Let OSp(V ) be the orthosymplectic supergroup on an orthosymplectic vector superspace V of superdimension (m|2n). Lehrer and Zhang showed that there is a surjective algebra homomorphism F r r : B r (m − 2n) → End OSp(V ) (V ⊗r ), where B r (m − 2n) is the Brauer algebra of degree r with parameter m − 2n. The second fundamental theorem of invariant theory in this setting seeks to describe the kernel KerF r r of F r r as a 2-sided ideal of B r (m − 2n). In this paper, we show that KerF r r = 0 if and only if r ≥ r c := (m + 1)(n + 1), and give a basis and a dimension formula for KerF r r . We show that KerF r r as a 2-sided ideal of B r (m − 2n) is generated by KerF rc rc for any r ≥ r c , and we provide an explicit set of generators for KerF rc rc . These generators coincide in the classical case with those obtained in recent papers of Lehrer and Zhang on the second fundamental theorem of invariant theory for the orthogonal and symplectic groups. As an application we obtain the necessary and sufficient conditions for the endomorphism algebra End osp(V) (V ⊗r ) over the orthosymplectic Lie superalgebra osp(V ) to be isomorphic to B r (m − 2n) r 2010 Mathematics Subject Classification. 16W22,15A72,17B20.We recall the FFT and SFT of invariant theory for the orthosymplectic supergroup given in [18,19]. We work over the complex number field C throughout this paper.2.1. Tensor representations of the orthosymplectic supergroup. Let V = V0 ⊕ V1 be a complex vector superspace with superdimension sdim(V ) = (m|ℓ), which means that dim(V0) = m and dim(V1) = ℓ. The parity [v] of any homogeneous element v ∈ Vī is defined by [v] :=ī (i = 0, 1). Assume that V admits a non-degenerate even bilinear formwhich is supersymmetric in the sense that (u, v) = (−1) [u][v] (v, u) for u, v ∈ V . This implies that the form is symmetric on V0×V0 and skew-symmetric on V1×V1, and satisfies (V0, V1) = 0 = (V1, V0). Therefore ℓ must be even.We refer to Harish-Chandra super pair [2,6] as the supergroup. Let osp(V ) be the orthosymplectic Lie superalgebra [15] preserving the bilinear form (2.1). Let O(V0) and Sp(V1) be the orthogonal and symplectic groups, which are the isometry algebraic groups preserving the restrictions of the form (2.1) to V0 and to V1 respectively. Then OSp(V ) 0 := O(V0)×Sp(V1) naturally acts on osp(V ) as automorphisms, and we have the Harish-Chandra super pair (OSp(V ) 0 , osp(V )). One may regard this as the orthosymplectic supergroup [6], and hereafter OSp(V ) denotes the Harish-Chandra super pair (OSp(V ) 0 , osp(V )).An OSp(V )-module M is defined to be a module for the Harish-Chandra super pair, namely, M is a vector superspace that is a module for both osp(V ) and OSp(V ) 0 (as an algebraic group) such that the two actions are compatible with the action of OSp(V ) 0 on osp(V ). The subspace of invariants is given byIf N is another OSp(V )-module, then Hom C (M, N) is naturally an OSp(V )-module with the action defined for any g ∈ OSp(V ) 0 and X ∈ osp(V ) on φ ∈ Hom C (M, N) by (g.φ)(v) = gφ(g −1 v), (X.φ)(v) = Xφ(v) ...

We develop the non-commutative polynomial version of the invariant theory for the quantum general linear supergroup U q pgl m|n q. A non-commutative U q pgl m|n q-module superalgebra P k|l r|s is constructed, which is the quantum analogue of the supersymmetric algebra over C k|l b C m|n ' C r|s b pC m|n q˚. We analyse the structure of the subalgebra of U q pgl m|n q-invariants in P k|l r|s by using the quantum super analogue of Howe duality.The subalgebra of U q pgl m|n q-invariants in P k|l r|s is shown to be finitely generated. We determine its generators and establish a surjective superalgebra homomorphism from a braided supersymmetric algebra onto it. This establishes the first fundamental theorem of invariant theory for U q pgl m|n q.We show that the above mentioned superalgebra homomorphism is an isomorphism if and only if m ě mintk, ru and n ě mintl, su, and obtain a PBW basis for the subalgebra of invariants in this case. When the homomorphism is not injective, we give a representation theoretical description of the generating elements of the kernel. This way we obtain the relations obeyed by the generators of the subalgebra of invariants, producing the second fundamental theorem of invariant theory for U q pgl m|n q.We consider the special case with n " 0 in greater detail, obtaining a complete treatment of the non-commutative polynomial version of the invariant theory for U q pgl m q. In particular, the explicit SFT proved here is believed to be new. We also recover the FFT and SFT of the invariant theory for the general linear superalgebra from the classical limit (i.e., q Ñ 1) of our results.

The quantum supergroup U q (osp(1|2n)) admits a finite dimensional spinor representation, which does not have a classical limit. We construct a realisation of this representation on the Fock space of q-fermions. We also generalise the construction to the infinite dimensional spinor representations of U q (osp(2m + 1|2n)) for m, n ≥ 1 by using both quantum bosons and fermions. This leads to a new realisation different from those obtained before by other researchers.Date: Nov 29, 2016.