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

摘要

We describe a linear equivariant isomorphism K from the enveloping algebra U(gl(n)) to the algebra CM-n,M-n congruent to Sym(gl(n)) of polynomials in the entries of a "generic" square matrix of order n. The isomorphism K maps any Capelli bitableau S vertical bar T in U(gl(n)) to the (determinantal) bitableau (S vertical bar T) in CM-n,M-n and any Capelli *-bitableau S vertical bar T* in U(gl(n)) to the (permanental) *-bitableau (S vertical bar T)* in CM-n,M-n. These results are far-reaching generalizations of the pioneering result of Koszul on the Capelli determinant in U(gl(n)). We introduce column Capelli bitableaux and *-bitableaux in Section 6; since they are mapped by the isomorphism K to monomials in CM-n,M-n, this isomorphism can be regarded as a sharpened version of the PBW isomorphism for the enveloping algebra U(gl(n)). Since the center zeta(n) of U(gl(n)) equals the subalgebra of invariants U(gl(n))(Adgl(n)), then