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A M Vershik - One of the best experts on this subject based on the ideXlab platform.
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the serpentine representation of the infinite symmetric group and the basic representation of the Affine Lie Algebra widehat mathfrak sl _2
Letters in Mathematical Physics, 2015Co-Authors: N V Tsilevich, A M VershikAbstract:We introduce and study the so-called serpentine representations of the infinite symmetric group \({\mathfrak{S}_\mathbb{N}}\), which turn out to be closely related to the basic representation of the Affine Lie Algebra \({\widehat{\mathfrak{sl}_2}}\) and representations of the Virasoro Algebra.
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the serpentine representation of the infinite symmetric group and the basic representation of the Affine Lie Algebra widehat mathfrak sl _2
arXiv: Representation Theory, 2014Co-Authors: N V Tsilevich, A M VershikAbstract:We introduce and study the so-called serpentine representations of the infinite symmetric group $\sinf$, which turn out to be closely related to the basic representation of the Affine Lie Algebra $\widehat{\mathfrak{sl}_2}$ and representations of the Virasoro Algebra. This is a new version of the manuscript of the same authors "On a relation between the basic representation of the Affine Lie Algebra $\widehat\sl$ and a Schur--Weyl representation of the infinite symmetric group" (arXiv:1403.1558)
N V Tsilevich - One of the best experts on this subject based on the ideXlab platform.
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the serpentine representation of the infinite symmetric group and the basic representation of the Affine Lie Algebra widehat mathfrak sl _2
Letters in Mathematical Physics, 2015Co-Authors: N V Tsilevich, A M VershikAbstract:We introduce and study the so-called serpentine representations of the infinite symmetric group \({\mathfrak{S}_\mathbb{N}}\), which turn out to be closely related to the basic representation of the Affine Lie Algebra \({\widehat{\mathfrak{sl}_2}}\) and representations of the Virasoro Algebra.
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the serpentine representation of the infinite symmetric group and the basic representation of the Affine Lie Algebra widehat mathfrak sl _2
arXiv: Representation Theory, 2014Co-Authors: N V Tsilevich, A M VershikAbstract:We introduce and study the so-called serpentine representations of the infinite symmetric group $\sinf$, which turn out to be closely related to the basic representation of the Affine Lie Algebra $\widehat{\mathfrak{sl}_2}$ and representations of the Virasoro Algebra. This is a new version of the manuscript of the same authors "On a relation between the basic representation of the Affine Lie Algebra $\widehat\sl$ and a Schur--Weyl representation of the infinite symmetric group" (arXiv:1403.1558)
Daisuke Sagaki - One of the best experts on this subject based on the ideXlab platform.
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tensor product decomposition theorem for quantum lakshmibai seshadri paths and standard monomial theory for semi infinite lakshmibai seshadri paths
Journal of Combinatorial Theory Series A, 2020Co-Authors: Satoshi Naito, Fumihiko Nomoto, Daisuke SagakiAbstract:Abstract Let λ be a (level-zero) dominant integral weight for an untwisted Affine Lie Algebra, and let QLS ( λ ) denote the set of quantum Lakshmibai-Seshadri (QLS) paths of shape λ. For an element w of a finite Weyl group W, the specializations at t = 0 and t = ∞ of the nonsymmetric Macdonald polynomial E w λ ( q , t ) are explicitly described in terms of QLS paths of shape λ and the degree function defined on them. Also, for (level-zero) dominant integral weights λ, μ, we have an isomorphism Θ : QLS ( λ + μ ) → QLS ( λ ) ⊗ QLS ( μ ) of crystals. In this paper, we study the behavior of the degree function under the isomorphism Θ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.
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tensor product decomposition theorem for quantum lakshmibai seshadri paths and standard monomial theory for semi infinite lakshmibai seshadri paths
arXiv: Quantum Algebra, 2018Co-Authors: Satoshi Naito, Fumihiko Nomoto, Daisuke SagakiAbstract:Let $\lambda$ be a (level-zero) dominant integral weight for an untwisted Affine Lie Algebra, and let $\mathrm{QLS}(\lambda)$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $\lambda$. For an element $w$ of a finite Weyl group $W$, the specializations at $t = 0$ and $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w \lambda}(q, t)$ are explicitly described in terms of QLS paths of shape $\lambda$ and the degree function defined on them. Also, for (level-zero) dominant integral weights $\lambda$, $\mu$, we have an isomorphism $\Theta : \mathrm{QLS}(\lambda + \mu) \rightarrow \mathrm{QLS}(\lambda) \otimes \mathrm{QLS}(\mu)$ of crystals. In this paper, we study the behavior of the degree function under the isomorphism $\Theta$ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.
