Twisted Demazure modules, fusion product decomposition and twisted 𝑄-systems
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Abstract
In this paper, we introduce a family of indecomposable finite-dimensional graded modules for the twisted current algebras. These modules are indexed by an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartAbsoluteValue upper R Superscript plus Baseline EndAbsoluteValue"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">|R^+|</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -tuple of partitions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi equals left-parenthesis xi Superscript alpha Baseline right-parenthesis Subscript alpha element-of upper R Sub Superscript plus"> <mml:semantics> <mml:mrow> <mml:mi> ξ </mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi> ξ </mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> α </mml:mi> </mml:mrow> </mml:msup> <mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> α </mml:mi> <mml:mo> ∈ </mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mo>+</mml:mo> </mml:msup> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\xi =(\xi ^{\alpha })_{\alpha \in R^+}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying a natural compatibility condition. We give three equivalent presentations of these modules and show that for a particular choice of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="xi"> <mml:semantics> <mml:mi> ξ </mml:mi> <mml:annotation encoding="application/x-tex">\xi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> these modules become isomorphic to Demazure modules in various levels for the twisted affine algebras. As a consequence we see that the defining relations of twisted Demazure modules can be greatly simplified. Furthermore, we investigate the notion of fusion products for twisted modules, first defined by Feigin and Loktev in 1999 for untwisted modules, and use the simplified presentation to prove a fusion product decomposition of twisted Demazure modules. As a consequence we prove that twisted Demazure modules can be obtained by taking the associated graded modules of (untwisted) Demazure modules for simply-laced affine algebras. Furthermore we give a semi-infinite fusion product construction for the irreducible representations of twisted affine algebras. Finally, we prove that the twisted <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -sytem defined by Hatayama et al. in 2001 extends to a non-canonical short exact sequence of fusion products of twisted Demazure modules.
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| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
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| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.002 |
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| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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