On the action of the Koszul map on the enveloping algebra of the general linear Lie algebra

Abstract

We describe a linear equivariant isomorphism $\mathcal{K}$ from the enveloping algebra $\mathbf{U}(gl(n))$ to the algebra $\mathbb{C}[M_{n,n}]\cong \mathbf{Sym}(gl(n))$ of polynomials in the entries of a “generic” square matrix of order n. The isomorphism $\mathcal{K}$ maps any Capelli bitableau $[S|T]$ in $\mathbf{U}(gl(n))$ to the (determinantal) bitableau $(S|T)$ in $\mathbb{C}[M_{n,n}]$ and any Capelli *-bitableau ${[S|T]}^{*}$ in $\mathbf{U}(gl(n))$ to the (permanental) *-bitableau ${(S|T)}^{*}$ in $\mathbb{C}[M_{n,n}].$ These results are far-reaching generalizations of the pioneering result of Koszul on the Capelli determinant in $\mathbf{U}(gl(n)).$ We introduce column Capelli bitableaux and *-bitableaux in Section 6; since they are mapped by the isomorphism $\mathcal{K}$ to monomials in $\mathbb{C}[M_{n,n}],$ this isomorphism can be regarded as a sharpened version of the PBW isomorphism for the enveloping algebra $\mathbf{U}(gl(n)).$ Since the center $\mathit{\zeta }(n)$ of $\mathbf{U}(gl(n))$ equals the subalgebra of invariants $\mathbf{U}{(gl(n))}^{A{d}_{gl(n)}},$ then $$\mathcal{K}\left[\mathit{\zeta }(n)\right]=\mathbb{C}{[M_{n,n}]}^{a{d}_{gl(n)}}.$$

Keywords:

• Capelli determinants
• central elements
• enveloping algebras
• Lie superalgebras
• Young tableaux

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 17B10
• 05E10
• 17B35

Notes

1 The symbol cdet denotes the column determinat of a matrix $A=[a_{ij}]$ with noncommutative entries: $\mathbf{cdet}(A)=\sum _{\sigma }\hspace{0.5em}{(-1)}^{\left|\sigma \right|}{a}_{\sigma (1),1}{a}_{\sigma (2),2}\cdots a_{\sigma (n),n}.$