We introduce a variant of the much-studied $Lie$ representation of the symmetric group $S_n$, which we denote by $Lie_n^{(2)}.$ Our variant gives rise to a decomposition of the regular representation as a sum of {exterior} powers of modules $Lie_n^{(2)}.$ This is in contrast to the theorems of Poincare-Birkhoff-Witt and Thrall which decompose the regular representation into a sum of symmetrised $Lie$ modules. We show that nearly every known property of $Lie_n$ has a counterpart for the module $Lie_n^{(2)},$ suggesting connections to the cohomology of configuration spaces via the character formulas of Sundaram and Welker, to the Eulerian idempotents of Gerstenhaber and Schack, and to the Hodge decomposition of the complex of injective words arising from Hochschild homology, due to Hanlon and Hersh.
{"title":"On a curious variant of the $S_n$-module Lie$_n$","authors":"S. Sundaram","doi":"10.5802/alco.127","DOIUrl":"https://doi.org/10.5802/alco.127","url":null,"abstract":"We introduce a variant of the much-studied $Lie$ representation of the symmetric group $S_n$, which we denote by $Lie_n^{(2)}.$ Our variant gives rise to a decomposition of the regular representation as a sum of {exterior} powers of modules $Lie_n^{(2)}.$ This is in contrast to the theorems of Poincare-Birkhoff-Witt and Thrall which decompose the regular representation into a sum of symmetrised $Lie$ modules. We show that nearly every known property of $Lie_n$ has a counterpart for the module $Lie_n^{(2)},$ suggesting connections to the cohomology of configuration spaces via the character formulas of Sundaram and Welker, to the Eulerian idempotents of Gerstenhaber and Schack, and to the Hodge decomposition of the complex of injective words arising from Hochschild homology, due to Hanlon and Hersh.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"69 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131965062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-05-05DOI: 10.1080/00927872.2020.1832105
Hau-wen Huang
Assume that $mathbb F$ is an algebraically closed field and $q$ is a nonzero scalar in $mathbb F$ that is not a root of unity. The universal Askey--Wilson algebra $triangle_q$ is a unital associative $mathbb F$-algebra generated by $A,B, C$ and the relations state that each of $$ A+frac{q BC-q^{-1} CB}{q^2-q^{-2}}, qquad B+frac{q CA-q^{-1} AC}{q^2-q^{-2}}, qquad C+frac{q AB-q^{-1} BA}{q^2-q^{-2}} $$ is central in $triangle_q$. The universal DAHA $mathfrak H_q$ of type $(C_1^vee,C_1)$ is a unital associative $mathbb F$-algebra generated by ${t_i^{pm 1}}_{i=0}^3$ and the relations state that begin{gather*} t_it_i^{-1}=t_i^{-1} t_i=1 quad hbox{for all $i=0,1,2,3$}; hbox{$t_i+t_i^{-1}$ is central} quad hbox{for all $i=0,1,2,3$}; t_0t_1t_2t_3=q^{-1}. end{gather*} It was given an $mathbb F$-algebra homomorphism $triangle_qto mathfrak H_q$ that sends begin{eqnarray*} A &mapsto & t_1 t_0+(t_1 t_0)^{-1}, B &mapsto & t_3 t_0+(t_3 t_0)^{-1}, C &mapsto & t_2 t_0+(t_2 t_0)^{-1}. end{eqnarray*} Therefore any $mathfrak H_q$-module can be considered as a $triangle_q$-module. Let $V$ denote a finite-dimensional irreducible $mathfrak H_q$-module. In this paper we show that $A,B,C$ are diagonalizable on $V$ if and only if $A,B,C$ act as Leonard triples on all composition factors of the $triangle_q$-module $V$.
