, 15, 16] of g (Theorem 5.5). Therefore, the latter also has an explicit L_{∞}-structure. See Section 5.1 for more discussion about this deformation complex.

A second example is given by the study of the case of Lie algebra morphisms. As in the associative case, there exists a 2-colored PROP Lie_{B→W} whose 2-colored algebras are morphism of Lie algebras. We obtain then an explicit expression for the L_{∞}-operations (Theorem 7.5 for k = 1, Theorem 8.2 for k = 2, and Theorem 8.4 for k ≥ 3). In particular the first operation l_{1} gives a complex isomorphic to the S-cohomology complex [9]. Hence this answers the question left open in [9] of the existence of such an L_{∞}-structure.

Another example of the L_{∞}-deformation complex is given in Section 9. There is a 2-colored operad Iso (Example 9.1) whose algebras are of the form F: U ⇆ V : G, in which U and V are chain complexes and F and G are mutually inverse chain maps. Using a modification of the results and constructions of earlier sections, we will write down explicitly the L_{∞}-operations on the deformation complex of a typical Iso-algebra T (Example 9.3).

Describing the L_{∞}-deformation complex requires the knowledge of a resolution of the corresponding colored PROP. So, while it will be interesting to work out the L_{∞}-deformation complex for morphisms of bialgebras, a description of a resolution of the PROP Bialg_{B→W} for morphisms of bialgebras is missing, though the minimal model of the bialgebra PROP Bialg is known [47].

Acknowledgment. We would like to thank Jim Stasheff and Bruno Vallette for reading the first version of the manuscript and many useful remarks.

2. Preliminaries on colored PROPs

Fix a ground field k, assumed to be of characteristic 0. This assumption is useful in considering models for operads or PROPs since it guarantees the existence of the ‘averagization’ of a nonequivariant map into an equivariant one. The characteristic zero assumption also simplifies concepts of Lie algebras and their generalizations.

In this section, we review some basic definitions about colored PROPs (and colored operads as their particular instances), their algebras, colored Σ-bimodules, and free colored PROPs. Examples of algebras over colored PROPs can be found at the end of this section.

2.1. Colored Σ-bimodule. Let ℭ be a nonempty set whose elements are called colors. A Σ-bimodule is a collection E = {E(m, n)}_{m,n≥0} of k-modules in which each E(m, n) is equipped with a left Σ_{m} and a right Σ_{n} actions that commute with each other.

1Be careful with the possible confusion with the complex associated to bialgebras. Both of these complexes share the same name.

A ℭ-colored Σ-bimodule is a Σ-bimodule E in which each E(m, n) admits a ℭ-colored decomposition into submodules,

(1)  E(m, n) = ⨁_{c_{i},d_{j}∈ℭ}E\binom{d_{1}, …, d_{m}}{c_{1}, …, c_{n}},

that is compatible with the Σ_{m}-Σ_{n}-actions. Elements of E(m, n) are said to have biarity (m, n). A morphism of ℭ-colored Σ-bimodules is a linear biequivariant map that respects the ℭ-colored decompositions (1).

ℭ-colored Σ-bimodules and their morphism are examples of ℭ-colored objects; more examples will follow. If ℭ has k elements, we will sometimes call ℭ-colored objects simply k-colored objects.

Definition 2.2. A ℭ-colored PROP ([27, 28], [35, Section 8]) is a ℭ-colored Σ-bimodule P = {P(m, n)} (so each P(m, n) admits a ℭ-colored decomposition (1)) that comes equipped with two operations: a horizontal composition

⊗: P\binom{d_{11}, …, d_{1m_{1}}}{c_{11}, …, c_{1n_{1}}} ⊗ ⋯ ⊗ P\binom{d_{s1}, …, d_{sm_{s}}}{c_{s1}, …, c_{sn_{s}}} →

P\binom{d_{11}, …, d_{sm_{s}}}{c_{11}, …, c_{sn_{s}}} ⊆ P(m_{1} + ⋯ + m_{s}, n_{1} + ⋯ + n_{s})

and a vertical composition

(2) ∘: P\binom{d_{1}, …, d_{m}}{c_{1}, …, c_{n}} ⊗ P\binom{b_{1}, …, b_{n}}{a_{1}, …, a_{k}} → P\binom{d_{1}, …, d_{m}}{a_{1}, …, a_{k}} ⊆ P(m, k),
(x, y) ↦ x ∘ y.

These two compositions are required to satisfy some associativity-type axioms. There is also a unit element 1_{c} ∈ P\binom{c}{c} for each color c. Moreover, the vertical composition x ∘ y in (2) is 0, unless

c_{i} = b_{i} for 1 ≤ i ≤ n.

Morphisms of ℭ-colored PROPs are unit-preserving morphisms of the underlying Σ-bimodules that commute with both the horizontal and the vertical compositions.

Colored operads are particular cases of colored PROPs having the property that P\binom{d_{1},...,d_{m}}{a_{1},...,a_{k}} = 0 for m ≥ 2. Note that colored PROPs can also be defined as ordinary (1-colored) PROPs over the semisimple algebra K = ⊕_{c∈ℭ}k_{c}, where each k_{c} is a copy of the ground field k [32, Section 2].

Example 2.3. The ℭ-colored endomorphism PROP End_{T}^{ℭ} of a ℭ-graded module T = ⊕_{c∈ℭ}T_{c} is the ℭ-colored PROP with

End_{T}^{ℭ} \binom{d_{1}, …, d_{m}}{c_{1}, …, c_{n}} = Hom_{k}(T_{c_{1}} ⊗ ⋯ ⊗ T_{c_{n}}, T_{d_{1}} ⊗ ⋯ ⊗ T_{d_{m}}).

The horizontal composition is given by tensor products of k-linear maps. The vertical composition is given by composition of k-linear maps with matching colors.

Definition 2.4. For a ℭ-colored PROP P, a P-algebra is a morphism of ℭ-colored PROPs

α: P → End_{T}^{ℭ}

for some ℭ-graded module T = ⊕_{c∈ℭ}T_{c}. In this case, we say that T is a P-algebra.

2.5. Pasting scheme for ℭ-colored PROPs. For natural m, n ≥ 1, let UGr^{ℭ}(m, n) be the set whose elements are pairs (G, ζ) such that:

(1) G ∈ UGr(m, n) is a directed (m, n)-graph [35, p. 38].

(2) For each vertex v ∈ Vert(G), the sets out(v) (outgoing edges from v) and in(v) (incoming edges to v) are labeled 1, …, q and 1, …, p, respectively, where #out(v) = q and #in(v) = p.

(3) ζ: edge(G) → ℭ is a function that assigns to each edge in G a color in ℭ. For any edge l ∈ edge(G), ζ(l) ∈ ℭ is called the color of l.

There is a ℭ-colored decomposition

UGr^{ℭ}(m, n) = ∐_{c_{i},d_{j}∈ℭ}UGr^{ℭ}\binom{d_{1}, …, d_{m}}{c_{1}, …, c_{n}},

where (G, ζ) ∈ UGr^{ℭ}\binom{d_{1},...,d_{m}}{c_{1},...,c_{n}} if and only if the input legs \{l_{in}^{1}, … l_{in}^{n}\} of G have colors c_{1}, …, c_{n} and the output legs {l_{out}^{1}, …, l_{out}^{m}} of G have colors d_{1}, …, d_{m}.

As in [35], UGr^{ℭ}(m, n) and UGr^{ℭ}\binom{d_{1},...,d_{m}}{c_{1},...,c_{n}} are the categories with colorrespecting isomorphisms as morphisms. Elements in UGr^{ℭ}(m, n) are called ℭ-colored directed (m, n)-graphs.

2.6. Decoration on colored directed graphs. Let E be a ℭ-colored Σbimodule and (G, ζ) be a ℭ-colored directed (m, n)-graph. Define

(3)  E(G, ζ) = ⨂_{v∈Vert(G)} E\binom{ζ(o_{v}^{1}), …, ζ(o_{v}^{q})}{ζ(i_{1}^{v}), …, ζ(i_{p}^{v})},

where in(v) = {i_{1}^{v}, …, i_{p}^{v}} and out(v) = {o_{v}^{1}, …, o_{v}^{q}}. Its elements are called E-decorated ℭ-colored directed (m, n)-graphs.

For an element Γ = ⊗_{v}e_{v} ∈ E(G, ζ), the element e_{v} ∈ E\binom{ζ(o_{v}^{1}),...,ζ(o_{v}^{q})}{ζ(i_{1}^{v}),...,ζ(i_{p}^{v})} corresponding to the vertex v ∈ Vert(G) is called the decoration of v.

In other words, E(G, ζ) is the space of decorations of the vertices of the ℭ-colored directed (m, n)-graph (G, ζ) with elements of E with matching biarity and colors.

2.7. Free colored PROP. Let E be a ℭ-colored Σ-bimodule. For c_{1}, …, c_{n}, d_{1}, …, d_{m} ∈ ℭ, define the module

F^{ℭ}(E)\binom{d_{1}, …, d_{m}}{c_{1}, …, c_{n}} = colim E(G, ζ),

where the colimit is taken over the category UGr^{ℭ}\binom{d_{1},...,d_{m}}{c_{1},...,c_{n}}. Then

F^{ℭ}(E) = \{F^{ℭ}(E)(m, n) = ⨁_{c_{i},d_{j}∈ℭ}F^{ℭ}(E)\binom{d_{1}, …, d_{m}}{c_{1}, …, c_{n}}\}

is a ℭ-colored PROP, in which the horizontal composition ⊗ is given by disjoint union of E-decorated ℭ-colored directed (m, n)-graphs. The vertical composition ∘ in F^{ℭ}(E) is given by grafting of ℭ-colored legs with matching colors.

Note that there is a natural ℤ_{≥0}-grading,

F^{ℭ}(E) = ⨁_{k≥0} F_{k}^{ℭ}(E),

where F_{k}^{ℭ}(E) is the submodule generated by the monomials involving k elements in E.

Proposition 2.8 (= ℭ-colored version of Proposition 57 in [35]). F^{ℭ}(E) = {F^{ℭ}(E)(m, n)} is the free ℭ-colored PROP generated by the ℭ-colored Σbimodule E. In other words, the functor F^{ℭ} is the left adjoint of the forgetful functor from ℭ-colored PROPs to ℭ-colored Σ-bimodules.

In particular, elements in the free ℭ-colored PROP F^{ℭ}(E) can be written as sums of E-decorated ℭ-colored directed graphs.

Convention 2.9. From now on, everything will be tacitly assumed to be ℭcolored with a suitable set of colors ℭ. When there is no danger of ambiguity, we will, for brevity, suppress ℭ from the notation.

Example 2.10 (Morphisms). Let P be an ordinary PROP (i.e., a 1-colored PROP). Then there is a 2-colored PROP P_{B→W} whose algebras are of the form f: U → V, in which U and V are P-algebras and f is a morphism of P-algebras [31, Example 1]. It can be constructed as the quotient

P_{B→W} = \frac{P_{B} ∗ P_{W} ∗ F(f)}{(f^{⊗m}x_{B} = x_{W}f^{⊗n} for all x ∈ P(m, n))},

where P_{B} and P_{W} are copies of P concentrated in the colors B and W, respectively, x_{B} and x_{W} are the respective copies of x in P_{B} and P_{W}, and F(f) is the free 2-colored PROP on the generator f: B → W. The star ∗ denotes the free product (= the coproduct) of 2-colored PROPs.

In the case that P is the operad As for associative algebras, cohomology of As_{B→W}-algebras (i.e., associative algebra morphisms) will be discussed in details in Sections 5 and 6.

Example 2.11 (Modules). There is a 2-colored operad AsMod whose algebras are of the form (A, M), where A is an associative algebra and M is a left A-module. It can be constructed as the quotient

AsMod = \frac{F(μ, λ)}{(μ(μ ⊗ 1_{A}) − μ(1_{A} ⊗ μ), λ(μ ⊗ 1_{M}) − λ(1_{A} ⊗ λ))}.

Here F(μ, λ) is the free 2-colored operad (with ℭ = {A, M}) on the generators,

μ ∈ F(μ, λ)\binom{A}{A, A} and λ ∈ F(μ, λ)\binom{M}{A, M},

which encode the multiplication in A and the left A-action on M, respectively.

If we depict the multiplication μ as • and the module action λ as • , then the associativity of μ is expressed by the diagram

(4)  \begin{aligned}& • \\ • \end{aligned} = _{••}

and the compatibility between the multiplication and the module action by

(5)  \begin{aligned}& • \\ • \end{aligned} = _{••} .

The diagrams in the above two displays should be interpreted as elements of the free colored PROP F(μ, λ), with the A-colored edges of the underlying graph represented by simple lines , and the M-colored edges by the double lines . We use the convention that the directed edges point upwards, i.e. the composition is performed from the bottom up.

Example 2.12 (Module-algebras). Let H = (H, μ_{H}, Δ_{H}) be a (co)associative bialgebra. An H-module-algebra is an associative algebra (A, μ_{A}) that is equipped with a left H-module structure such that the multiplication map on A becomes an H-module morphism. In other words, the module-algebra axiom

x(ab) = ∑_{(x)} (x_{(1)}a)(x_{(2)}b)

holds for x ∈ H and a, b ∈ A, where Δ_{H}(x) = ∑_{(x)} x_{(1)} ⊗ x_{(2)} using the Sweedler’s notation for comultiplication.

