Nilpotency of the Unit Group of a Finite-Dimensional Associative Algebra

Wenxue Huang
Zhongshan University/Generation 5 Inc.

 Let $A$ be a finite-dimensional (unitary) associative algebra over an
algebraically closed field.  Then the multiplicative semigroup of $A$ is
a linear algebraic monoid, of which the unit group is a linear algebraic
group.   Several characterizations of nilpotency of the unit group, in
terms of Lie algebra, idempotent, the normalizer of a maximal torus,
subgroup or subalgebra, are discussed.