(University of Chile, Santiago)
Friday 4th February, 2.05-2.55pm, Carslaw 375
From non-commutative determinants to *-analogues
The problem of constructing a determinant mapping for matrices over a
non-commutative base field may be tackled in a manner dual to
Dieudonné's approach, i.e. imposing restrictions at the source of the
"determinant to be" mapping instead of identifying values in its target.
It turns out, however, that even in the modest case of 2 by 2 matrices over a
base ring A, the problem is still too hard, unless one asks that our
non-commuting entries satisfy some sort of commutation relation.
One interesting example is given by (ba)* = a*b*, where * stands for an
(anti)involution in A. This leads us to *-analogues of the classical groups
GL(2,A) and SL(2,A), reminiscent of q-analogues, which play a
significant role in representation theory and seem to afford interesting
new examples for the case of non-semisimple A's.