First fundamental theorems of invariant theory for quantum supergroups
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First fundamental theorems of invariant theory for quantum supergroups Gustav I. Lehrer1 · Hechun Zhang2 · Ruibin Zhang1 Dedicated to the memory of Stefan Papadima Received: 21 January 2019 / Revised: 18 June 2019 / Accepted: 25 June 2019 © Springer Nature Switzerland AG 2019
Abstract Let Uq (g) be the quantum supergroup of glm|n or the modified quantum supergroup of ospm|2n over the field of rational functions in q, and let Vq be the natural module for Uq (g). There exists a unique tensor functor associated with Vq , from the category of ribbon graphs to the category of finite dimensional representations of Uq (g), which preserves ribbon category structures. We show that this functor is full in the cases g = glm|n or osp2+1|2n . For g = osp2|2n , we show that the space HomUq (g) (Vq⊗r, Vq⊗s ) is spanned by images of ribbon graphs if r + s < 2(2n + 1). The proofs involve an equivalence of module categories for two versions of the quantisation of U(g). Keywords Quantum supergroup · Tensor invariant · Ribbon graph category Mathematics Subject Classification 17B37 · 15A72 · 18D10
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Quantum supergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
This work was supported by the Australian Research Council and the National Science Foundation of China. Part of this work was carried out during stays of H. Zhang and R. Zhang at the School of Mathematical Sciences, the University of Science and Technology of China.
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Gustav I. Lehrer [email protected] Hechun Zhang [email protected] Ruibin Zhang [email protected]
1
School of Mathematics and Statistics, The University of Sydney, Sydney, NSW 2006, Australia
2
Department of Mathematical Sciences, Tsinghua University, Beijing, People’s Republic of China
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G.I. Lehrer et al. 3 Invariant theory of the quantum general linear supergroup . . . . . . . . . . . . . . . . . . . . . . . 4 Invariant theory of the quantum orthosymplectic supergroup . . . . . . . . . . . . . . . . . . . . . . A Appendix: Categories of ribbon graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B Appendix: Braided quasi Hopf superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction Quantum supergroups [5,50,70] are a class of quasi-triangular [8] Hopf superalgebras introduced in the early 90s, which have since been studied quite extensively; see e.g., [26,37,52,56,58,60,71] for results on their finite dimensional irreducible representations. Quantum supergroups have been applied to obtain interesting results in several areas, most notably in the study of Yang–Baxter type integrable models [4,5,67], construction of topological invariants of knots and 3-manifolds [34,57,61,62] and development of quantum supergeometry [35,63,65]. It is the quasi-triangu
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