The Niltriangular Subalgebra of the Chevalley Algebra: the Enveloping Algebra, Ideals, and Automorphisms

Abstract—The enveloping algebra of the niltriangular subalgebra $N\Phi (K)$ of the Chevalley algebra of type ${{A}_{{n - 1}}}$ is the algebra of niltriangular $n \times n$ matrices over K. The enveloping algebras R of other types constructed so far are nonassociative. For classical types, an explicit description of automorphisms of the rings R over any commutative associative ring with an identity is given; in the case where K is a field, all ideals in R are also listed. The enumeration of ideals in R for $K = GF(q)$ leads to a solution of a combinatorial problem concerning ideals of the algebras $N\Phi (K)$.