##
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.