This algebraic structure arises often in algebraic topology [2], quantum groups [21], Lie and Hopf algebras theory [7, 40, 49], and group representations [1]. For example, in algebraic topology, the complex cobordism ring MU^{∗}(X) of a topological space X is an S-module-algebra, where S is the Landweber–Novikov algebra [26, 43] of stable cobordism operations.

Another important example of a module-algebra arises in the theory of Lie algebras. Finite-dimensional simple sl(2, ℂ)-modules are, up to isomorphism, the highest weight modules V (n) (n ≥ 0) [20, Theorem 7.2]. There is a U(sl(2, ℂ))-module-algebra structure on the polynomial algebra ℂ[x, y] such that the submodule ℂ[x, y]_{n} of homogeneous polynomials of degree n is isomorphic to the highest weight module V (n) [21, Theorem V.6.4]. In other words, all the finite-dimensional simple sl(2, ℂ)-modules can be encoded inside a single U(sl(2, ℂ))-module-algebra.

There is a 2-colored PROP ModAlg whose algebras are of the form (H, A), where H is a (co)associative bialgebra and A is an H-module-algebra. (The third author first learned about this fact from Bruno Vallette in private correspondence.) It can be constructed as the quotient (with ℭ = {H, A})

(6)  ModAlg = F(μ_{H}, Δ_{H}, μ_{A}, λ)/I,

where F = F(μ_{H}, Δ_{H}, μ_{A}, λ) is the free 2-colored PROP on the generators:

μ_{H} ∈ F\binom{H}{H, H}, Δ_{H} ∈ F\binom{H, H}{H}, μ_{A} ∈ F\binom{A}{A, A}, and λ ∈ F\binom{A}{H, A},

which encode the multiplication and comultiplication in H, the multiplication in A, and the H-module structure on A, respectively. The ideal I is generated by the elements:

μ_{H}(μ_{H} ⊗ 1_{H}) − μ_{H}(1_{H} ⊗ μ_{H}) (associativity of μ_{H}),

(Δ_{H} ⊗ 1_{H})Δ_{H} − (1_{H} ⊗ Δ_{H})Δ_{H} (coassociativity of Δ_{H}),
Δ_{H}μ_{H} − μ_{H}^{⊗2}(2 3)Δ_{H}^{⊗2} (compatibility of μ_{H} and Δ_{H}),
μ_{A}(μ_{A} ⊗ 1_{A}) − μ_{A}(1_{A} ⊗ μ_{A}) (associativity of μ_{A}),
λ(μ_{H} ⊗ 1_{A}) − λ(1_{H} ⊗ λ) (H-module axiom),
λ(1_{H} ⊗ μ_{A}) − μ_{A}λ^{⊗2}(2 3)(Δ_{H} ⊗ 1_{A}^{⊗2}) (module-algebra axiom).

Here (2 3) ∈ Σ_{4} is the permutation that switches 2 and 3.

If we draw the multiplication μ_{H} as • , the comultiplication Δ_{H} as • , the multiplication μ_{A} as • , and the H-module action λ as • , then the bialgebra axioms for H are expressed by

\begin{aligned}& • \\ • \end{aligned} = _{••} , • • = •• and • • =  • • • •  ,

The associativity of μ_{H} is given by the obvious -colored version of (4), the H-module axiom by (5), and the module-algebra axiom by

•• = • • • • .

Variants of module-algebras, including module-co/bialgebras and comodule-(co/bi)algebras are algebras over similar 2-colored PROPs. Deformations, in the classical sense [13], of module-algebras and its variants were studied in [53, 54].

Example 2.13 (Entwining structures). An entwining structure [4, 6] is a tuple (A, C, ψ), in which A = (A, μ) is an associative algebra, C = (C, Δ) is a coassociative coalgebra, and ψ: C ⊗ A → A ⊗ C, such that the following two entwining axioms are satisfied:

ψ(Id_{C} ⊗μ) = (μ ⊗ Id_{C})(Id_{A} ⊗ψ)(ψ ⊗ Id_{A}),
(Id_{A} ⊗Δ)ψ = (ψ ⊗ Id_{C})(C ⊗ ψ)(Δ ⊗ Id_{A}).

(7)

If we symbolize μ by • , Δ by • and ψ by • , then the entwining axioms can be written as

•• = • • • and •• = • • • .

This algebraic structure arises in the study of coalgebra-Galois extension and its dual notion, algebra-Galois coextension [5], generalizing the Hopf-Galois extension of [23].

There is a 2-colored PROP Ent whose algebras are entwining structures. It can be constructed as the quotient

Ent = F(μ, Δ, ψ)/I

of the free 2-colored PROP F = F(μ, Δ, ψ) (with ℭ = {A, C}) on the generators:

μ ∈ F\binom{A}{A, A}, Δ ∈ F\binom{C, C}{C}, and ψ ∈ F\binom{A, C}{C, A}.

The ideal I is generated by the elements expressing the associativity of μ, the coassociativity of Δ, and the two entwining axioms (7).

Example 2.14 (Yetter–Drinfel’d modules). A Yetter–Drinfel’d module [56] (also called a crossed bimodule and quantum Yang–Baxter module) over a (co)associative bialgebra (H, μ, Δ) is a vector space M together with a left H-module action ω: H ⊗M → M and a right H-comodule coaction ρ: M → M ⊗ H that satisfy the Yetter–Drinfel’d condition,

(8) (Id_{M} ⊗μ) ∘ (ρ ⊗ Id_{H}) ∘ τ ∘ (Id_{H} ⊗ω) ∘ (Δ ⊗ Id_{M}) = (ω ⊗ μ) ∘ (Id_{H} ⊗τ ⊗ Id_{H}) ∘ (Δ ⊗ ρ),

where τ is the twist isomorphism H ⊗ M ≅ M ⊗ H. If we depict μ as • , Δ as • , ω as • , and ρ as • , then

• • • •  =  • • • •  .

Yetter–Drinfel’d modules were introduced by Yetter [56], and are studied further in [25, 44, 45, 46, 48], among others. If the bialgebra H is a finitedimensional Hopf algebra, then the left-modules over its Drinfel’d double D(H) are exactly the Yetter–Drinfel’d modules over H. These objects play important roles in the theory of quantum groups and mathematical physics. Indeed, a finite-dimensional Yetter–Drinfel’d module M gives rise to a solution of the quantum Yang–Baxter equation [25, 45] (i.e., an R-matrix [21, Chapter VIII]). Conversely, through the so-called FRT construction [8, 21], every R-matrix on a finite-dimensional vector space gives rise to a Yetter– Drinfel’d module over some bialgebra. Cohomology for Yetter–Drinfel’d modules and their morphisms over a fixed bialgebra have been studied in [44] and [55], respectively.

There is a 2-colored PROP YD whose algebras are of the form (H, M), where H is a bialgebra and M is a Yetter–Drinfel’d module over H. It can be constructed as the quotient

YD = F(μ, Δ, ω, ρ)/I

of the free 2-colored PROP F = F(μ, Δ, ω, ρ) (with ℭ = {H, M}) on the generators:

μ ∈ F\binom{H}{H, H}, Δ ∈ F\binom{H, H}{H}, ω ∈ F\binom{M}{H, M}, and ρ ∈ F\binom{M, H}{M}.

The ideal I is generated by elements expressing the bialgebra axioms for μ and Δ, the left H-module axiom for ω, the right H-comodule axiom for ρ, and the Yetter–Drinfel’d condition (8).

Example 2.15 (Hopf modules). A Hopf module over a bialgebra (H, μ, Δ) is a vector space M together with a left H-module action ω: H ⊗ M → M and a right H-comodule coaction ρ: M → M ⊗ H that satisfy the Hopf module condition:

(9)  ρ ∘ ω = (ω ⊗ μ) ∘ (Id_{H} ⊗τ ⊗ Id_{H}) ∘ (Δ ⊗ ρ).

There is a 2-colored PROP HopfMod whose algebras are of the form (H, M), in which H is a bialgebra and M is a Hopf module over H. It admits the same construction as YD, except that the Yetter–Drinfel’d condition (8) is replaced by the Hopf module condition (9) in the ideal I of relations.

3. Minimal models and cohomology

In this section, we define (i) minimal models for colored PROPs and (ii) cohomology for algebras over a colored PROP based on minimal models. Since minimal models are (at least in most cases) known to be unique up to isomorphism, the cohomology based on minimal models is unique already on the chain level. We will therefore require minimality whenever possible, though an arbitrary cofibrant resolution should give the same cohomology. There are, however, important PROPs that do not have a minimal model, as the colored operad Iso considered in Section 9.

First we recall the notions of modules and derivations for colored PROPs.

3.1. Modules. For a ℭ-colored PROP P and a ℭ-colored Σ-bimodule U, a P-module structure on U [30, p. 203] consists of the following operations:

\begin{aligned}∘ = ∘_{l} & : P\binom{d_{1}, …, d_{m}}{c_{1}, …, c_{n}}⊗U\binom{b_{1}, …, b_{n}}{a_{1}, …, a_{k}} → U\binom{d_{1}, …, d_{m}}{a_{1}, …, a_{k}}, \\ ∘ = ∘_{r} & : U\binom{d_{1}, …, d_{m}}{c_{1}, …, c_{n}}⊗P\binom{b_{1}, …, b_{n}}{a_{1}, …, a_{k}} → U\binom{d_{1}, …, d_{m}}{a_{1}, …, a_{k}}, \\ ⊗ = ⊗_{l} & : P\binom{d_{1}, …, d_{m_{1}}}{c_{1}, …, c_{n_{1}}}⊗U\binom{b_{1}, …, b_{m_{2}}}{a_{1}, …, a_{n_{2}}} → U\binom{d_{1}, …, d_{m_{1}}, b_{1}, …, b_{m_{2}}}{c_{1}, …, c_{n_{1}}, a_{1}, …, a_{n_{2}}}, \\ ⊗ = ⊗_{r} & : U\binom{d_{1}, …, d_{m_{1}}}{c_{1}, …, c_{n_{1}}}⊗P\binom{b_{1}, …, b_{m_{2}}}{a_{1}, …, a_{n_{2}}} → U\binom{d_{1}, …, d_{m_{1}}, b_{1}, …, b_{m_{2}}}{c_{1}, …, c_{n_{1}}, a_{1}, …, a_{n_{2}}}. \end{aligned}

As usual, the vertical operations ∘_{l} and ∘_{r} are trivial unless b_{i} = c_{i} for 1 ≤ i ≤ n. The following compatibility axioms are also imposed on the four operations:

\begin{aligned}f ∘ (g ∘ h) & = (f ∘ g) ∘ h, \\ f ⊗ (g ⊗ h) & = (f ⊗ g) ⊗ h, \\ (f_{1} ∘ f_{2}) ⊗ (g_{1} ∘ g_{2}) & = (f_{1} ⊗ g_{1}) ∘ (f_{2} ⊗ g_{2}). \end{aligned}

Here exactly one of f, g, and h lies in U and the other two lie in P. Likewise, exactly one of f_{1}, f_{2}, g_{1}, and g_{2} lies in U and the other three lie in P.

We note that P-modules can also be defined as abelian group objects in the category PROP/P of ℭ-colored PROPs over P.

For example, if β: P → Q is a morphism of ℭ-colored PROPs, then Q becomes a P-module via β in the obvious way.

3.2. Derivations. Given a P-module U, a derivation P → U is a ℭ-colored Σ-bimodule morphism d: P → U that satisfies the usual derivation property with respect to both the vertical operations ∘ and the horizontal operations ⊗ [30, p. 204]. Denote by Der(P, U) the vector space of derivations P → U.

Proposition 3.3 (= ℭ-colored version of Proposition 3 in [36]). Let U be an F^{ℭ}(E)-module for some ℭ-colored Σ-bimodule E. Then there is a canonical isomorphism

(10)  Der(F^{ℭ}(E), U) ≅ Hom_{Σ}^{ℭ}(E, U),

where Hom_{Σ}^{ℭ}(E, U) denotes the vector space of ℭ-colored Σ-bimodule morphisms E → U.

In one direction, isomorphism (10) takes a derivation θ ∈ Der(F^{ℭ}(E), U) to its restriction θ|_{E} to the space E of generators. In the other direction, it takes a map φ: E → U ∈ Hom_{Σ}^{ℭ}(E, U) to its unique extension Ex(φ): F^{ℭ}(E) → U as a derivation such that Ex(φ)|_{E} = φ.

Following [29, 34], we make the following definition.

Definition 3.4. Let P be a ℭ-colored PROP. A minimal model of P is a differential graded ℭ-colored PROP (F^{ℭ}(E), ∂) for some ℭ-colored Σ-bimodule E together with a homology isomorphism

ρ: (F^{ℭ}(E), ∂) → (P, 0)

such that the following minimality condition is satisfied:

∂(E) ⊆ ⨁_{k≥2} F_{k}^{ℭ}(E).

In other words, the image of E under ∂ consists of decomposables.

3.5. Cohomology. Here we define cohomology of an algebra over a colored PROP following [30, 36].

Let P be a ℭ-colored PROP, and let (F^{ℭ}(E), ∂) ⟶^{ρ} (P, 0) be a minimal model of P. Let P ⟶^{α} End_{T}^{ℭ} be a P-algebra structure on T = ⊕_{c∈ℭ}T_{c}. Consider End_{T}^{ℭ} as an F^{ℭ}(E)-module via the morphism

β = αρ: F^{ℭ}(E) → End_{T}^{ℭ} .

