  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)).
