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)).
Categories: math.RA