Then the map

Der(F^{ℭ}(E), End_{T}^{ℭ}) ⟶^{δ} Der(F^{ℭ}(E), End_{T}^{ℭ})
θ ↦ θ∂

is well-defined and is a differential (δ^{2} = 0) because ∂^{2} = 0.

Definition 3.6. In the above setting, define the cochain complex

(11)  C_{P}^{∗}(T; T) = ↑ Der(F^{ℭ}(E), End_{T}^{ℭ})^{−∗},

where the degree +1 differential δ_{P} is induced by δ, ↑ denotes suspension, and −∗ denotes reversed grading. We call (C_{P}^{∗}(T; T), δ_{P}) the deformation complex of T. Its cohomology,

H_{P}^{∗}(T; T) = H(C_{P}^{∗}(T; T), δ_{P}),

is called the cohomology of T with coefficients in itself.

Note that if ℭ = {∗}, i.e., P is an ordinary (1-colored) PROP, then (C_{P}^{∗}(T; T), δ_{P}) and H_{P}^{∗}(T; T) defined above coincide with the definitions in [30, 36].

4. L_{∞}-structure on C_{P}^{∗}(T; T) and deformations

In this section, we observe that the deformation complex (C_{P}^{∗}(T; T), δ_{P}) (Definition 3.6) of an algebra T over a colored PROP P has the natural structure of an L_{∞}-algebra (Theorem 4.2). The relationship between this L_{∞}-algebra and deformations of T is discussed in Section 4.3. An explicit construction of the L_{∞}-operations l_{k} in (C_{P}^{∗}(T; T), δ_{P}) is given in Section 4.4. This construction will first be applied in Sections 5 and 6 to obtain very explicit formulas for the operations l_{k} in the deformation complex of an associative algebra morphism.

First we recall the notion of an L_{∞}-algebra.

Definition 4.1 (Definition 2.1 in [24], Example 3.90 in [37]). An L_{∞}- structure on a ℤ-graded module V consists of a sequence of operations (δ = l_{1}, l_{2}, l_{3}, …) with

l_{n}: V^{⊗n} → V

of degree 2 − n such that each l_{n} is anti-symmetric and the condition

(12) ∑_{i+j=n+1} ∑_{σ} χ(σ)(−1)^{i(j−1)}l_{j} (l_{i}(x_{σ(1)}, ..., x_{σ(i)}), x_{σ(i+1)}, ..., x_{σ(n)}) = 0

holds for n ≥ 1. Here σ runs through all the (i, n − i)-unshuffles for i ≥ 1, and

χ(σ) = sgn(σ) · ε(σ; x_{1}, …, x_{n}),

where ε(σ; x_{1}, …, x_{n}) is the Koszul sign given by

x_{1} ∧ ⋯ ∧ x_{n} = ε(σ; x_{1}, …, x_{n}) · x_{σ(1)} ∧ ⋯ ∧ x_{σ(n)}.

In this case, we call (V, δ, l_{2}, l_{3}, …) an L_{∞}-algebra. The anti-symmetry of l_{n} means that

l_{n}(x_{σ(1)}, …, x_{σ(n)}) = χ(σ)l_{n}(x_{1}, …, x_{n})

for σ ∈ Σ_{n} and x_{1}, …, x_{n} ∈ V.

Theorem 4.2. In the setting of §3.5, there exists an L_{∞}-algebra structure (δ_{P}, l_{2}, l_{3}, …) on C_{P}^{∗}(T; T) capturing deformations of colored P-algebras in the sense of §4.3 below. This L_{∞}-structure induces a graded Lie algebra structure on the cohomology H_{P}^{∗}(T; T).

Proof. This is the ℭ-colored version of [36, Theorem 1], whose proof, with some very minor modifications, applies to the ℭ-colored setting as well. In fact, Sections 3 and 4 in [36] (which contain the proof of Theorem 1 in that paper) apply basically verbatim to the ℭ-colored setting. An explicit “graphical” construction of the operations l_{k} will be given below (§4.4). □

4.3. Deformations of colored PROP algebras. Section 5 in [36] concerning deformations of algebras over a PROP also applies to the ℭ-colored setting without change. In particular, deformations of an algebra T over a ℭ-colored PROP P (i.e., F^{ℭ}(E)-algebra structures on T) correspond to elements κ ∈ C_{P}^{1}(T; T) that satisfy the Quantum Master Equation [36, Eq.(4)]:

(13)  0 = δ_{P}(κ) + \frac{1}{2!}l_{2}(κ, κ) − \frac{1}{3!}l_{3}(κ, κ, κ) − \frac{1}{4!}l_{4}(κ, κ, κ, κ) + · · · .

In other words, the L_{∞}-algebra

(C_{P}^{∗}(T; T), δ_{P}, l_{2}, l_{3}, …)

in Theorem 4.2 controls the deformations of T as a P-algebra. As explained in [36, Introduction], this L_{∞}-algebra is an L_{∞}-version of the Deligne groupoid [18, 19] governing deformations that are described by the usual Master Equation (also known as the Maurer–Cartan Equation):

0 = dκ + \frac{1}{2}[κ, κ].

When P is a properad [50], there is another approach to studying L_{∞}- deformations of P-algebras due to Merkulov and Vallette [39]. Their approach is based on a generalization of Van der Laan’s homotopy (co)operads [51] to homotopy (co)properads. They show that the deformation complex (C_{P}^{∗}(T; T), δ_{P}) inherits a L_{∞}-algebra structure from a homotopy properad (Theorem 28 of [39]). Vallette recently informed the third author in private correspondence that the paper [39] can also be extended to the colored setting.

4.4. Construction of the operations l_{k} on C_{P}^{∗}(T; T). Here we describe how the operations l_{k} in Theorem 4.2 are constructed, again following [36, Section 2] closely.

Suppose that F_{1}, …, F_{k} ∈ Hom_{Σ}^{ℭ}(E, End_{T}^{ℭ}) and that Γ ∈ E(G, ζ) is an E-decorated ℭ-colored directed (m, n)-graph (3) with underlying ℭ-colored graph (G, ζ) ∈ UGr^{ℭ}(m, n). Let v_{1}, …, v_{k} ∈ Vert(G) be k distinct vertices in G. Consider the End_{T}^{ℭ} -decorated ℭ-colored directed (m, n)-graph

Γ_{{β}}^{{v_{1},...,v_{k}}}[F_{1}, …, F_{k}] ∈ End_{T}^{ℭ} (G, ζ)

obtained from Γ by:

(1) replacing the decoration e_{v_{i}} ∈ E of the vertex v_{i} by F_{i}(e_{v_{i}}) ∈ End_{T}^{ℭ} for 1 ≤ i ≤ k, and

(2) replacing the decoration e_{v} ∈ E of any vertex v ∉ {v_{1}, …, v_{k}} by β(e_{v}) = αρ(e_{v}).

The graph Γ_{{β}}^{{v_{1},...,v_{k}}}[F_{1}, …, F_{k}] is visualized in Figure 1 which is a colored version of a picture taken from [36]. Using the ℭ-colored PROP structure on End_{T}^{ℭ} (Example 2.3), the graph Γ_{{β}}^{{v_{1},...,v_{k}}}[F_{1}, …, F_{k}] produces an element

γ (Γ_{\{β\}}^{\{v_{1},...,v_{k}\}}[F_{1}, …, F_{k}]) ∈ End_{T}^{ℭ} \binom{ζ(l_{out}^{1}), …, ζ(l_{out}^{m})}{ζ(l_{in}^{1}), …, ζ(l_{in}^{n})} ⊆ End_{T}^{ℭ} (m, n).

Here {l_{out}^{1}, …, l_{out}^{m}} and {l_{in}^{1}, …, l_{in}^{n}} are the output and input legs, respectively, of (G, ζ).

Now pick cochains f_{1}, …, f_{k} ∈ C_{P}^{∗}(T; T) corresponding to F_{1}, …, F_{k} ∈ Hom_{Σ}^{ℭ}(E, End_{T}^{ℭ}) under the isomorphism (10):

(14) C_{P}^{∗}(T; T) = ↑ Der(F^{ℭ}(E), End_{T}^{ℭ})^{−∗} ≅ ↑ Hom_{Σ}^{ℭ}(E, End_{T}^{ℭ})^{−∗}.

[figure: ⋯; ⋯; β ... ... •; β ... ... •; β ... ... •; F_{k} ... ... •; F_{2} ... ... •; F_{3} ... ... •; F_{1} ... ... •]

Figure 1. End_{T}^{ℭ} -decorated graph Γ_{{β}}^{{v_{1},...,v_{k}}}[F_{1}, …, F_{k}]. Vertices labelled F_{i} are decorated by F_{i}(e_{v_{i}}), 1 ≤ i ≤ k, the remaining vertices are decorated by β(e_{v}).

If ξ ∈ E\binom{d_{1},...,d_{m}}{c_{1},...,c_{n}}, then ∂(ξ) ∈ F^{ℭ}(E)\binom{d_{1},...,d_{m}}{c_{1},...,c_{n}} can be written as a finite sum

∂(ξ) = ∑_{s∈S_{ξ}} Γ_{s},

where each Γ_{s} ∈ E(G, ζ) for some (G, ζ) ∈ UGr^{ℭ}\binom{d_{1},...,d_{m}}{c_{1},...,c_{n}}. Define

l_{k}(f_{1}, …, f_{k})(ξ) ∈ End_{T}^{ℭ} \binom{d_{1}, …, d_{m}}{c_{1}, …, c_{n}}

to be the element

(15) l_{k}(f_{1}, …, f_{k})(ξ) =^{def} (−1)^{ν(f_{1},...,f_{k})} ∑_{s∈S_{ξ}} ∑_{(v_{1},...,v_{k})} γ (Γ_{s,{β}}^{{v_{1},...,v_{k}}}[F_{1}, …, F_{k}]),

where (v_{1}, …, v_{k}) runs through all the k-tuples of distinct vertices in the underlying graph of Γ_{s}. The sign on the right-hand side of (15) is given by

(16)  ν(f_{1}, …, f_{k}) =^{def} (k − 1)|f_{1}| + (k − 2)|f_{2}| + ⋯ + |f_{k−1}|.

Since ξ is arbitrary, (15) specifies an element

(17)  l_{k}(f_{1}, …, f_{k}) ∈ Hom_{Σ}^{ℭ}(E, End_{T}^{ℭ}) ≅ C_{P}^{∗}(T; T).

The arguments in Sections 3-4 in [36] ensure that (17) is indeed welldefined. We note that the L_{∞} axiom (12) for the operations l_{k} constructed above is a consequence of ∂^{2} = 0. Also, an obvious modification of the above construction applies to free cofibrant, not necessarily minimal, models as well. We will see an instance of such a generalization in Section 9.

5. Deformation complex of an associative algebra morphism

In this section and Section 6, we illustrate the L_{∞}-deformation theory of colored PROP algebras (Section 4) in the case of associative algebra morphisms. Let g: U → V be an associative algebra morphism, and set T = U ⊕V as a 2-colored graded module. Let As_{B→W} denote the 2-colored operad encoding associative algebra morphisms (Example 2.10). The morphism g: U → V can be regarded as an As_{B→W}-algebra structure on T.

The purposes of this section are (i) to express the differential δ_{As_{B→W}} in C_{As_{B→W}}^{∗}(T; T) (11) in terms of the Hochschild differential (Theorem 5.5), and (ii) to observe that the cochain complex (C_{As_{B→W}}^{∗}(T; T), δ_{As_{B→W}}) is isomorphic to the Gerstenhaber–Schack cochain complex (C_{GS}^{∗+1}(g; g), d_{GS}) of the morphism g [14, 15, 16] (Theorem 5.5). This isomorphism allows us to transfer the L_{∞}-structure on (C_{As_{B→W}}^{∗}(T; T), δ_{As_{B→W}}) to the Gerstenhaber–Schack cochain complex (C_{GS}^{∗+1}(g; g), d_{GS}) (Corollary 5.6).

The materials in this section and Section 6 can be easily dualized to obtain an explicit L_{∞}-structure on the deformation complex of a morphism of coassociative coalgebras. The associated deformation theory of coalgebra morphisms is the one constructed in [52].

5.1. Background. Deformation of an associative algebra morphism g, in the classical sense of Gerstenhaber [13], was studied by Gerstenhaber and Schack in [14, 15, 16]. In the case of a single associative algebra A, the deformation complex is the Hochschild cochain complex C^{∗}(A; A) of A with coefficients in itself, which has the structure of a differential graded Lie algebra [12]. On the other hand, the work of Gerstenhaber and Schack [14, 15, 16] left open the question of what structure the deformation complex (C_{GS}^{∗}(g; g), d_{GS}) of g possesses. Borisov answered this question in [3] by showing that (C_{GS}^{∗}(g; g), d_{GS}), while not a differential graded Lie algebra, is isomorphic to the underlying cochain complex of an L_{∞}-algebra.

With our approach based on minimal models, we are able to write down all the L_{∞}-operations l_{k} on C_{As_{B→W}}^{∗}(T; T) explicitly (with l_{1} in Theorem 5.5 and l_{k} (k ≥ 2) in Section 6). In particular, all the higher l_{k} (k ≥ 3) can be written in terms of a certain generalized “comp” operation (28), which extends the classical ∘_{i} operation in the Hochschild cochain complex [12]. As far as we know, these higher operations l_{k} have never been explicitly written down before. We believe that this example of associative algebra morphisms will serve as a guide for obtaining explicit formulas for the L_{∞}- operations in the deformation complexes of other kinds of morphisms and general diagrams.

