
Journal of Lie Theory 18 (2008), No. 3, 581594 Copyright Heldermann Verlag 2008 A Central Element in the Universal Enveloping Algebra of Type D_{n} via Minor Summation Formula of Pfaffians Takashi Hashimoto Dept. of Informatics, Graduate School of Engineering, Tottori University, Tottori 6808552, Japan thashi@ike.tottoriu.ac.jp It is well known that the universal enveloping algebra U(o_{2n}) of the orthogonal Lie algebra o_{2n} of size even has a central element given explicitly in terms of Pfaffian of a certain matrix which is alternating along the antidiagonal (we call such a matrix "antialternating" for short) whose entries are in U(o_{2n}). In this paper, we establish a minor summation formula of Pfaffian of the noncommutative antialternating matrix, as well as of commutative antialternating matrix. As an application, we show that the eigenvalues of the Pfaffiantype central element on the highest weight representations of o_{2n} can be easily computed. Keywords: Orthogonal Lie algebra, noncommutative Pfaffian, minor summation formula. MSC: 17B35, 15A15 [ Fulltextpdf (190 KB)] for subscribers only. 