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<p>Webs are combinatorial objects used to encode homomorphisms between representations of Lie (super)algebras and related objects. This paper extends the theory of webs to the quantum group of type <i>Q</i>; i.e., we define a monoidal supercategory of quantum type <i>Q</i> webs and show it admits a full, essentially surjective functor onto the monoidal supercategory of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(U_{q}(\frak{q}_{n})\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>U</mi> <mrow> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow> <mi mathvariant="fraktur">q</mi> </mrow> <mrow> <mi mathvariant="fraktur">n</mi> </mrow> </msub> <mo mathvariant="fraktur" stretchy="false">)</mo> </math></EquationSource> </InlineEquation>-modules generated by the quantum symmetric powers of the natural representation and their duals. We also show that a certain subcategory of the web category is a ribbon category and discuss applications to the representation theory of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(U_{q}(\frak{q}_{n})\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msub> <mi>U</mi> <mrow> <mi>q</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mrow> <mi mathvariant="fraktur">q</mi> </mrow> <mrow> <mi mathvariant="fraktur">n</mi> </mrow> </msub> <mo mathvariant="fraktur" stretchy="false">)</mo> </math></EquationSource> </InlineEquation> and to invariants of oriented, framed links.</p>

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Quantum webs of type Q

  • 期刊论文

  • 发表日期：

    2025年11月25日

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  • Gordon C. Brown,
  • Nicholas J. Davidson,
  • Jonathan R. Kujawa

摘要

Webs are combinatorial objects used to encode homomorphisms between representations of Lie (super)algebras and related objects. This paper extends the theory of webs to the quantum group of type Q; i.e., we define a monoidal supercategory of quantum type Q webs and show it admits a full, essentially surjective functor onto the monoidal supercategory of \(U_{q}(\frak{q}_{n})\) U q ( q n ) -modules generated by the quantum symmetric powers of the natural representation and their duals. We also show that a certain subcategory of the web category is a ribbon category and discuss applications to the representation theory of \(U_{q}(\frak{q}_{n})\) U q ( q n ) and to invariants of oriented, framed links.