5.2. The Gerstenhaber–Schack complex (C_{GS}^{∗}(g; g), d_{GS}). Here we recall the Gerstenhaber–Schack cochain complex (C_{GS}^{∗}(g; g), d_{GS}) [14, 15, 16]. Fix a morphism g: U → V of associative algebras. We also consider V as a U-bimodule via g. Then

(18) C_{GS}^{n}(g; g) =^{def} Hom(U^{⊗n}, U) ⊕ Hom(V^{⊗n}, V) ⊕ Hom(U^{⊗n−1}, V)

for n ≥ 1. A typical element in C_{GS}^{n}(g; g) is denoted by (x_{U}, x_{V} , x_{g}) with x_{U} ∈ Hom(U^{⊗n}, U), x_{V} ∈ Hom(V^{⊗n}, V), and x_{g} ∈ Hom(U^{⊗n−1}, V). Its differential is defined as

d_{GS}^{n}(x_{U}, x_{V} , x_{g}) =^{def} (bx_{U}, bx_{V} , gx_{U} − x_{V} g^{⊗n} − bx_{g}),

where b denotes the Hochschild differential in Hom(U^{⊗∗}, U), Hom(V^{⊗∗}, V), or Hom(U^{⊗∗}, V).

5.3. The minimal model of As_{B→W}. Here we recall from [31, 32] the minimal model of the 2-colored operad As_{B→W} that encodes associative algebra morphisms. The 2-colored operad As_{B→W} can be represented as (Example 2.10)

As_{B→W} = \frac{As_{B} ∗ As_{W} ∗F(f)}{(fμ = νf^{⊗2})},

where μ and ν denote the generators in As_{B}(1, 2) and As_{W}(1, 2), respectively, which encode the multiplications in the domain and the target.

Let E be the 2-colored Σ-bimodule with the following generators:

μ_{n}: B^{⊗n} → B of degree n − 2 and biarity (1, n) (n ≥ 2),
ν_{n}: W^{⊗n} → W of degree n − 2 and biarity (1, n) (n ≥ 2), and
f_{n}: B^{⊗n} → W of degree n − 1 and biarity (1, n) (n ≥ 1).

Then the minimal model for As_{B→W} is

(F(E), ∂) ⟶^{α} As_{B→W},

where

α(μ_{n}) = \begin{cases}μ & if n = 2, \\ 0 & otherwise, \end{cases} α(ν_{n}) = \begin{cases}ν & if n = 2, \\ 0 & otherwise, \end{cases}

and

α(f_{n}) = \begin{cases}f & if n = 1, \\ 0 & otherwise. \end{cases}

The differential ∂ is given by:

(19a) ∂(μ_{n}) = ∑_{\begin{matrix}i+j = n+1 \\ i,j ≥ 2 \end{matrix}} ∑_{s=0}^{n−j}(−1)^{i+s(j+1)}μ_{i} ∘_{s+1} μ_{j},

(19b) ∂(ν_{n}) = ∑_{\begin{matrix}i+j = n+1 \\ i,j ≥ 2 \end{matrix}} ∑_{s=0}^{n−j}(−1)^{i+s(j+1)}ν_{i} ∘_{s+1} ν_{j},

(s + 1)th input · · · · · · · · · •^{μ_{j}} •^{μ_{i}}
Figure 2. The graph corresponding to μ_{i} ∘_{s+1} μ_{j}.

(19c) ∂(f_{n}) = − ∑_{l=2}^{n} ∑_{r_{1}+···+r_{l}=n}(−1)^{∑_{1≤i<j≤l} r_{i}(r_{j}+1)}ν_{l}(f_{r_{1}} ⊗ ⋯ ⊗ f_{r_{l}})

− ∑_{\begin{matrix}i+j = n+1 \\ i ≥ 1, j ≥ 2 \end{matrix}} ∑_{s=0}^{n−j}(−1)^{i+s(j+1)}f_{i} ∘_{s+1} μ_{j}.

Here

(20)  μ_{i} ∘_{s+1} μ_{j} =^{def} μ_{i} (1_{B}^{⊗s} ⊗ μ_{j} ⊗ 1_{B}^{⊗i−s−1}) ,

which “plugs” μ_{j} into the (s + 1)st input of μ_{i} (see Figure 2), and similarly for ν_{i} ∘_{s+1} ν_{j} and f_{i} ∘_{s+1} μ_{j}.

5.4. The cochain complex (C_{As_{B→W}}^{∗}(T; T), δ_{As_{B→W}}). Suppose that T = U ⊕ V as a 2-colored graded module and that g: U → V is a morphism of associative algebras represented by the morphism ρ: As_{B→W} → End_{T} of 2-colored operads. Then the canonical isomorphism (14) says in this case,

C_{As_{B→W}}^{∗}(T; T) = ↑ Der(F(E), End_{T})^{−∗} ≅ ↑ Hom_{Σ}^{{B,W}}(E, End_{T})^{−∗}.

Under this isomorphism, an element θ ∈ C_{As_{B→W}}^{n}(T; T) is uniquely determined by the tuple

(21)  (θ_{U}, θ_{V} , θ_{g}) =^{def} (θ(μ_{n+1}), θ(ν_{n+1}), θ(f_{n})) ∈ C_{GS}^{n+1}(g; g).

This establishes a linear isomorphism

C_{As_{B→W}}^{n}(T; T) ≅ C_{GS}^{n+1}(g; g),
θ ↔ (θ_{U}, θ_{V} , θ_{g}) .

Denote by δ_{GS} the differential on the graded module C_{GS}^{∗}(g; g) induced by δ_{As_{B→W}}. The identification (21) provides an isomorphism

(C_{As_{B→W}}^{∗}(T; T), δ_{As_{B→W}}) ⟶^{≅} (C_{GS}^{∗+1}(g; g), δ_{GS})

of cochain complexes.

Theorem 5.5. For θ ∈ C_{As_{B→W}}^{n−1}(T; T), we have

δ_{GS} (θ_{U}, θ_{V} , θ_{g}) = (22)

((−1)^{n+1}bθ_{U}, (−1)^{n+1}bθ_{V} , gθ_{U} − θ_{V} g^{⊗n} − (−1)^{n}bθ_{g}) ,

in which b denotes the appropriate Hochschild differential. In particular, there is a cochain complex isomorphism

(23)  (C_{As_{B→W}}^{∗−1}(T; T), δ_{As_{B→W}}) ≅ (C_{GS}^{∗}(g; g), d_{GS})

given by

θ ∈ C_{As_{B→W}}^{n−1}(T; T) ↦ ((−1)^{\frac{n(n+1)}{2}} θ_{U}, (−1)^{\frac{n(n+1)}{2}} θ_{V} , (−1)^{\frac{(n−1)n}{2}} θ_{g}) .

Since (C_{As_{B→W}}^{∗}(T; T), δ_{As_{B→W}} = l_{1}, l_{2}, l_{3}, …) is, by Theorem 4.2, an L_{∞}- algebra, we can use the cochain complex isomorphism (23) to transfer the higher brackets l_{k} (k ≥ 2) to (C_{GS}^{∗+1}(g; g), d_{GS}).

Corollary 5.6. There is an L_{∞}-algebra structure (d_{GS} = l_{1}, l_{2}, l_{3}, …) on C_{GS}^{∗+1}(g; g) governing deformations of the associative algebra morphism g.

Proof of Theorem 5.5. Since δ_{As_{B→W}} = l_{1} in the L_{∞}-algebra and since the degree of l_{1} is +1, we have

δ_{GS} (θ_{U}, θ_{V} , θ_{g}) = (l_{1}(θ)(μ_{n+1}), l_{1}(θ)(ν_{n+1}), l_{1}(θ)(f_{n}))

by the identification (21). Therefore, to prove (22), it suffices to show:

\begin{aligned}l_{1}(θ)(μ_{n+1}) & = (−1)^{n+1}bθ_{U}, \\ l_{1}(θ)(ν_{n+1}) & = (−1)^{n+1}bθ_{V} , and \\ l_{1}(θ)(f_{n}) & = gθ_{U} − θ_{V} g^{⊗n} − (−1)^{n}bθ_{g}. \end{aligned} ; (24)

From the description (15) of l_{k}, the computation of l_{1}(θ)(μ_{n+1}) starts with ∂(μ_{n+1}) (19a). As an E-decorated 2-colored directed (1, n + 1)-graph, the term μ_{i} ∘_{s+1} μ_{j} in ∂(μ_{n+1}) has two vertices, whose decorations are μ_{i} and μ_{j}, see Figure 2. Therefore, the expression (15), when applied to the current situation, gives

l_{1}(θ)(μ_{n+1}) = (25)

∑_{\begin{matrix}i+j = n+2 \\ i,j ≥ 2 \end{matrix}} ∑_{s=0}^{n+1−j}(−1)^{i+s(j+1)} \{θ(μ_{i}) ∘_{s+1} β(μ_{j}) + β(μ_{i}) ∘_{s+1} θ(μ_{j})\} .

Note that, since θ ∈ C_{As_{B→W}}^{n−1} (T; T),

θ(μ_{i}) = \begin{cases}0 & if i ≠ n, \\ θ_{U} & if i = n, \end{cases} and β(μ_{j}) = ρ(α(μ_{j})) = \begin{cases}0 & if j ≠ 2, \\ μ_{U} & if j = 2, \end{cases}

where μ_{U}: U^{⊗2} → U is the multiplication on U. It follows that (25) reduces to

l_{1}(θ)(μ_{n+1}) = ∑_{s=0}^{n−1}(−1)^{n+s(2+1)}θ_{U} ∘_{s+1} μ_{U} + ∑_{s=0}^{1}(−1)^{2+s(n+1)}μ_{U} ∘_{s+1} θ_{U}

= (−1)^{n+1}μ_{U}(−, θ_{U}) + μ_{U}(θ_{U}, −)

+ (−1)^{n+1} ∑_{s=1}^{n}(−1)^{s}θ_{U}(Id_{U}^{⊗s−1} ⊗μ_{U} ⊗ Id_{U}^{⊗n−s})

= (−1)^{n+1}bθ_{U},

which is the first condition in (24).

The previous paragraph applies verbatim to l_{1}(θ)(ν_{n+1}) (with ν_{l} replacing μ_{l} everywhere), since the definition of ∂(ν_{∗}) (19b) admits the same formula as that of ∂(μ_{∗}). Therefore, it remains to show the last condition in (24).

In ∂(f_{n}) (19c), the term ν_{l}(f_{r_{1}} ⊗ ⋯ ⊗ f_{r_{l}}) (respectively, f_{i} ∘_{s+1} μ_{j}) is an E-decorated 2-colored directed (1, n)-graph with l + 1 (respectively, 2) vertices. Since

β(f_{j}) = ρ(α(f_{j})) = \begin{cases}g: U → V & if j = 1, \\ 0 & otherwise, \end{cases}

the same kind of analysis as above gives

l_{1}(θ)(f_{n}) = −θ_{V} g^{⊗n} − (−1)^{(n−1)(1+1)}μ_{V} (θ_{g} ⊗ g) − (−1)^{n−1+1}μ_{V} (g ⊗ θ_{g})

− ∑_{s=0}^{n−2}(−1)^{n−1+s(2+1)}θ_{g} ∘_{s+1} μ_{U} − (−1)^{1+0}gθ_{U}

= gθ_{U} − θ_{V} g^{⊗n}

− (−1)^{n} {μ_{V} (g ⊗ θ_{g}) + (−1)^{n}μ_{V} (θ_{g} ⊗ g) + ∑_{s=1}^{n−1}(−1)^{s}θ_{g} ∘_{s} μ_{U}}

= gθ_{U} − θ_{V} g^{⊗n} − (−1)^{n}bθ_{g}.

Here μ_{V} : V^{⊗2} → V denotes the multiplication on V. This establishes the last condition in (24) and finishes the proof of Theorem 5.5. □

6. The higher brackets in C_{As_{B→W}}^{∗}(T; T)

We keep the same setting and notations as in the previous section. The purpose of this section is to describe the L_{∞}-operations l_{k} on C_{As_{B→W}}^{∗}(T; T) for k ≥ 2. The cases k = 2 (Theorem 6.2) and k ≥ 3 (Theorem 6.4) are treated separately. As an immediate consequence of our explicit formula for l_{k} (k ≥ 3), we observe that, when applied to the tensor powers of C_{As_{B→W}}^{≤q}(T; T) for some fixed q ≥ 0, only δ_{As_{B→W}}(T; T) = l_{1}, l_{2}, …, l_{q+2} can be nontrivial (Corollary 6.5).

6.1. The operation l_{2}. First we deal with the case k = 2. Pick elements θ ∈ C_{As_{B→W}}^{n−1}(T; T) and ω ∈ C_{As_{B→W}}^{m−1}(T; T). Under the identification (21), θ and ω correspond to

(θ_{U}, θ_{V} , θ_{g}) ∈ C_{GS}^{n}(g; g) and (ω_{U}, ω_{V} , ω_{g}) ∈ C_{GS}^{m}(g; g),

respectively.

Since l_{2} has degree 0, the element l_{2}(θ, ω) lies in C_{As_{B→W}}^{(n+m−1)−1}(T; T). Under the identification (21), l_{2}(θ, ω) is uniquely determined by

(l_{2}(θ, ω)(μ_{n+m−1}), l_{2}(θ, ω)(ν_{n+m−1}), l_{2}(θ, ω)(f_{n+m−2})) ∈ C_{GS}^{n+m−1}(g; g).

