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Journal of Lie Theory 18 (2008), No. 3, 581--594
Copyright Heldermann Verlag 2008

A Central Element in the Universal Enveloping Algebra of Type Dn via Minor Summation Formula of Pfaffians

Takashi Hashimoto
Dept. of Informatics, Graduate School of Engineering, Tottori University, Tottori 680-8552, Japan

It is well known that the universal enveloping algebra U(o2n) of the orthogonal Lie algebra o2n of size even has a central element given explicitly in terms of Pfaffian of a certain matrix which is alternating along the anti-diagonal (we call such a matrix "anti-alternating" for short) whose entries are in U(o2n). In this paper, we establish a minor summation formula of Pfaffian of the noncommutative anti-alternating matrix, as well as of commutative anti-alternating matrix. As an application, we show that the eigenvalues of the Pfaffian-type central element on the highest weight representations of o2n can be easily computed.

Keywords: Orthogonal Lie algebra, noncommutative Pfaffian, minor summation formula.

MSC: 17B35, 15A15

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