Abstract. We establish a noncommutative analogue of the first fundamental theorem of classical invariant theory. For each quantum group associated with a classical Lie algebra, we construct a noncommutative associative algebra whose underlying vector space forms a module for the quantum group and whose algebraic structure is preserved by the quantum group action. The subspace of invariants is shown to form a subalgebra, which is finitely generated. We determine generators of this subalgebra of invariants and determine their commutation relations. In each case considered, the noncommutative modules we construct are flat deformations of their classical commutative analogues. Thus by taking the limit as q → 1, our results imply the first fundamental theorem of classical invariant theory, and therefore generalise them to the noncommutative case.

In a previous work we established a super Schur-Weyl-Brauer duality between the orthosymplectic supergroup of superdimension (m|2n) and the Brauer algebra with parameter m − 2n. This led to a proof of the first fundamental theorem of invariant theory, using some elementary algebraic supergeometry, and based upon an idea of Atiyah. In this work we use the same circle of ideas to prove the second fundamental theorem for the orthosymplectic supergroup. The proof uses algebraic supergeometry to reduce the problem to the case of the general linear supergroup, which is understood. The main result has a succinct formulation in terms of Brauer diagrams. Our proof includes new proofs of the corresponding second fundamental theorems for the classical orthogonal and symplectic groups, as well as their quantum analogues. These new proofs are independent of the Capelli identities, which are replaced by algebraic geometric arguments.2010 Mathematics Subject Classification. 16W22,15A72,17B20. 1 2 G.I. LEHRER AND R.B. ZHANG 6.1. The othogonal case: G = O(m, C) 21 6.2. The symplectic case 22 References 23 → Λ(N) for the direct limit; this is the Grassmann algebra.Since Λ is evidently a Z/2Z-graded algebra (graded by partity of the degree), we may form V = V C ⊗ C Λ. This is a Z 2 -graded Λ-module, and we write V = V0 ⊕ V1. For a homogeneous element v ∈ V , we denote by [v] the parity of v.The Λ-module End Λ (V ) of Λ-endomorphisms of V is Z/2Z-graded, and we write E := End Λ (V )0.The group GL(V ) is the group of invertible elements of E, that isThis is very much in the spirit of the physics literaure [20,26].