 

W. S. Gant and
Ben Williams
Spaces of generators for the 2 x 2 complex matrix algebra
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Published: 
June 3, 2024. 
Keywords: 
Matrix algebras; spaces of generators; generators of Azumaya algberas. 
Subject [2020]: 
14F25 (Primary); 15A99, 16S15, 55R40 (Secondary). 


Abstract
This paper studies the space of rtuples of 2 x 2 complex matrices that generate
the full matrix algebra, considered up to changeofbasis. We show that when r is 2,
this space is homotopy equivalent to the quotient of a product of a circle and a sphere by an involution. When r is greater than 2, we determine the rational cohomology of the space in degrees less than 4r6. As an application, we use the machinery of [5] to prove that for all natural numbers d, there exists a ring R of Krull dimension d and a degree2 Azumaya algebra A over R that cannot be generated by
fewer than 2[ d/4 ] + 2 elements.


Acknowledgements
We acknowledge the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC), RGPIN201603780, RGPIN202102603.
Cette recherche a ete financee par le Conseil de recherches en sciences naturelles et en genie du Canada (CRSNG), RGPIN201603780, RGPIN202102603.


Author information
W. S. Gant
Department of Mathematics
University of British Columbia
Vancouver, BC V6T 1Z2, Canada
wsgant@math.ubc.ca
Ben Williams
Department of Mathematics
University of British Columbia
Vancouver, BC V6T 1Z2, Canada
tbjw@math.ubc.ca

