# The Koszul map K

## TL;DR

This paper introduces an equivariant isomorphism linking the universal enveloping algebra of gl(n) with polynomial functions on matrices, generalizing Koszul's classical results and sharpening the PBW theorem.

## Contribution

It constructs a linear equivariant isomorphism between U(gl(n)) and polynomial matrix algebra, extending Koszul's Capelli determinant results and refining the PBW isomorphism.

## Key findings

- Maps Capelli bitableaux to determinantal bitableaux
- Establishes an isomorphism that generalizes Koszul's theorem
- Provides a sharpened PBW isomorphism for U(gl(n))

## 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}} $[S|T]^*$ in $\mathbf{U}(gl(n))$ to the {\textit{(permanental) *-bitableau}} $(S|T)^*$ 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 $\boldsymbol{\zeta}(n)$ of $\mathbf{U}(gl(n))$ equals the subalgebra of invariants $\mathbf{U}(gl(n))^{Ad_{gl(n)}}$, then $$ \mathcal{K} \big[ \boldsymbol{\zeta}(n) \big] = {\mathbb C}[M_{n,n}]^{ad_{gl(n)}}. $$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)).