Zhongshan University/Generation 5 Inc.
Let $A$ be a finite-dimensional (unitary) associative algebra
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.