Invariants of the orthosymplectic Lie superalgebra and super Pfaffians
type: Item
instance of: scholarly article
title: Invariants of the orthosymplectic Lie superalgebra and super Pfaffians
zbMATH Open document ID: 1381.15020
published in: Mathematische Zeitschrift
publication date: 2017-09-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1507.01329
review text: Given a complex orthosymplectic superspace \(V\), the main goal of the authors is to develop the invariant theory of the orthosymplectic Lie superalgebra \(\mathfrak{osp}(V )\), after having presented in [\textit{G. I. Lehrer} and \textit{R. B. Zhang}, Commun. Math. Phys. 349, No. 2, 661--702 (2017; Zbl 1360.22027); J. Eur. Math. Soc. (JEMS) 17, No. 9, 2311--2351 (2015; Zbl 1328.14079)] the invariant theory for the orthosymplectic supergroup \(\mathrm{OSp}(V )\). This theory starts with Theorem 2.5 which describes the \(\mathfrak{gl}_N\)-module structure of \(S(N)^{\mathrm{OSp}(V)}\), the subalgebra of invariants with respect to \(\mathrm{OSp}(V )\). Section 3 presents two equivalent constructions of a super Pfaffian. In Section 4, Theorem 4.5 is the first fundamental theorem of invariant theory for the orthosymplectic Lie superalgebra \(\mathfrak{osp}(V )\). It states that the super Pfaffians and the quadratic invariants of the orthosymplectic supergroup generate the algebra of all invariants of the orthosymplectic Lie superalgebra acting on the superalgebra \(S(N)\). Finally, the last section provides a complete understanding of the subspace \((V ^{\otimes N})^{\mathfrak{osp}(V )}\) of \(\mathfrak{osp}(V )\)-invariants in \(V ^{\otimes N}\) for any \(N\).
reviewed by: Gema Maria Diaz Toca
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 14A22
Mathematics Subject Classification ID: 14L24
Mathematics Subject Classification ID: 14M30
Mathematics Subject Classification ID: 17B60
Mathematics Subject Classification ID: 58A17
zbMATH DE Number: 6780317
zbMATH Keywords: super algebraic geometry
zbMATH Keywords: orthosymplectic group
zbMATH Keywords: super Pfaffian
zbMATH Keywords: polynomial invariant
zbMATH Keywords: tensor invariant
zbMATH Keywords: orthosymplectic Lie superalgebra
author: Gus I. Lehrer
author: R. B. Zhang
MaRDI profile type: Publication
OpenAlex ID: W2199992752
arXiv ID: 1507.01329
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DOI: 10.1007/S00209-016-1789-4
description: scientific article