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path model for a level zero extremal weight module over a quantum Affine Algebra ii
Advances in Mathematics, 2006Co-Authors: Satoshi Naito, Daisuke SagakiAbstract:Abstract Let ϖ i be a level-zero fundamental weight for an Affine Lie Algebra g over Q , and let B ( ϖ i ) be the crystal of all Lakshmibai–Seshadri paths of shape ϖ i . First, we prove that the crystal graph of B ( ϖ i ) is connected. By combining this fact with the main result of our previous work, we see that B ( ϖ i ) is, as a crystal, isomorphic to the crystal base B ( ϖ i ) of the extremal weight module V ( ϖ i ) over a quantum Affine Algebra U q ( g ) over Q (q) of extremal weight ϖ i . Next, we obtain an explicit description of the decomposition of the crystal B ( m ϖ i ) of all Lakshmibai–Seshadri paths of shape m ϖ i into connected components. Furthermore, we prove that B ( m ϖ i ) is, as a crystal, isomorphic to the crystal base B ( m ϖ i ) of the extremal weight module V ( m ϖ i ) over U q ( g ) of extremal weight m ϖ i .
Robert Mcrae - One of the best experts on this subject based on the ideXlab platform.
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non negative integral level Affine Lie Algebra tensor categories and their associativity isomorphisms
Communications in Mathematical Physics, 2016Co-Authors: Robert McraeAbstract:For a finite-dimensional simple Lie Algebra \({\mathfrak{g}}\), we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the Affine Lie Algebra \({{\widehat{\mathfrak{g}}}}\) at a fixed level \({\ell\in\mathbb{N}}\) with a certain tensor category of finite-dimensional \({\mathfrak{g}}\)-modules. More precisely, the category of level l standard \({{\widehat{\mathfrak{g}}}}\)-modules is the module category for the simple vertex operator Algebra \({L_{\widehat{\mathfrak{g}}}(\ell, 0)}\), and as is well known, this category is equivalent as an abelian category to \({\mathbf{D}(\mathfrak{g},\ell)}\), the category of finite-dimensional modules for the Zhu’s Algebra \({A{(L_{\widehat{\mathfrak{g}}}(\ell, 0))}}\), which is a quotient of \({U(\mathfrak{g})}\). Our main result is a direct construction using Knizhnik–Zamolodchikov equations of the associativity isomorphisms in \({\mathbf{D}(\mathfrak{g},\ell)}\) induced from the associativity isomorphisms constructed by Huang and Lepowsky in \({{L_{\widehat{\mathfrak{g}}}(\ell, 0) - \mathbf{mod}}}\). This construction shows that \({\mathbf{D}(\mathfrak{g},\ell)}\) is closely related to the Drinfeld category of \({U(\mathfrak{g})}\)[[h]]-modules used by Kazhdan and Lusztig to identify categories of \({{\widehat{\mathfrak{g}}}}\)-modules at irrational and most negative rational levels with categories of quantum group modules.
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Intertwining operators among modules for Affine Lie Algebra and lattice vertex operator Algebras which respect integral forms
Journal of Pure and Applied Algebra, 2015Co-Authors: Robert McraeAbstract:We define an integral intertwining operator among modules for a vertex operator Algebra to be an intertwining operator which respects integral forms in the modules, and we show that an intertwining operator is integral if it is integral when restricted to generators of the integral forms in the modules. We apply this result to classify integral intertwining operators which respect certain natural integral forms in modules for Affine Lie Algebra and lattice vertex operator Algebras.
