University of Sydney Algebra Seminar

Ruibin Zhang (University of Sydney)

Friday 1 May, 12-1pm, Place: Carslaw 375

Invariants of the orthosymplectic Lie superalgebra and the super Pfaffian

The invariant theory of the orthosymplectic Lie superalgebra \(\mathfrak{osp}(V)\) is more intricate than that of the orthosymplectic supergroup \(OSp(V)\). For example, the endomorphism algebra \(End_{OSp(V)}(V^{\otimes r})\) over the orthosymplectic supergroup is a quotient of the Brauer algebra of degree \(r\) (with parameter equal to the superdimension of \(V\)) as proved recently, but the endomorphism algebra \(End_{\mathfrak{osp}(V)}(V^{\otimes r})\) over the orthosymplectic superalgebra contains elements which are not Brauer-like. We will describe explicitly some \(\mathfrak{osp}(V)\)-invariants which are referred to as super Pfaffians, and show that the \(OSp(V)\)-invariants and super Pfaffians together generate all the \(\mathfrak{osp}(V)\)-invariants. This is joint work with Gus Lehrer.