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Mixed quantum skew Howe duality and link invariants of type \(A\). (English) Zbl 1469.57013
The paper under review defines a ribbon category \(\mathsf{Sp}(\beta)\) which depends on a parameter \(\beta\). For \(\beta=m-n\), this category describes the monoidal category of representations of \(U_q(\mathfrak{gl}_{m\vert n})\) generated by exterior powers of the vector representation and their duals.
The main result of the paper is an extension of skew Howe duality to a more general picture where both exterior powers of the vector representation and of its dual appear at the same time, and where one can make sense of the limit \(m\to\infty\) by taking a generic value of the parameter \(\beta\). This allows one to realize \(\mathsf{Sp}(\beta)\) as a direct limit of quotients of a dual idempotented quantum group. Consequently, the category \(\mathsf{Sp}(\beta)\) gives a unified natural setting for defining the colored \(\mathfrak{gl}_{m\vert n}\) link invariant (for \(\beta=m-n\)) and the colored HOMFLY-PT polynomial (for \(\beta\) generic).

MSC:
57K16 Finite-type and quantum invariants, topological quantum field theories (TQFT)
17B35 Universal enveloping (super)algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20G43 Schur and \(q\)-Schur algebras
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
Citations:
Zbl 1423.57018
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