Geodesic
A Central Element in the Universal Enveloping Algebra of Type Dn via Minor Summation Formula of Pfaffians
Journal of Lie Theory, Volume 18 (2008) no. 3, pp. 581-594

See the original article notice from the Heldermann Verlag source

Zbl arXiv
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.
Classification: 17B35, 15A15
Keywords: Orthogonal Lie algebra, noncommutative Pfaffian, minor summation formula 
T. Hashimoto. A Central Element in the Universal Enveloping Algebra of Type Dn via Minor Summation Formula of Pfaffians. Journal of Lie Theory, Volume 18 (2008) no. 3, pp. 581-594. http://geodesic.mathdoc.fr/item/JOLT_2008_18_3_a5/
@article{JOLT_2008_18_3_a5,
     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},
     zbl = {1210.17018},
     url = {http://geodesic.mathdoc.fr/item/JOLT_2008_18_3_a5/}
}
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