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

Voir la notice de l'article provenant de la source Heldermann Verlag

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
Mots-clés : 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},
     publisher = {mathdoc},
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T. Hashimoto . A Central Element in the Universal Enveloping Algebra of Type Dn via Minor Summation Formula of Pfaffians. Journal of Lie theory, Tome 18 (2008) no. 3, pp. 581-594. http://geodesic.mathdoc.fr/item/JLT_2008_18_3_JLT_2008_18_3_a5/