Classification of maximal transitive prolongations of super-Poincar\'e algebras
| Autor: | Altomani, Andrea, Santi, Andrea |
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| Rok vydání: | 2012 |
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| Zdroj: | Adv. Math. 265 (2014), 60-96 |
| Druh dokumentu: | Working Paper |
| Popis: | Let $V$ be a complex vector space with a non-degenerate symmetric bilinear form and $\mathbb S$ an irreducible module over the Clifford algebra $Cl(V)$ determined by this form. A supertranslation algebra is a $\mathbb Z$-graded Lie superalgebra $\mathfrak m=\mathfrak{m}_{-2}\oplus\mathfrak{m}_{-1}$, where $\mathfrak{m}_{-2}=V$ and $\mathfrak{m}_{-1}=\mathbb S\oplus\cdots\oplus\mathbb{S}$ is the direct sum of an arbitrary number $N\geq 1$ of copies of $\mathbb S$, whose bracket $[\cdot,\cdot]|_{\mathfrak{m}_{-1}\otimes \mathfrak{m}_{-1}}:\mathfrak{m}_{-1}\otimes\mathfrak{m}_{-1}\rightarrow\mathfrak{m}_{-2}$ is symmetric, $\mathfrak{so}(V)$-equivariant and non-degenerate (that is the condition "$s\in\mathfrak{m}_{-1}, [s,\mathfrak{m}_{-1}]=0$" implies $s=0$). We consider the maximal transitive prolongations in the sense of Tanaka of supertranslation algebras. We prove that they are finite-dimensional for $\dim V\geq3$ and classify them in terms of super-Poincar\'e algebras and appropriate $\mathbb Z$-gradings of simple Lie superalgebras. Comment: 32 pages, v2: general presentation improved, corrected several typos. Proofs and results unchanged. Final version to appear in Adv. Math |
| Databáze: | arXiv |
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