Theorem 6.2. With the notations above, we have

l_{2}(θ, ω)(μ_{n+m−1}) = (26a)

− ∑_{s=1}^{n}(−1)^{(s+1)(m+1)}θ_{U} ∘_{s} ω_{U} − (−1)^{n+m} ∑_{s=1}^{m}(−1)^{(s+1)(n+1)}ω_{U} ∘_{s} θ_{U},

l_{2}(θ, ω)(ν_{n+m−1}) = (26b)

− ∑_{s=1}^{n}(−1)^{(s+1)(m+1)}θ_{V} ∘_{s} ω_{V} − (−1)^{n+m} ∑_{s=1}^{m}(−1)^{(s+1)(n+1)}ω_{V} ∘_{s} θ_{V} ,

l_{2}(θ, ω)(f_{n+m−2}) = (26c)

− ∑_{s=1}^{n−1}(−1)^{(s+1)(m+1)}θ_{g} ∘_{s} ω_{U} − (−1)^{n+m} ∑_{s=1}^{m−1}(−1)^{(s+1)(n+1)}ω_{g} ∘_{s} θ_{U}

+(−1)^{n} ∑_{i=1}^{n}(−1)^{(i−1)m}θ_{V} ∘_{i} ω_{g} + ∑_{j=1}^{m}(−1)^{jn}ω_{V} ∘_{j} θ_{g}

+(−1)^{nm+n+m}θ_{g} ⌣ ω_{g} + (−1)^{nm}ω_{g} ⌣ θ_{g}.

In the above theorem, the notation is as in (20) and (21), except that

\begin{aligned}θ_{V} ∘_{i} ω_{g} & = θ_{V} (g^{⊗i−1} ⊗ ω_{g} ⊗ g^{⊗n−i}), \\ ω_{V} ∘_{j} θ_{g} & = ω_{V} (g^{⊗j−1} ⊗ θ_{g} ⊗ g^{⊗m−j}), \\ θ_{g} ⌣ ω_{g} & = μ_{V} (θ_{g} ⊗ ω_{g}), \end{aligned} ; (27)

and similarly for ω_{g} ⌣ θ_{g}, θ_{g} ∘_{s} ω_{U}, and ω_{g} ∘_{s} θ_{U}. In other words, θ_{V} ∘_{i} ω_{g} is obtained by plugging ω_{g} into the ith input of θ_{V} and g into the other (n − 1) inputs of θ_{V}. Likewise, θ_{g} ⌣ ω_{g} is simply the usual cup-product of θ_{g} and ω_{g}.

The proof will be given at the end of this section. Note that Gerstenhaber and Schack did construct a bracket [−, −] on their cochain complex (C_{GS}^{∗}(g; g), d_{GS}) (see, e.g., the graded commutator bracket of the operation [14, p. 11 (9)] or [16, pp. 158-159]). It is straightforward to check that the linear isomorphism (23) is compatible with [−, −] and l_{2} as well.

6.3. The operations l_{k} for k ≥ 3. Now consider the cases k ≥ 3. Pick elements θ_{s} ∈ C_{As_{B→W}}^{n_{s}−1}(T; T) (1 ≤ s ≤ k). Each θ_{s} corresponds, via the identification (21), to the tuple

(θ_{s,U}, θ_{s,V} , θ_{s,g}) = (θ_{s}(μ_{n_{s}}), θ_{s}(ν_{n_{s}}), θ_{s}(f_{n_{s}−1})) ∈ C_{GS}^{n_{s}}(g; g).

Since l_{k} has degree 2 − k, the element l_{k}(θ_{1}, …, θ_{k}) lies in C_{As_{B→W}}^{t−1}(T; T), where

t = 3 − 2k + ∑_{s=1}^{k} n_{s}.

Under the identification (21), l_{k}(θ_{1}, …, θ_{k}) is uniquely determined by

(l_{k}(θ_{1}, …, θ_{k})(μ_{t}), l_{k}(θ_{1}, …, θ_{k})(ν_{t}), l_{k}(θ_{1}, …, θ_{k})(f_{t−1})) ∈ C_{GS}^{t}(g; g).

Now we extend the first ∘_{i} operation in (27) as follows. Fix s ∈ {1, …, k}. Let

a = (a_{1}, …, â_{s}, …, a_{k})

be a (k − 1)-tuple of distinct points in the set {1, …, n_{s}}. Then we define

(28)  θ_{s,V} ∘_{a} (θ_{1,g}, …, θ̂_{s,g}, …, θ_{k,g}) ∈ Hom(U^{⊗t−1}, V)

to be the element obtained by plugging θ_{j,g} (1 ≤ j ≤ k, j ≠ s) into the a_{j}th input of θ_{s,V} and g into the other (n_{s} − (k − 1)) inputs of θ_{s,V}. Also define the sign

(−1)^{a} = (−1)^{∑_{1≤i<j≤ns} r_{i}(r_{j}+1)},

where

r_{a} = \begin{cases}|θ_{j}| = n_{j} − 1 & if a = a_{j} ∈ \{a_{1}, …, â_{s}, …, a_{k}\}, \\ 1 & otherwise. \end{cases}

Theorem 6.4. For k ≥ 3 and notations as above, we have

(29a) l_{k}(θ_{1}, …, θ_{k})(μ_{t}) = 0,
(29b)  l_{k}(θ_{1}, …, θ_{k})(ν_{t}) = 0, and
(29c) l_{k}(θ_{1}, …, θ_{k})(f_{t−1}) =

− (−1)^{ν(θ_{1},...,θ_{k})} ∑_{s=1}^{k} ∑_{a} (−1)^{a} θ_{s,V} ∘_{a} (θ_{1,g}, …, θ̂_{s,g}, …, θ_{k,g}).

Here ν(θ_{1}, …, θ_{k}) is defined in (16) and, for each s, a = (a_{1}, …, â_{s}, …, a_{k}) runs through all the (k − 1)-tuples of distinct points in the set {1, …, n_{s}}.

Corollary 6.5. Suppose that k ≥ 3 and that θ_{s} ∈ C_{As_{B→W}}^{n_{s}−1}(T; T), 1≤s≤k. If

n_{s} < k − 1 for 1 ≤ s ≤ k,

then

l_{k}(θ_{1}, …, θ_{k}) = 0.

In other words, for each q ≥ 0 and any k ≥ q + 3, the operation

l_{k}: (C_{As_{B→W}}^{≤q}(T; T))^{⊗k} → C_{As_{B→W}}^{∗}(T; T)

is trivial.

Proof of Theorem 6.2. To prove (26a), first note that

∂(μ_{n+m−1}) = ∑_{\begin{matrix}i+j = n+m \\ i,j ≥ 2 \end{matrix}} ∑_{s=0}^{n+m−1−j}(−1)^{i+s(j+1)}μ_{i} ∘_{s+1} μ_{j}.

Since the E-decorated 2-colored directed (1, n + m − 1)-graph μ_{i} ∘_{s+1} μ_{j} has two vertices, we have l_{2}(θ, ω)(μ_{n+m−1})

= (−1)^{|θ|} ∑_{\begin{matrix}i+j = n+m \\ i,j ≥ 2 \end{matrix}} ∑_{s=0}^{n+m−1−j} (−1)^{i+s(j+1)}\{θ(μ_{i})∘_{s+1}ω(μ_{j}) + ω(μ_{i})∘_{s+1}θ(μ_{j})\}

= (−1)^{n−1} (∑_{s=0}^{n−1}(−1)^{n+s(m+1)}θ_{U} ∘_{s+1}ω_{U} + ∑_{s=0}^{m−1}(−1)^{m+s(n+1)}ω_{U} ∘_{s+1}θ_{U}).

This is exactly (26a) after a shift of the summation indexes.

Since ∂(ν_{n+m−1}) has the same defining formula as ∂(μ_{n+m−1}) (with ν_{l} replacing μ_{l} everywhere), the reasoning in the previous paragraph also applies to l_{2}(θ, ω)(ν_{n+m−1}) to establish (26b).

To prove (26c), first note that

(30) ∂(f_{n+m−2}) = − ∑_{l=2}^{n+m−2} ∑_{r_{1}+···+r_{l} = n+m−2} (−1)^{∑_{1≤i<j≤l} r_{i}(r_{j}+1)}ν_{l}(f_{r_{1}} ⊗⋯⊗f_{r_{l}})

− ∑_{\begin{matrix}i+j = n+m−1 \\ i≥1, j≥2 \end{matrix}} ∑_{s=0}^{n+m−2−j}(−1)^{i+s(j+1)}f_{i} ∘_{s+1} μ_{j}.

An argument essentially identical to the first paragraph of this proof can be applied to the terms f_{i} ∘_{s+1} μ_{j}. This gives rise to the sums

(31) − ∑_{s=1}^{n−1}(−1)^{(s+1)(m+1)}θ_{g} ∘_{s} ω_{U} − (−1)^{n+m} ∑_{s=1}^{m−1}(−1)^{(s+1)(n+1)}ω_{g} ∘_{s} θ_{U}

in l_{2}(θ, ω)(f_{n+m−2}).

In (30), the E-decorated 2-colored directed (1, n + m − 2)-graph Γ = ν_{l}(f_{r_{1}} ⊗ ⋯ ⊗ f_{r_{l}}) has l + 1 vertices, say, v_{top}, v_{bot}^{1}, …, v_{bot}^{l}, with decorations ν_{l}, f_{r_{1}}, …, f_{r_{l}}, respectively. In this graph Γ, the only pairs of distinct vertices are (v_{top}, v_{bot}^{∗}), (v_{bot}^{∗}, v_{top}), and (v_{bot}^{i}, v_{bot}^{j}) (i ≠ j). The corresponding elements in l_{2}(θ, ω)(f_{n+m−2}) (without the signs) are:

(1) θ(ν_{l})(β(f_{r_{1}}) ⊗ ⋯ ⊗ ω(f_{r_{i}}) ⊗ ⋯ β(f_{r_{l}})) (1 ≤ i ≤ l), which is 0 unless l = n, r_{i} = m − 1, and all the other r_{∗} = 1;

(2) ω(ν_{l})(β(f_{r_{1}}) ⊗ ⋯ ⊗ θ(f_{r_{j}}) ⊗ ⋯ ⊗ β(f_{r_{l}})) (1 ≤ j ≤ l), which is 0 unless l = m, r_{j} = n − 1, and all the other r_{∗} = 1;

(3) β(ν_{l})(β(f_{r_{1}}) ⊗ ⋯ ⊗ θ(f_{r_{i}}) ⊗ ⋯ ⊗ ω(f_{r_{j}}) ⊗ ⋯ ⊗ β(f_{r_{l}})), which is 0 unless l = 2 and (r_{1}, r_{2}) = (n − 1, m − 1);

(4) β(ν_{l})(β(f_{r_{1}}) ⊗ ⋯ ⊗ ω(f_{r_{i}}) ⊗ ⋯ ⊗ θ(f_{r_{j}}) ⊗ ⋯ ⊗ β(f_{r_{l}})), which is 0

unless l = 2 and (r_{1}, r_{2}) = (m − 1, n − 1). Taking all the signs into account, we obtain in l_{2}(θ, ω)(f_{n+m−2}) the following sums:

\begin{matrix}− (−1)^{|θ|} ∑_{i=1}^{n}(−1)^{(i−1)(m−1+1)}θ_{V} (g^{⊗i−1} ⊗ ω_{g} ⊗ g^{⊗n−i}) \\ − (−1)^{|θ|} ∑_{j=1}^{m}(−1)^{(j−1)(n−1+1)}ω_{V} (g^{⊗j−1} ⊗ θ_{g} ⊗ g^{⊗m−j}) \\ + (−1)^{|θ|}(−1)^{(n−1)(m−1)} \{(−1)^{n}μ_{V} (θ_{g} ⊗ ω_{g}) + (−1)^{m}μ_{V} (ω_{g} ⊗ θ_{g})\} . \end{matrix} ; (32)

The required result (26c) is now obtained by combining (31) and (32). This finishes the proof of Theorem 6.2. □

Proof of Theorem 6.4. The computation of l_{k}(θ_{1}, …, θ_{k})(μ_{t}) involves the choice of k ≥ 3 distinct vertices in the graphs μ_{i} ∘_{s+1} μ_{j}, each of which has only two vertices. It follows that

l_{k}(θ_{1}, …, θ_{k})(μ_{t}) = 0,

which is (29a). The same argument establishes (29b). Moreover, the same reasoning also shows that the terms f_{i} ∘_{s+1} μ_{j} in ∂(f_{t−1}) cannot contribute nontrivially to l_{k}(θ_{1}, …, θ_{k})(f_{t−1}).

The remaining statement (29c) is now proved by an argument very similar to the last paragraph in the proof of Theorem 6.2. There is one major difference: In order for the term ν_{l}(f_{r_{1}} ⊗ ⋯ ⊗ f_{r_{l}}) in ∂(f_{t−1}) to contribute nontrivially to l_{k}(θ_{1}, …, θ_{k})(f_{t−1}), the vertex v_{top} (with decoration ν_{l}) must be chosen as one of the k distinct vertices because k ≥ 3 and β(ν_{l}) = 0 for l ≥ 3. It follows that each nontrivial term in l_{k}(θ_{1}, …, θ_{k})(f_{t−1}) has the form (28), except for the sign, which is

−(−1)^{ν(θ_{1},...,θ_{k})}(−1)^{a}.

The desired condition (29c) now follows. □

7. Deformation complex of a Lie algebra morphism

In this section and Section 8, we give a second illustration of the L_{∞}- deformation theory of colored PROP algebras (Section 4) in the case of Lie algebra morphisms. The parallelism of the analysis in the associative and Lie cases shows the unifying character of this approach.

