Casimir Elements for Some Graded Lie Algebras and Superalgebras |
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Authors: | Yuri Bahturin Alexander Molev |
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Affiliation: | (1) Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John s, NF, A1C5S7, Canada;(2) School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia |
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Abstract: | We consider a class of Lie algebras L such that L admits a grading by a finite Abelian group so that each nontrivial homogeneous component is one-dimensional. In particular, this class contains simple Lie algebras of types A, C and D where in C and D cases the rank of L is a power of 2. We give a simple construction of a family of central elements of the universal enveloping algebra U(L). We show that for the A-type Lie algebras the elements coincide with the Gelfand invariants and thus generate the center of U(L). The construction can be extended to Lie superalgebras with the additional assumption that the group grading is compatible with the parity grading. |
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Keywords: | graded Lie algebra Casimir element |
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