{"title":"The universal DAHA of type $(C_1^vee,C_1)$ and Leonard triples","authors":"Hau-wen Huang","doi":"10.1080/00927872.2020.1832105","DOIUrl":"https://doi.org/10.1080/00927872.2020.1832105","url":null,"abstract":"Assume that $mathbb F$ is an algebraically closed field and $q$ is a nonzero scalar in $mathbb F$ that is not a root of unity. The universal Askey--Wilson algebra $triangle_q$ is a unital associative $mathbb F$-algebra generated by $A,B, C$ and the relations state that each of $$ A+frac{q BC-q^{-1} CB}{q^2-q^{-2}}, qquad B+frac{q CA-q^{-1} AC}{q^2-q^{-2}}, qquad C+frac{q AB-q^{-1} BA}{q^2-q^{-2}} $$ is central in $triangle_q$. The universal DAHA $mathfrak H_q$ of type $(C_1^vee,C_1)$ is a unital associative $mathbb F$-algebra generated by ${t_i^{pm 1}}_{i=0}^3$ and the relations state that begin{gather*} t_it_i^{-1}=t_i^{-1} t_i=1 quad hbox{for all $i=0,1,2,3$}; hbox{$t_i+t_i^{-1}$ is central} quad hbox{for all $i=0,1,2,3$}; t_0t_1t_2t_3=q^{-1}. end{gather*} It was given an $mathbb F$-algebra homomorphism $triangle_qto mathfrak H_q$ that sends begin{eqnarray*} A &mapsto & t_1 t_0+(t_1 t_0)^{-1}, B &mapsto & t_3 t_0+(t_3 t_0)^{-1}, C &mapsto & t_2 t_0+(t_2 t_0)^{-1}. end{eqnarray*} Therefore any $mathfrak H_q$-module can be considered as a $triangle_q$-module. Let $V$ denote a finite-dimensional irreducible $mathfrak H_q$-module. In this paper we show that $A,B,C$ are diagonalizable on $V$ if and only if $A,B,C$ act as Leonard triples on all composition factors of the $triangle_q$-module $V$.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"56 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114984058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study universal enveloping Hopf algebras of Lie algebras in the category of weakly complete vector spaces over the real and complex field.
研究了实复域上弱完备向量空间范畴中李代数的普适包络Hopf代数。
{"title":"On Weakly Complete Universal Enveloping Algebras of pro-Lie algebras","authors":"K. Hofmann, L. Kramer","doi":"10.14760/OWP-2020-10","DOIUrl":"https://doi.org/10.14760/OWP-2020-10","url":null,"abstract":"We study universal enveloping Hopf algebras of Lie algebras in the category of weakly complete vector spaces over the real and complex field.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127876854","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-25DOI: 10.1142/s0219498821501668
C. Ringel
Let A be a finite-dimensional algebra. If A is self-injective, then all modules are reflexive. Marczinzik recently has asked whether A has to be self-injective in case all the simple modules are reflexive. Here, we exhibit an 8-dimensional algebra which is not self-injective, but such that all simple modules are reflexive (actually, for this example, the simple modules are the only non-projective indecomposable modules which are reflexive). In addition, we present some properties of simple reflexive modules in general. Marczinzik had motivated his question by providing large classes of algebras such that any algebra in the class which is not self-injective has simple modules which are not reflexive. However, as it turns out, most of these classes have the property that any algebra in the class which is not self-injective has simple modules which are not even torsionless.
{"title":"Simple reflexive modules over finite-dimensional algebras","authors":"C. Ringel","doi":"10.1142/s0219498821501668","DOIUrl":"https://doi.org/10.1142/s0219498821501668","url":null,"abstract":"Let A be a finite-dimensional algebra. If A is self-injective, then all modules are reflexive. Marczinzik recently has asked whether A has to be self-injective in case all the simple modules are reflexive. Here, we exhibit an 8-dimensional algebra which is not self-injective, but such that all simple modules are reflexive (actually, for this example, the simple modules are the only non-projective indecomposable modules which are reflexive). In addition, we present some properties of simple reflexive modules in general. Marczinzik had motivated his question by providing large classes of algebras such that any algebra in the class which is not self-injective has simple modules which are not reflexive. However, as it turns out, most of these classes have the property that any algebra in the class which is not self-injective has simple modules which are not even torsionless.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116727733","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathfrak{g}_0$ be a simple Lie algebra of type ADE and let $U'_q(mathfrak{g})$ be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group $B(mathfrak{g}_0)$ on the quantum Grothendieck ring $K_t(mathfrak{g})$ of Hernandez-Leclerc's category $C_{mathfrak{g}}^0$. Focused on the case of type $A_{N-1}$, we construct a family of monoidal autofunctors ${mathscr{S}_i}_{iin mathbb{Z}}$ on a localization $T_N$ of the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{infty}$. Under an isomorphism between the Grothendieck ring $K(T_N)$ of $T_N$ and the quantum Grothendieck ring $K_t({A^{(1)}_{N-1}})$, the functors ${mathscr{S}_i}_{1le ile N-1}$ recover the action of the braid group $B(A_{N-1})$. We investigate further properties of these functors.