Let g: U → V be a Lie algebra morphism, and set T = U ⊕ V as a 2colored graded module. Let Lie_{B→W} denote the 2-colored operad encoding Lie algebra morphisms. The purposes of this section are (i) to express the differential δ_{Lie_{B→W}} in C_{Lie_{B→W}}^{∗} (T; T) (11) in terms of the Chevalley–Eilenberg differential (Theorem 7.5), and (ii) to observe that the cochain complex (C_{Lie_{B→W}}^{∗} (T; T), δ_{Lie_{B→W}}) is isomorphic to the S-cohomology cochain complex (Λ^{∗}(U, V), Δ^{∗}) of the morphism g [9, 17] (Corollary 7.6). This isomorphism allows us to transfer the L_{∞}-structure on (C_{Lie_{B→W}}^{∗} (T; T), δ_{Lie_{B→W}}) to the S-cohomology cochain complex (Λ^{∗}(U, V), Δ^{∗}) (Corollary 7.7).

7.1. Background. The question of deformation of morphisms of Lie algebras was treated for the first time by Nijenhuis and Richardson in [42]. The approach chosen was not the classical method of Gerstenhaber [13], but the use of the formalization of deformation theory in terms of graded algebras on the space of cochains developed by Nijenhuis and Richardson in [41]. The starting point was then the graded Lie algebra on cochains and the differential which were guessed, the deformation theory being only a corollary. As drawbacks, the algebras were not allowed to be deformed and the notion of equivalent deformations was not natural. In order to cure these two problems, the first author reexamined this problem from the classical point of view of Gerstenhaber and introduced in [9] the S-cohomology, concluding his work by addressing the question of the description of a structure for its deformation complex. Later, in [17], Gerstenhaber, Giaquinto and Schack showed that this construction is completely parallel to the one given in [14] which leads to Diagram cohomology of associative algebras, and hence gave the diagrammatic description of S-cohomology.

7.2. The S-Cohomology complex (Λ^{n}(U, V ), Δ^{n}). Here we recall the S-cochain complex (Λ^{∗}(U, V), Δ^{∗}) [9]. We modify slightly the notations from [9] to be coherent with the present notations.

Fix a morphism g: U → V of Lie algebras. We also consider V as a left U-module via g. Then

Λ^{n}(U, V) =^{def} Hom(U^{∧n}, U) ⊕ Hom(V^{∧n}, V) ⊕ Hom(U^{∧n−1}, V)

for n ≥ 1. We will also denote a typical element in Λ^{n}(U, V) by (x_{U}, x_{V} , x_{g}) with x_{U} ∈ Hom(U^{∧n}, U), x_{V} ∈ Hom(V^{∧n}, V), and x_{g} ∈ Hom(U^{∧n−1}, V). Its differential is defined as

(33) Δ^{n}(x_{U}, x_{V} , x_{g}) =^{def} (bx_{U}, bx_{V} , (−1)^{n−1}gx_{U} − (−1)^{n−1}x_{V} g^{⊗n} + bx_{g}),

where b denotes the Chevalley–Eilenberg differential in one of the spaces Hom(U^{∧∗}, U), Hom(V^{∧∗}, V), or Hom(U^{∧∗}, V).

7.3. The minimal model of Lie_{B→W}. Here we construct the minimal model of the 2-colored operad Lie_{B→W} that encodes Lie algebra morphisms. This definition is very similar to the one in the associative category, except for the definition of the differential which differs slightly. Moreover one has to be careful with respect to the symmetry which is a new feature of the Lie category.

The 2-colored operad Lie_{B→W} can be represented as

Lie_{B→W} = \frac{Lie_{B} ∗ Lie_{W} ∗F(f)}{(fμ = νf^{⊗2})},

where μ and ν denote the generators in Lie_{B}(1, 2) and Lie_{W} (1, 2), respectively, which encode the multiplications in the domain and the target.

Let E be the 2-colored Σ-bimodule with the following skew symmetric generators:

μ_{n}: B^{⊗n} → B of degree n − 2 and biarity (1, n) (n ≥ 2)
ν_{n}: W^{⊗n} → W of degree n − 2 and biarity (1, n) (n ≥ 2), and
f_{n}: B^{⊗n} → W of degree n − 1 and biarity (1, n) (n ≥ 1).

Then the minimal model for Lie_{B→W} is

(F(E), ∂) ⟶^{α} Lie_{B→W} ,

where

α(μ_{n}) = \begin{cases}μ & if n = 2, \\ 0 & otherwise, \end{cases} α(ν_{n}) = \begin{cases}ν & if n = 2, \\ 0 & otherwise, \end{cases}

and

α(f_{n}) = \begin{cases}f & if n = 1, \\ 0 & otherwise. \end{cases}

The differential ∂ is given by:

(34a) ∂(μ_{n}) = ∑_{\begin{matrix}i+j = n+1 \\ i,j ≥ 2 \end{matrix}}(−1)^{j(i−1)} ∑_{σ∈S_{j,i−1}} sgn(σ)μ_{i} ∘ (μ_{j} ⊗ Id^{⊗^{i−1}}) ∘ σ,

(34b) ∂(ν_{n}) = ∑_{\begin{matrix}i+j = n+1 \\ i,j ≥ 2 \end{matrix}}(−1)^{j(i−1)} ∑_{σ∈S_{j,i−1}} sgn(σ)ν_{i} ∘ (ν_{j} ⊗ Id^{⊗^{i−1}}) ∘ σ,

(34c) ∂(f_{n}) = ∑_{l=2}^{n} ∑_{\begin{matrix}r_{1}+···+r_{l}=n \\ r_{1} ≤ ··· ≤ r_{l} \end{matrix}}(−1)^{ε} ∑_{σ∈S_{r_{1},...,r_{l}}^{<}} sgn(σ)ν_{l}(f_{r_{1}} ⊗ ⋯ ⊗ f_{r_{l}}) ∘ σ

− ∑_{\begin{matrix}i+j = n+1 \\ i ≥ 1,j ≥ 2 \end{matrix}}(−1)^{j(i−1)} ∑_{σ∈S_{j,i−1}} sgn(σ)f_{i} ∘ (μ_{j} ⊗ Id^{⊗^{i−1}}) ∘ σ,

where S_{j,i−1} denotes the set of j, i − 1 unshuffles and S_{r_{1},...,r_{l}}^{<} the set of r_{1}, …, r_{l}-unshuffles satisfying σ(r_{1} + ⋯ + r_{i−1} + 1) < σ(r_{1} + ⋯ + r_{i} + 1) if r_{i} = r_{i+1}. The sign in the first line of (34c) is given by

(35)  ε = ε(r_{1}, …, r_{l}) := \frac{l(l − 1)}{2} + ∑_{i=1}^{l−1} r_{i}(l − i).

It is also assumed in this notation that r_{i} ≤ r_{i+1}. We refer to [10] for the proof that it is a minimal model.

One may alternatively write the above formulas with the summations running over the entire symmetric groups, with coefficients involving factorials. This would reflect the convention in describing the morphism of L_{∞}-algebras used for instance in [22]. The graded anti-symmetry of the structure operations allows one to bring these formulas into the above ‘reduced’ form.

7.4. The cochain complex (C_{Lie_{B→W}}^{∗} (T; T), δ_{Lie_{B→W}} ). Suppose that T = U⊕V as a 2-colored graded module and that g: U → V is a morphism of Lie algebras represented by the morphism ρ: Lie_{B→W} → End_{T} of 2-colored operads. Then the canonical isomorphism (14) says in this case,

C_{Lie_{B→W}}^{∗} (T; T) = ↑ Der(F(E), End_{T})^{−∗} ≅ ↑ Hom_{Σ}^{{B,W }}(E, End_{T})^{−∗}.

Under this isomorphism, an element θ ∈ C_{Lie_{B→W}}^{n} (T; T) is determined by the tuple

(36)  (θ_{U}, θ_{V} , θ_{g}) =^{def} (θ(μ_{n+1}), θ(ν_{n+1}), θ(f_{n})) ∈ Λ^{n+1}(U, V).

This establishes a linear isomorphism

C_{Lie_{B→W}}^{n} (T; T) ≅ Λ^{n+1}(U, V),
θ ↔ (θ_{U}, θ_{V} , θ_{g}) .

Denote by δ the differential on the graded module Λ^{∗}(U, V) induced by δ_{Lie_{B→W}}. The identification (36) provides an isomorphism

(37)  (C_{Lie_{B→W}}^{∗} , δ_{Lie_{B→W}}) ⟶^{≅} (Λ^{∗+1}(U, V), δ)

of cochain complexes.

Theorem 7.5. For θ ∈ C_{Lie_{B→W}}^{n−1} (T; T), we have

(38)  δ (θ_{U}, θ_{V} , θ_{g}) = (bθ_{U}, bθ_{V} , −bθ_{g} + θ_{V} g^{⊗n} − gθ_{U}) ,

in which b denotes the appropriate Chevalley–Eilenberg differential.

One can then compare this complex (38) with the S-cohomology (33).

Corollary 7.6. There is a cochain complex isomorphism

π = (π^{∗}, π^{∗}, π^{∗}): (Λ^{∗}(U, V), δ) ⟶^{≅} (Λ^{∗}(U, V), Δ)

given by

\begin{cases}π^{n} & = Id, \\ π^{n} & = (−1)^{n−1} Id . \end{cases}

Combined with (37), we obtain an isomorphism

(39)  (C_{Lie_{B→W}}^{∗} (T; T), δ_{Lie_{B→W}}) ⟶^{≅} (Λ^{∗}(U, V), Δ)

of cochain complexes.

Since (C_{Lie_{B→W}}^{∗} (T; T), δ_{Lie_{B→W}} = l_{1}, l_{2}, l_{3}, …) is an L_{∞}-algebra (Theorem 4.2), we can use the cochain complex isomorphism (39) to transfer the higher brackets l_{k} (k ≥ 2) to (Λ^{∗}(U, V), Δ).

Corollary 7.7. There is an L_{∞}-algebra structure (Δ = l_{1}, l_{2}, l_{3}, …) on Λ^{∗}(U, V) capturing deformations of the Lie algebra morphism g.

Proof of Theorem 7.5. Since δ = l_{1} in the L_{∞}-algebra and since the degree of l_{1} is +1, we have

δ (θ_{U}, θ_{V} , θ_{g}) = (l_{1}(θ)(μ_{n+1}), l_{1}(θ)(ν_{n+1}), l_{1}(θ)(f_{n}))

by the identification (36). Therefore, to prove (38), it suffices to show:

\begin{aligned}l_{1}(θ)(μ_{n+1}) & = bθ_{U}, \\ l_{1}(θ)(ν_{n+1}) & = bθ_{V} , and \\ l_{1}(θ)(f_{n}) & = −bθ_{g} + θ_{V} g^{⊗n} − gθ_{U}. \end{aligned} ; (40)

From the description (15) of the operation l_{k}, the computation of the term l_{1}(θ)(μ_{n+1}) starts with ∂(μ_{n+1}) (34a). As an E-decorated 2-colored directed (1, n + 1)-graph, the term μ_{i} ∘_{s+1} μ_{j} in ∂(μ_{n+1}) has two vertices, whose decorations are μ_{i} and μ_{j}. Therefore, the expression (15), when applied to the current situation and using notation (20), gives

l_{1}(θ)(μ_{n+1}) = (41)

∑_{\begin{matrix}i+j = n+2 \\ i,j ≥ 2, σ∈S_{j,i−1} \end{matrix}} (−1)^{j(i−1)}sgn(σ) \{θ(μ_{i}) ∘_{1} β(μ_{j}) + β(μ_{i}) ∘_{1} θ(μ_{j})\} ∘ σ.

Note that, since θ ∈ C_{Lie_{B→W}}^{n−1} (T; T),

θ(μ_{i}) = \begin{cases}0 & if i ≠ n, \\ θ_{U} & if i = n, \end{cases} and β(μ_{j}) = ρ(α(μ_{j})) = \begin{cases}0 & if j ≠ 2, \\ μ_{U} & if j = 2, \end{cases}

where μ_{U}: U^{∧2} → U is the multiplication on U. It follows that (41) reduces to

\begin{aligned}l_{1}(θ)(μ_{n+1}) & = \underbrace{(−1)^{2(n−1)} ∑_{σ∈S_{2,n−1}} sgn(σ)(θ_{U} ∘_{1} μ_{U}) ∘ σ}_{(a)} \\ & + \underbrace{(−1)^{n(2−1)} ∑_{σ∈S_{n,1}} sgn(σ)(μ_{U} ∘_{1} θ_{U}) ∘ σ}_{(b)}. \end{aligned}

In particular, applied to elements of U, (a) and (b) give:

(a)(x_{1}, ..., x_{n+1}) = (−1)^{s+t−1} ∑_{1≤s<t≤n+1} θ_{U}(μ_{U}(x_{s}, x_{t}), x_{1}, ..., x̂_{s}, ..., x̂_{t}, ..., x_{n+1}),
(b)(x_{1}, ..., x_{n+1}) = (−1)^{n(2−1)} ∑_{1≤s≤n+1} (−1)^{n+1−s}μ_{U}(θ_{U}(x_{1}, ..., x̂_{s}, ..., x_{n+1}), x_{s})

= (−1)^{−s} ∑_{1≤s≤n+1} μ_{U}(x_{s}, θ_{U}(x_{1}, ..., x̂_{s}, ..., x_{n+1})).

