The Brauer category and invariant theory

Abstract

A category of Brauer diagrams, analogous to Turaev's tangle category, is introduced, a presentation of the category is given, and full tensor functors are constructed from this category to the category of tensor representations of the orthogonal group O$(V)$ or the symplectic group Sp$(V)$ over any field of characteristic zero. The first and second fundamental theorems of invariant theory for these classical groups are generalised to the category theoretic setting. The major outcome is that we obtain presentations for the endomorphism algebras of the module $V^{\otimes r}$, which are new in the classical symplectic case and in the orthogonal and symplectic quantum case, while in the orthogonal classical case, the proof we give here is more natural than in our earlier work. These presentations are obtained by appending to the standard presentation of the Brauer algebra of degree $r$ one additional relation. This relation stipulates the vanishing of a single element of the Brauer algebra which is quasi-idempotent, and which we describe explicitly both in terms of diagrams and algebraically. In the symplectic case, if dim $V=2n$, the element is precisely the central idempotent in the Brauer subalgebra of degree $n+1$, which corresponds to its trivial representation. Since this is the Brauer algebra of highest degree which is semisimple, our generator is an exact analogue for the Brauer algebra of the Jones idempotent of the Temperley-Lieb algebra. In the orthogonal case the additional relation is also a quasi-idempotent in the integral Brauer algebra. Both integral and quantum analogues of these results are given, the latter of which involve the BMW algebras.