A central element in the universal enveloping algebra of type D_n via minor summation formula of Pfaffians

TL;DR

This paper develops minor summation formulas for Pfaffians of anti-alternating matrices in the universal enveloping algebra of type D_n, facilitating eigenvalue computations of central elements on highest weight modules.

Contribution

It introduces new minor summation formulas for Pfaffians of noncommutative and commutative anti-alternating matrices, advancing understanding of central elements in type D_n Lie algebras.

Findings

Derived minor summation formulas for Pfaffians of anti-alternating matrices.

Enabled straightforward eigenvalue calculations for central elements on highest weight modules.

Extended formulas to both noncommutative and commutative matrix cases.

Abstract

It is known that the universal enveloping algebra of the orthogonal Lie algebra of size even has a central element expressed in terms of Pfaffian of a certain matrix alternating along the anti-diagonal (which we call anti-alternating for short) whose entries are in the univerasal enveloping algebra. In this paper, we establish minor summation formulae of Pfaffian for the noncommutative anti-alternating matrix, as well as for commutative anti-alternating matrix. As an application, we show that the eigenvalues on the highest weight modules of the central element given by the Pfaffian can be easily computed.