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

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

2006-02-03

arxiv.org/abs/math/0602055v2

Mathematics

Representation Theory

version 1.1(corrected typos and added a reference), 17 pages, no figures

Scientific paper

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.

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