The Koszul map K

Abstract

The Bitableax correspondence isomorphism/Koszul map Theorem (BCK Theorem, for short, Theorem 6.5 below) describes a relevant pair of mutually inverse vector space isomorphisms, the Koszul map K : U(gl(n))-> Sym(gl(n)) and the bitableaux correspondence iWe describe a linear \emph{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 {\textit{Capelli bitableau}} $[S|T]$ in $\mathbf{U}(gl(n))$ to the {\textit{(determinantal) bitableau}} $(S|T)$ in $\mathbb{C}[M_{n,n}]$ and any {\textit{Capelli *-bitableau}} $[S|T{]}^{\ast }$ in $\mathbf{U}(gl(n))$ to the {\textit{(permanental) *-bitableau}} $(S|T{)}^{\ast }$ in $\mathbb{C}[M_{n,n}]$. These results are far-reaching generalizations of the pioneering result of J.-L. Koszul [19] on the Capelli determinant in $\mathbf{U}(gl(n))$ (see, e.g. [24], [27]). We introduce {\textit{column}} Capelli bitableaux and *-bitableaux in Section 6; since they are mapped by the isomorphism $\mathcal{K}$ to {\textit{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 $\mathbf{\zeta }(n)$ of $\mathbf{U}(gl(n))$ equals the subalgebra of invariants $\mathbf{U}(gl(n){)}^{A{d}_{gl(n)}}$, then $$\mathcal{K}[\mathbf{\zeta }(n)]=\mathbb{C}[M_{n,n}{]}^{a{d}_{gl(n)}}.$$somorphism B : Sym(gl(n)) -> U(gl(n)) that deeply link the enveloping algebra U(gl(n)) of the general linear Lie algebra gl(n) and the symmetric algebra Sym(gl(n)). The BCK Theorem can be regarded as a sharpened version of the PBW Theorem for the enveloping algebra U(gl(n)).

arXiv admin note: text overlap with arXiv:1807.10045