2021/02/13 by Dimitar Grantcharov, Grantcharov, Dimitar, Luke Robitaille +1
Mathematics · Chemistry · #Algebraic structures and combinatorial models #Advanced Topics in Algebra #Molecular spectroscopy and chirality
paper · pdf · doi:10.48550/arxiv.2102.07027
In this paper we study properties of a homomorphism ρ from the universal enveloping algebra U=U(\mathfrakgl(n+1)) to a tensor product of an algebra \mathcal D'(n) of differential operators and U(\mathfrakgl(n)). We find a formula for the image of the Capelli determinant of \mathfrakgl(n+1) under ρ, and, in particular, of the images under ρ of the Gelfand generators of the center Z(\mathfrakgl(n+1)) of U. This formula is proven by relating ρ to the corresponding Harish-Chandra isomorphisms, and, alternatively, by using a purely computational approach. Furthermore, we define a homomorphism from \mathcal D'(n) ⊗ U(\mathfrakgl(n)) to an algebra containing U as a subalgebra, so that σ(ρ(u)) - u ∈ G1 U, for all u ∈ U, where G1 = ∑i=0n Eii.