Invariants of the orthosymplectic Lie superalgebra and super Pfaffians
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 \(N\).
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