A Central Element in the Universal Enveloping Algebra of Type D<sub>n</sub> via Minor Summation Formula of Pfaffians

T. Hashimoto

doi:10.5802/jolt.512

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

T. Hashimoto

Journal of Lie Theory, Volume 18 (2008) no. 3, pp. 581-594

Abstract

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 anti-diagonal (we call such a matrix "anti-alternating" for short) whose entries are in U(o_2n). 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 o_2n can be easily computed.

arXiv Zbl

DOI: 10.5802/jolt.512

Classification: 17B35, 15A15
Keywords: Orthogonal Lie algebra, noncommutative Pfaffian, minor summation formula

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     author = {T. Hashimoto},
     title = {A {Central} {Element} in the {Universal} {Enveloping} {Algebra} of {Type} {D\protect\textsubscript{n}} via {Minor} {Summation} {Formula} of {Pfaffians}},
     journal = {Journal of Lie Theory},
     pages = {581--594},
     year = {2008},
     volume = {18},
     number = {3},
     doi = {10.5802/jolt.512},
     zbl = {1210.17018},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.512/}
}

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T. Hashimoto. A Central Element in the Universal Enveloping Algebra of Type D_n via Minor Summation Formula of Pfaffians. Journal of Lie Theory, Volume 18 (2008) no. 3, pp. 581-594. doi: 10.5802/jolt.512

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