Quantum Supergroups III. Twistors

Abstract

We establish direct connections at several levels between quantum groups and supergroups associated to bar-consistent super Cartan datum by constructing an automorphism (called twistor) in the setting of covering quantum groups. The canonical bases of the halves of quantum groups and supergroups are shown to match under the twistor up to powers of a square root of -1. We further show that the modified quantum group and supergroup are isomorphic over the rational function field adjoint with a square root of -1, by constructing a twistor on the modified covering quantum group. An equivalence of categories of weight modules for quantum groups and supergroups follows.

v1 19 pages; v2 added acknowledgement, mild changes, to appear in CMP

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• Quiver Hecke superalgebras
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Cited by in corpus (18)

• An odd categorification of quantum sl(2)
• Trialities of orthosymplectic $\mathcal{W}$-algebras
• The differential graded odd nilHecke algebra
• Super Kac-Moody 2-categories
• Irreducible Characters of Kac-Moody Lie superalgebras
• Quantum supergroups II. Canonical basis
• Quantum Shuffles and Quantum Supergroups of Basic Type
• Length-two representations of quantum affine superalgebras and Baxter operators
• DG structures on odd categorified quantum sl(2)
• Quantum osp(1|2n) knot invariants are the same as quantum so(2n+1) knot invariants
• Two-parameter quantum algebras, canonical bases and categorifications
• Higher level $q$-oscillator representations for $U_q(C_n^{(1)}),U_q(C^{(2)}(n+1))$ and $U_q(B^{(1)}(0,n))$
• Quantum supergroups VI. Roots of $1$
• An identification of Lusztig's modified forms of quantum algebras and their analogues
• Drinfeld double of deformed quantum algebras
• Quantum supergroups IV. The modified form
• Character formulae in category $\mathcal{O}$ for exceptional Lie superalgebra $G(3)$
• A geometric setting for quantum osp(1|2)