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non negative integral level Affine Lie Algebra tensor categories and their associativity isomorphisms
arXiv: Quantum Algebra, 2015Co-Authors: Robert McraeAbstract:For a finite-dimensional simple Lie Algebra $\mathfrak{g}$, we use the vertex tensor category theory of Huang and Lepowsky to identify the category of standard modules for the Affine Lie Algebra $\hat{\mathfrak{g}}$ at a fixed level $\ell\in\mathbb{N}$ with a certain tensor category of finite-dimensional $\mathfrak{g}$-modules. More precisely, the category of level $\ell$ standard $\hat{\mathfrak{g}}$-modules is the module category for the simple vertex operator Algebra $L_{\hat{\mathfrak{g}}}(\ell,0)$, and as is well known, this category is equivalent as an abelian category to $\mathbf{D}(\mathfrak{g},\ell)$, the category of finite-dimensional modules for the Zhu's Algebra $A(L_{\hat{\mathfrak{g}}}(\ell,0))$, which is a quotient of $U(\mathfrak{g})$. Our main result is a direct construction using Knizhnik-Zamolodchikov equations of the associativity isomorphisms in $\mathbf{D}(\mathfrak{g},\ell)$ induced from the associativity isomorphisms constructed by Huang and Lepowsky in $L_{hat{\mathfrak{g}}}(\ell,0)-\mathbf{mod}$. This construction shows that $\mathbf{D}(\mathfrak{g},\ell)$ is closely related to the Drinfeld category of $U(\mathfrak{g})[[\hbar]]$-modules used by Kazhdan and Lusztig to identify categories of $\hat{\mathfrak{g}}$-modules at irrational and most negative rational levels with categories of quantum group modules.
Satoshi Naito - One of the best experts on this subject based on the ideXlab platform.
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tensor product decomposition theorem for quantum lakshmibai seshadri paths and standard monomial theory for semi infinite lakshmibai seshadri paths
Journal of Combinatorial Theory Series A, 2020Co-Authors: Satoshi Naito, Fumihiko Nomoto, Daisuke SagakiAbstract:Abstract Let λ be a (level-zero) dominant integral weight for an untwisted Affine Lie Algebra, and let QLS ( λ ) denote the set of quantum Lakshmibai-Seshadri (QLS) paths of shape λ. For an element w of a finite Weyl group W, the specializations at t = 0 and t = ∞ of the nonsymmetric Macdonald polynomial E w λ ( q , t ) are explicitly described in terms of QLS paths of shape λ and the degree function defined on them. Also, for (level-zero) dominant integral weights λ, μ, we have an isomorphism Θ : QLS ( λ + μ ) → QLS ( λ ) ⊗ QLS ( μ ) of crystals. In this paper, we study the behavior of the degree function under the isomorphism Θ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.
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tensor product decomposition theorem for quantum lakshmibai seshadri paths and standard monomial theory for semi infinite lakshmibai seshadri paths
arXiv: Quantum Algebra, 2018Co-Authors: Satoshi Naito, Fumihiko Nomoto, Daisuke SagakiAbstract:Let $\lambda$ be a (level-zero) dominant integral weight for an untwisted Affine Lie Algebra, and let $\mathrm{QLS}(\lambda)$ denote the quantum Lakshmibai-Seshadri (QLS) paths of shape $\lambda$. For an element $w$ of a finite Weyl group $W$, the specializations at $t = 0$ and $t = \infty$ of the nonsymmetric Macdonald polynomial $E_{w \lambda}(q, t)$ are explicitly described in terms of QLS paths of shape $\lambda$ and the degree function defined on them. Also, for (level-zero) dominant integral weights $\lambda$, $\mu$, we have an isomorphism $\Theta : \mathrm{QLS}(\lambda + \mu) \rightarrow \mathrm{QLS}(\lambda) \otimes \mathrm{QLS}(\mu)$ of crystals. In this paper, we study the behavior of the degree function under the isomorphism $\Theta$ of crystals through the relationship between semi-infinite Lakshmibai-Seshadri (LS) paths and QLS paths. As an application, we give a crystal-theoretic proof of a recursion formula for the graded characters of generalized Weyl modules.
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path model for a level zero extremal weight module over a quantum Affine Algebra ii
Advances in Mathematics, 2006Co-Authors: Satoshi Naito, Daisuke SagakiAbstract:Abstract Let ϖ i be a level-zero fundamental weight for an Affine Lie Algebra g over Q , and let B ( ϖ i ) be the crystal of all Lakshmibai–Seshadri paths of shape ϖ i . First, we prove that the crystal graph of B ( ϖ i ) is connected. By combining this fact with the main result of our previous work, we see that B ( ϖ i ) is, as a crystal, isomorphic to the crystal base B ( ϖ i ) of the extremal weight module V ( ϖ i ) over a quantum Affine Algebra U q ( g ) over Q (q) of extremal weight ϖ i . Next, we obtain an explicit description of the decomposition of the crystal B ( m ϖ i ) of all Lakshmibai–Seshadri paths of shape m ϖ i into connected components. Furthermore, we prove that B ( m ϖ i ) is, as a crystal, isomorphic to the crystal base B ( m ϖ i ) of the extremal weight module V ( m ϖ i ) over U q ( g ) of extremal weight m ϖ i .