Therefore, by the definition of the Chevalley–Eilenberg differential, we have

l_{1}(θ)(μ_{n+1}) = bθ_{U},

which is the first condition in (40).

The previous paragraph applies verbatim to l_{1}(θ)(ν_{n+1}) (with ν_{l} replacing μ_{l} everywhere), since the definition of ∂(ν_{∗}) (34b) admits the same formula as that of ∂(μ_{∗}). Therefore, it remains to show the last condition in (40).

In ∂(f_{n}), the term ν_{l}(f_{r_{1}}⊗⋯⊗f_{r_{l}}) (respectively, f_{i}∘_{1}μ_{j}) is an E-decorated 2-colored directed (1, n)-graph with l + 1 (respectively, 2) vertices. Since

β(f_{j}) = ρ(α(f_{j})) = \begin{cases}g: U → V & if j = 1, \\ 0 & otherwise, \end{cases}

the same kind of analysis as above gives

l_{1}(θ)(f_{n}) = \underbrace{∑_{σ∈S_{1,...,1}^{<}}sgn(σ)θ_{V} (g^{⊗n}) ∘ σ}_{(s_{1})} + \underbrace{∑_{σ∈S_{1,n−1}}sgn(σ)μ_{V} (g ⊗ θ_{g}) ∘ σ}_{(s_{2})}

− (−1)^{2(n−1)}\underbrace{∑_{σ∈S_{2,n−2}} sgn(σ)θ_{g}(μ_{U} ⊗ Id^{⊗^{n−2}}) ∘ σ}_{(s_{3})}

− (−1)^{n(1−1)}\underbrace{∑_{σ∈S_{n,0}} sgn(σ)gθ_{U} ∘ σ}_{(s_{4})}.

(42)

We now apply the middle summands of (42) on elements and rearrange them in order to recognize the Chevalley–Eilenberg differential. We have

(s_{2})(x_{1} ⊗ ··· ⊗ x_{n}) = ∑_{1≤s≤n}(−1)^{s−1}μ_{V} (g ⊗ θ_{g})(x_{s} ⊗ x_{1} ⊗ ··· ⊗ x̂_{s} ⊗ ··· ⊗ x_{n}),

so

((s_{2}) + (s_{3}))(x_{1} ⊗ ⋯ ⊗ x_{n}) =

+ ∑_{1≤s≤n}(−1)^{s−1}μ_{V} (g(x_{s}), θ_{g}(x_{1}, …, x̂_{s}, …, x_{n}))

− ∑_{1≤s<t≤n}(−1)^{s−1+t−2}θ_{g}(μ_{U}(x_{s}, x_{t}), x_{1}, …, x̂_{s}, …, x̂_{t}, …, x_{n})

= −bθ_{g}(x_{1} ⊗ ⋯ ⊗ x_{n}).

Considering the fact that both S_{n,0} and S_{1,...,1}^{<} consist of a single element, the trivial permutation, one finally gets

l_{1}(θ)(f_{n}) = −bθ_{g} + θ_{V} g^{⊗n} − gθ_{U},

which ends the proof of Theorem 7.5. □

8. The higher brackets in C_{Lie_{B→W}}^{∗} (T; T)

We keep the same setting and notations as in the previous section. The purpose of this section is to make the L_{∞}-operations l_{k} on C_{Lie_{B→W}}^{∗} (T; T) explicit for k ≥ 2. The cases k = 2 (Theorem 8.2) and k ≥ 3 (Theorem 8.4) are treated separately. As an immediate consequence of our explicit formula for l_{k} (k ≥ 3), we observe that, when applied to the tensor powers of C_{Lie_{B→W}}^{≤q} (T; T) for some fixed q ≥ 0, only δ_{Lie_{B→W}} (T; T) = l_{1}, l_{2}, …, l_{q+2} can be nontrivial (Corollary 8.5).

8.1. The operation l_{2}. First we deal with the case k = 2. Pick elements θ ∈ C_{Lie_{B→W}}^{n} (T; T) and ω ∈ C_{Lie_{B→W}}^{m} (T; T). Under the identification (36), θ and ω correspond to

(θ_{U}, θ_{V} , θ_{g}) ∈ Λ^{n}(U, V) and (ω_{U}, ω_{V} , ω_{g}) ∈ Λ^{m}(U, V),

respectively.

Since l_{2} has degree 0, the element l_{2}(θ, ω) lies in C_{Lie_{B→W}}^{n+m} (T; T). Under the identification (36), l_{2}(θ, ω) is determined by

(l_{2}(θ, ω)(μ_{n+m+1}), l_{2}(θ, ω)(ν_{n+m+1}), l_{2}(θ, ω)(f_{n+m})) ∈ Λ^{n+m}(U, V).

Theorem 8.2. With the notations above, we have

(43a) l_{2}(θ, ω)(μ_{n+m+1}) = (−1)^{mn}( ∑_{σ∈S_{m+1,n}} sgn(σ)θ_{U} ∘_{1} ω_{U} ∘ σ

+ (−1)^{m+n} ∑_{σ∈S_{n+1,m}} sgn(σ)ω_{U} ∘_{1} θ_{U} ∘ σ),

(43b) l_{2}(θ, ω)(ν_{n+m+1}) = (−1)^{mn}( ∑_{σ∈S_{m+1,n}} sgn(σ)θ_{V} ∘_{1} ω_{V} ∘ σ

+ (−1)^{m+n} ∑_{σ∈S_{n+1,m}} sgn(σ)ω_{V} ∘_{1} θ_{V} ∘ σ),

l_{2}(θ, ω)(f_{n+m}) = (43c)

(−1)^{m(n−1)}( ∑_{σ∈S_{m+1,n−1}} sgn(σ)θ_{g}∘_{1}ω_{U} ∘σ + ∑_{σ∈S_{n+1,m−1}} sgn(σ)ω_{g}∘_{1}θ_{U} ∘σ)

(−1)^{n}( ∑_{σ∈S_{1,...,1,m}^{<}} sgn(σ)θ_{V} (g^{⊗n}⊗ω_{g})∘σ+ ∑_{σ∈S_{1,...,1,n}^{<}} sgn(σ)ω_{V} (g^{⊗m}⊗θ_{g})∘σ)

− ∑_{σ∈S_{n,m}^{<}}sgn(σ)μ_{V} (θ_{g}⊗ω_{g})∘σ−(−1)^{n+m} ∑_{σ∈S_{m,n}^{<}}sgn(σ)μ_{V} (ω_{g}⊗θ_{g})∘σ.

In the above theorem, we use the notation of (20) and of S^{<} as defined after (34c). One should remark that depending on whether m < n or n < m, the first or the second summand in the last bracketed expression above is zero. The proof of the theorem will be given at the end of this section.

8.3. The operations l_{k} for k ≥ 3. Now consider the cases k ≥ 3. Pick elements θ_{s} ∈ C_{Lie_{B→W}}^{n_{s}} (T; T) (1 ≤ s ≤ k). Each θ_{s} corresponds, via the identification (36), to the tuple

(θ_{s,U}, θ_{s,V} , θ_{s,g}) = (θ_{s}(μ_{n_{s}+1}), θ_{s}(ν_{n_{s}+1}), θ_{s}(f_{n_{s}})) ∈ ⋀^{n_{s}}(U; V).

Since l_{k} has degree 2 − k, the element l_{k}(θ_{1}, …, θ_{k}) lies in C_{Lie_{B→W}}^{t} (T; T), where

t = −k + 2 + ∑_{s=1}^{k} n_{s}.

Under the identification (36), l_{k}(θ_{1}, …, θ_{k}) is determined by

(l_{k}(θ_{1}, …, θ_{k})(μ_{t+1}), l_{k}(θ_{1}, …, θ_{k})(ν_{t+1}), l_{k}(θ_{1}, …, θ_{k})(f_{t})) ∈ ⋀^{t}(U; V).

Now we extend the ∘_{i} operation as follows. Fix s ∈ {1, …, k}. Let

a′ = (a_{1}, …, â_{s}, …, a_{k})

be a (k−1)-tuple of distinct points in the set {1, …, n_{s} +1}. Then we define

θ_{s,V} ∘_{a′} (θ_{1,g}, …, θ̂_{s,g}, …, θ_{k,g}) ∈ Hom(U^{⊗t}, V)

to be the element obtained by plugging θ_{j,g} (1 ≤ j ≤ k, j ≠ s) into the a_{j}th input of θ_{s,V} and g into the other (n_{s} + 2 − k) inputs of θ_{s,V}. Also define the coefficient

(−1)^{a′} = (−1)^{\frac{(ns+1)(ns)}{2} +∑_{i=ns+1} r_{i}(n_{s}+1−i)},

where

r_{a} = \begin{cases}|θ_{j}| = n_{j} & if a = a_{j} ∈ \{a_{1}, …, â_{s}, …, a_{k}\}, \\ 1 & otherwise. \end{cases}

Let us remark that the set {r_{1}, …, r_{n_{s}+1}} satisfies r_{1} + ⋯ + r_{n_{s}+1} = t. One says that a′ is admissible if this set also satisfies r_{1} ≤ ⋯ ≤ r_{n_{s}+1}. One denotes by A the set of admissible a′.

Theorem 8.4. For k ≥ 3 and notations as above, we have

l_{k}(θ_{1}, …, θ_{k})(μ_{t+1}) = 0, l_{k}(θ_{1}, …, θ_{k})(ν_{t+1}) = 0, and l_{k}(θ_{1}, …, θ_{k})(f_{t}) =

(−1)^{ν(θ_{1},...,θ_{k})}∑_{s=1}^{k} ∑_{a′∈A} ∑_{σ∈S_{r_{1},...,r_{ns+1}}^{<}} sgn(σ)(−1)^{a′}θ_{s,V} ∘_{a′} (θ_{1,g}, ..., θ̂_{s,g}, ..., θ_{k,g})∘σ,

with ν(θ_{1}, …, θ_{k}) defined as in (16), and S^{<} defined after (34c).

Corollary 8.5. Suppose k ≥ 3 and θ_{s} ∈ C_{Lie_{B→W}}^{n_{s}} (T; T) (1 ≤ s ≤ k). If

n_{s} < k − 1 for 1 ≤ s ≤ k,

then

l_{k}(θ_{1}, …, θ_{k}) = 0.

In other words, for each q ≥ 0 and any k ≥ q + 3, the operation

l_{k}: (C_{Lie_{B→W}}^{≤q} (T; T))^{⊗k} → C_{Lie_{B→W}}^{∗} (T; T)

is trivial.

Proof of Theorem 8.2. To prove (43a), first note that

∂(μ_{n+m+1}) = ∑_{\begin{matrix}i+j = n+m+2 \\ i,j ≥ 2 \end{matrix}}(−1)^{j(i−1)} ∑_{σ∈S_{j,i−1}} sgn(σ)μ_{i} ∘ (μ_{j} ⊗ Id^{i−1}) ∘ σ.

Since the E-decorated 2-colored directed (1, n+m+1)-graph μ_{i}∘(μ_{j} ⊗Id^{i−1}) has two vertices, we have

l_{2}(θ, ω)(μ_{n+m+1}) =

(−1)^{|θ|} ∑_{\begin{matrix}i+j = n+m+2 \\ i,j ≥ 2 \end{matrix}} (−1)^{j(i−1)} ∑_{σ∈S_{j,i−1}}sgn(σ)

{θ(μ_{i}) ∘ (ω(μ_{j}) ⊗ g^{⊗^{i−1}}) + ω(μ_{i}) ∘ (θ(μ_{j}) ⊗ g^{⊗^{i−1}})} ∘ σ

= (−1)^{mn} ∑_{σ∈S_{m+1,n}} sgn(σ)θ_{U} ∘ (ω_{U} ⊗ g^{⊗^{n}}) ∘ σ

+(−1)^{(n+1)m+n} ∑_{σ∈S_{n+1,m}} sgn(σ)ω_{U} ∘ (θ_{U} ⊗ g^{⊗^{m}}) ∘ σ.

Since ∂(ν_{n+m+1}) has the same defining formula as ∂(μ_{n+m+1}) (with ν_{l} replacing μ_{l} everywhere), the reasoning in the previous paragraph also applies to l_{2}(θ, ω)(ν_{n+m+1}) to establish (43b).

To prove (43c), first note that

∂(f_{n+m}) = ∑_{l=2}^{n+m} ∑_{\begin{matrix}r_{1}+···+r_{l}=n+m \\ r_{1} ≤ ··· ≤ r_{l} \end{matrix}}(−1)^{ε} ∑_{σ∈S_{r_{1},...,r_{l}}^{<}} sgn(σ)ν_{l}(f_{r_{1}} ⊗ ⋯ ⊗ f_{r_{l}}) ∘ σ

− ∑_{\begin{matrix}i+j = n+m+1 \\ i ≥ 1,j ≥ 2 \end{matrix}}(−1)^{j(i−1)} ∑_{σ∈S_{j,i−1}} sgn(σ)f_{i} ∘ (μ_{j} ⊗ Id^{⊗^{i−1}}) ∘ σ,

where ε is as in (35).

An argument essentially identical to the first paragraph of this proof can be applied to the terms f_{i} ∘ (μ_{j} ⊗ Id^{i−1}). This gives rise to the sums

\begin{matrix}(−1)^{m(n−1)} ∑_{σ∈S_{m+1,n−1}} sgn(σ)θ_{g} ∘ (ω_{U} ⊗ g^{⊗^{n−1}}) ∘ σ \\ + (−1)^{m(n−1)} ∑_{σ∈S_{n+1,m−1}} sgn(σ)ω_{g} ∘ (θ_{U} ⊗ g^{⊗^{m−1}}) ∘ σ \end{matrix} ; (45)

in l_{2}(θ, ω)(f_{n+m}).

