Mathematics – Representation Theory
Scientific paper
1998-09-06
Mathematics
Representation Theory
54 pages
Scientific paper
Let $\Gamma$ be a countable abelian semigroup and $A$ be a countable abelian group satisfying a certain finiteness condition. Suppose that a group $G$ acts on a $(\Gamma \times A)$-graded Lie superalgebra ${\frak L}=\bigoplus_{(\alpha,a) \in \Gamma\times A} {\frak L}_{(\alpha,a)}$ by Lie superalgebra automorphisms preserving the $(\Gamma\times A)$-gradation. In this paper, we show that the Euler-Poincar\'e principle yields the generalized denominator identity for ${\frak L}$ and derive a closed form formula for the supertraces $\text{str}(g|{\frak L}_{(\alpha,a)})$ for all $g\in G$,$(\alpha,a) \in \Gamma\times A$. We discuss the applications of our supertrace formula to various classes of infinite dimensional Lie superalgebras such as free Lie superalgebras and generalized Kac-Moody superalgebras. In particular, we determine the decomposition of free Lie superalgebras into a direct sum of irreducible $GL(n) \times GL(k)$-modules, and the supertraces of the Monstrous Lie superalgebras with group actions. Finally, we prove that the generalized characters of Verma modules and the irreducible highest weight modules over a generalized Kac-Moody superalgebra ${\frak g}$ corresponding to the Dynkin diagram automorphism $\sigma$ are the same as the usual characters of Verma modules and irreducible highest weight modules over the orbit Lie superalgebra $\breve{\frak g}={\frak g}(\sigma)$ determined by $\sigma$.
Kang Seok-Jin
Kwon Jae-Hoon
No associations
LandOfFree
Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Graded Lie Superalgebras, Supertrace Formula, and Orbit Lie Superalgebras will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-333947