In (45), the E-decorated 2-colored directed (1, n + m)-graph

Γ = ν_{l}(f_{r_{1}} ⊗ ⋯ ⊗ f_{r_{l}})

has l+1 vertices, say, v_{top}, v_{bot}^{1}, …, v_{bot}^{l}, with decorations ν_{l}, f_{r_{1}}, …, f_{r_{l}}, respectively. In this graph Γ, the only pairs of distinct vertices are (v_{top}, v_{bot}^{∗}), (v_{bot}^{∗}, v_{top}), and (v_{bot}^{i}, v_{bot}^{j}) (i ≠ j). The corresponding elements (without the signs) in l_{2}(θ, ω)(f_{n+m}) are:

(1) θ(ν_{l})(β(f_{r_{1}}) ⊗ ⋯ ⊗ ω(f_{r_{i}}) ⊗ ⋯ β(f_{r_{l}})) (1 ≤ i ≤ l), which is 0 unless l = n + 1, r_{n+1} = m, and all the other r_{∗} = 1; (2) ω(ν_{l})(β(f_{r_{1}}) ⊗ ⋯ ⊗ θ(f_{r_{j}}) ⊗ ⋯ ⊗ β(f_{r_{l}})) (1 ≤ j ≤ l), which is 0 unless

l = m + 1, r_{m+1} = n, and all the other r_{∗} = 1;

(3) β(ν_{l})(β(f_{r_{1}}) ⊗ ⋯ ⊗ θ(f_{r_{i}}) ⊗ ⋯ ⊗ ω(f_{r_{j}}) ⊗ ⋯ ⊗ β(f_{r_{l}})), which is 0 unless l = 2 and (r_{1}, r_{2}) = (n, m);

(4) β(ν_{l})(β(f_{r_{1}}) ⊗ ⋯ ⊗ ω(f_{r_{i}}) ⊗ ⋯ ⊗ θ(f_{r_{j}}) ⊗ ⋯ ⊗ β(f_{r_{l}})), which is 0

unless l = 2 and (r_{1}, r_{2}) = (m, n). Taking all the signs into account, we also obtain the following sums in l_{2}(θ, ω)(f_{n+m}):

(−1)^{|θ|} ∑_{σ∈S_{1,...,1,m}^{<}} sgn(σ)θ_{V} (g^{⊗n} ⊗ ω_{g}) ∘ σ,

(−1)^{|θ|} ∑_{σ∈S_{1,...,1,n}^{<}} sgn(σ)ω_{V} (g^{⊗m} ⊗ θ_{g}) ∘ σ,

(−1)^{|θ|}(−1)^{1+n} ∑_{σ∈S_{n,m}^{<}}sgn(σ)μ_{V} (θ_{g} ⊗ ω_{g}) ∘ σ,

(−1)^{|θ|}(−1)^{1+m} ∑_{σ∈S_{m,n}^{<}}sgn(σ)μ_{V} (ω_{g} ⊗ θ_{g}) ∘ σ.

(46)

The required result (43c) is now obtained by combining (45) and (46). This finishes the proof of Theorem 8.2. □

Proof of Theorem 8.4. This proof is identical to the proof of Theorem 6.4 if one shifts the indices, replacing t by t + 1, s + 1 by 1, and (−1)^{a} by −(−1)^{a′} □

9. Deformations of diagrams

Recall that a diagram in a category 𝒞 is a functor ℱ : 𝒟 → 𝒞 from a small category 𝒟 to 𝒞; the category 𝒟 is called the shape of the diagram ℱ. Diagrams of shape 𝒟 can equivalently be described as algebras over an Ob(𝒟)-colored operad D which has only elements of arity 1 (one input, one output) and

D\binom{d}{c} := Mor_{𝒟} (c, d), for c, d ∈ Ob(𝒟).

The operadic composition in D equals the categorial composition of 𝒟. It is clear that 𝒟-diagrams in 𝒞 are precisely Ob(𝒟)-colored D-algebras in 𝒞.

The Ob(𝒟)-colored operad D as defined above lives in the category of sets. Since we will be primarily interested in diagrams in the category of k-vectors spaces, we may as well consider the k-linear operad generated by D or assume from the very beginning that D is given by

D\binom{d}{c} := Span_{k} (Mor_{𝒟} (c, d)) , for c, d ∈ Ob(𝒟),

where Span_{k}(−) denotes the k-linear span. We will call colored operads of the above form diagram operads.

Example 9.1. In this example we describe the diagram operad Iso associated to the category Iso consisting of two objects and two mutually inverse maps between these objects. Let f : B → W, g : W → B be two degree-zero generators. Then

Iso := \frac{F(f, g)}{(fg = 1_{W}, gf = 1_{B})},

where F(f, g) denotes the free {B, W}-colored operad on the set {f, g} and (fg = 1_{W}, gf = 1_{B}) the operadic ideal generated by fg − 1_{W} and gf − 1_{B}.

Algebras over Iso consist of two mutually inverse degree zero chain maps F : U → V and G : V → U. In other words, Iso-algebras are diagrams

(47)  [figure: F; G; V; U; , F G = 1 and GF = 1.]

A typical diagram operad D does not admit a minimal model. For instance, a hypothetical minimal model of the operad Iso from Example 9.1 shall have generators f_{0}, g_{0} for f and g, but also a generator, say f_{1}, whose boundary kills the difference f_{0}g_{0} − 1_{W}, i.e. satisfying

f_{0}g_{0} − 1_{W} = ∂f_{1}.

The “constant” 1_{W} however defies any thinkable notion of minimality.

This phenomenon is related to the fact that a typical diagram operad D, such as Iso, is not augmented, by which we mean that it does not admit an operad morphism D → i to the terminal Ob(𝒟)-colored operad i. In the next example we will see that sometimes there still exists a cofibrant resolution whose size is that of a minimal model.

Example 9.2. A small cofibrant resolution of Iso was described in [33, Theorem 9]. It is a graded colored differential operad

R_{iso} := (F(f_{0}, f_{1}, …; g_{0}, g_{1}, …), d),

with generators of two types,

(i) generators \{f_{n}\}_{n≥0}, deg(f_{n}) = n, \{ \begin{matrix}f_{n} : B → W if n is even, \\ f_{n} : B → B if n is odd, \end{matrix}

(ii) generators \{g_{n}\}_{n≥0}, deg(g_{n}) = n, \{ \begin{matrix}g_{n} : W → B if n is even, \\ g_{n} : W → W if n is odd. \end{matrix}

The differential ∂ is given by

\begin{matrix}∂f_{0} := 0, & ∂g_{0} := 0, \\ ∂f_{1} := g_{0}f_{0} − 1, & ∂g_{1} := f_{0}g_{0} − 1 \end{matrix}

and, on remaining generators, by the formula

∂f_{2m} := ∑_{0≤i<m}(f_{2i}f_{2(m−i)−1} − g_{2(m−i)−1}f_{2i}), m ≥ 0,
∂f_{2m+1} := ∑_{0≤j≤m} g_{2j}f_{2(m−j)} − ∑_{0≤j<m} f_{2j+1}f_{2(m−j)−1}, m ≥ 1,
∂g_{2m} := ∑_{0≤i<m}(g_{2i}g_{2(m−i)−1} − f_{2(m−i)−1}g_{2i}), m ≥ 0,
∂g_{2m+1} := ∑_{0≤j≤m} f_{2j}g_{2(m−j)} − ∑_{0≤j<m} g_{2j+1}g_{2(m−j)−1}, m ≥ 1.

The above resolution is “minimal’ in the following sense. Consider a oneparametric family Iso_{ε} of {B, W}-colored operads defined by

Iso_{ε} := \frac{F(f, g)}{(fg = ε · 1_{W}, gf = ε · 1_{B})},

where ε is a formal parameter. The operad Iso_{0} clearly describes couples (F, G) of maps F : U → V and G : V → U such that F G = 0 and GF = 0, while Iso_{ε} is, at a generic ε, isomorphic to the operad Iso. In other words, Iso is a deformation of Iso_{0}. It turns out that Iso_{0} is an augmented colored operad that admits a minimal model whose generators are the same as the generators of R_{iso}; see [33, Theorem 10].

An obvious generalization of the machinery developed in the previous sections applies verbatim to resolutions of diagram operads. One typically gets an L_{∞}-algebra with a nontrivial ‘curvature’ l_{0}; see [36, Section 5] for the terminology and definitions. The corresponding Maurer–Cartan equation then involves the l_{0}-term.

Example 9.3. In this example we describe the L_{∞}-deformation complex (C_{Iso}^{∗}(T, T), l_{0}, l_{1}, l_{2}, …) for a diagram T as in (47). As we already observed, this deformation complex has, as a consequence of the presence of 1 in the formulas for ∂f_{1} and ∂g_{1} in R_{iso}, a nontrivial l_{0}. On the other hand, since the differential ∂ on the generators of R_{iso} does not have higher than quadratic terms, all l_{k}’s are trivial, for k ≥ 3.

Formula (11) applied to the resolution R_{iso} from Example 9.2 gives the underlying cochain complex

C_{Iso}^{n}(T, T) = \begin{cases}Hom(U, V) ⊕ Hom(V, U) & for n ≥ 1 odd, and \\ Hom(U, U) ⊕ Hom(V, V) & for n ≥ 1 even. \end{cases}

Formula (15) makes sense also for k = 0 and describes l_{0} ∈ C_{Iso}^{2}(T, T) as the direct sum of the identity maps Id_{U} ⊕ Id_{V} ∈ Hom(U, U) ⊕ Hom(V, V).

Likewise, one obtains the following formulas for the operation

l_{1} : C_{Iso}^{∗}(T, T) → C_{Iso}^{∗+1}(T, T).

If α ⊕ β ∈ C_{Iso}^{n}(T, T), n ≥ 1 odd, then

l_{1}(α ⊕ β) = (Gα + βF) ⊕ (αG + F β) ∈ C_{Iso}^{n+1}(T, T).

For γ ⊕ δ ∈ C_{Iso}^{n}(T, T), n ≥ 1 even, we have

l_{1}(γ ⊕ δ) = (F γ − δF) ⊕ (Gδ − γG) ∈ C_{Iso}^{n+1}(T, T).

The bracket

l_{2} : C_{Iso}^{m}(T, T) ⊗ C_{Iso}^{n}(T, T) → C_{Iso}^{m+n}(T, T)

is given as follows. For α′ ⊕β′ ∈ C_{Iso}^{m}(T, T), α′′ ⊕β′′ ∈ C_{Iso}^{n}(T, T), m, n odd, we have

l_{2}(α′ ⊕ β′, α′′ ⊕ β′′) = (β′α′′ + β′′α′) ⊕ (α′β′′ + α′′β′) ∈ C_{Iso}^{m+n}(T, T).

For α ⊕ β ∈ C_{Iso}^{m}(T, T), γ ⊕ δ ∈ C_{Iso}^{n}(T, T), m odd, n even, we have

l_{2}(α ⊕ β, γ ⊕ δ) = −l_{2}(γ ⊕ δ, α ⊕ β) = (αγ − δα) ⊕ (βδ − γβ) ∈ C_{Iso}^{m+n}(T, T).

Finally, for γ′ ⊕ δ′ ∈ C_{Iso}^{m}(T, T), γ′′ ⊕ δ′′ ∈ C_{Iso}^{n}(T, T), m, n even, we have

l_{2}(γ′ ⊕ δ′, γ′′ ⊕ δ′′) = −l_{2}(γ′′ ⊕ δ′′, γ′ ⊕ δ′) =
(γ′γ′′ − γ′′γ′) ⊕ (δ′δ′′ − δ′′δ′) ∈ C_{Iso}^{m+n}(T, T).

The higher l_{k}’s, k ≥ 3, are trivial.

Observe that, for each w ∈ C_{Iso}^{∗}(T, T), l_{2}(l_{0}, w) = 0; therefore, by [36, Section 5], l_{1}^{2} = 0. In other words, l_{1} is a differential and the standard analysis of deformation theory applies. For instance, there exists the canonical element χ := F ⊕ G ∈ C_{Iso}^{1}(T, T) such that

l_{1}(w) = l_{2}(χ, w), w ∈ C_{Iso}^{∗}(T, T).

As expected [36, 51], Iso-algebras are solutions of the Maurer–Cartan equation in the L_{∞}-complex of the trivial D-algebra Tø with F = 0, G = 0. Indeed, if κ = Φ ⊕ Ψ ∈ C_{Iso}^{1}(T, T), then the Maurer–Cartan equation

−l_{0} + \frac{1}{2}l_{2}(κ, κ) = 0

expands into

−(Id_{U} ⊕ Id_{V}) + \frac{1}{2}(2ΨΦ ⊕ 2ΦΨ) = 0,

which says precisely that Φ and Ψ are mutually inverse isomorphisms.

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Mathematics Research Unit, 162A, avenue de la faiencerie L-1511, Luxembourg, Grand duchy of Luxembourg yael.fregier@uni.lu

Mathematical Institute of the Academy, Žitná 25, 115 67 Prague 1, The Czech Republic markl@math.cas.cz

Department of Mathematics, The Ohio State University at Newark, 1179 University Drive, Newark, OH 43055, USA dyau@math.ohio-state.edu

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