设$mathfrak{g}_0$为ADE型的简单李代数，设$U'_q(mathfrak{g})$为对应的非扭曲量子仿射代数。我们证明了编织群$B(mathfrak{g}_0)$在Hernandez-Leclerc范畴$C_{mathfrak{g}}^0$的量子Grothendieck环$K_t(mathfrak{g})$上存在一个作用。在类型为$A_{N-1}$的情况下，我们在类型为$A_{infty}$的quiver Hecke代数上有限维梯度模类的一个局部化$T_N$上构造了一类单形自函子${mathscr{S}_i}_{iin mathbb{Z}}$。在$T_N$的Grothendieck环$K(T_N)$与量子Grothendieck环$K_t({A^{(1)}_{N-1}})$之间的同构下，函子${mathscr{S}_i}_{1le ile N-1}$恢复了编织群$B(A_{N-1})$的作用。我们进一步研究了这些函子的性质。
{"title":"Braid group action on the module category of quantum\u0000 affine algebras","authors":"M. Kashiwara, Myungho Kim, Se-jin Oh, E. Park","doi":"10.3792/PJAA.97.003","DOIUrl":"https://doi.org/10.3792/PJAA.97.003","url":null,"abstract":"Let $mathfrak{g}_0$ be a simple Lie algebra of type ADE and let $U'_q(mathfrak{g})$ be the corresponding untwisted quantum affine algebra. We show that there exists an action of the braid group $B(mathfrak{g}_0)$ on the quantum Grothendieck ring $K_t(mathfrak{g})$ of Hernandez-Leclerc's category $C_{mathfrak{g}}^0$. Focused on the case of type $A_{N-1}$, we construct a family of monoidal autofunctors ${mathscr{S}_i}_{iin mathbb{Z}}$ on a localization $T_N$ of the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{infty}$. Under an isomorphism between the Grothendieck ring $K(T_N)$ of $T_N$ and the quantum Grothendieck ring $K_t({A^{(1)}_{N-1}})$, the functors ${mathscr{S}_i}_{1le ile N-1}$ recover the action of the braid group $B(A_{N-1})$. We investigate further properties of these functors.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128186810","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the first part of this paper, we discuss the classical W-algebra $mathcal{W}(mathfrak{g}, F)$ associated with a Lie superalgebra $mathfrak{g}$ and the nilpotent element $F$ in an $mathfrak{sl}_2$-triple. We find a generating set of $mathcal{W}(mathfrak{g}, F)$ and compute the Poisson brackets between them. In the second part, which is the main part of the paper, we discuss supersymmetric classical W-algebras. We introduce two different constructions of a supersymmetric classical W-algebra $mathcal{W}(mathfrak{g}, f)$ associated with a Lie superalgebra $mathfrak{g}$ and an odd nilpotent element $f$ in a subalgebra isomorphic to $mathfrak{osp}(1|2)$. The first construction is via the SUSY classical BRST complex and the second is via the SUSY Drinfeld-Sokolov Hamiltonian reduction. We show that these two methods give rise to isomorphic SUSY Poisson vertex algebras. As a supersymmetric analogue of the first part, we compute explicit generators and Poisson brackets between the generators.
{"title":"Structures of (supersymmetric) classical W-algebras","authors":"U. Suh","doi":"10.1063/5.0010006","DOIUrl":"https://doi.org/10.1063/5.0010006","url":null,"abstract":"In the first part of this paper, we discuss the classical W-algebra $mathcal{W}(mathfrak{g}, F)$ associated with a Lie superalgebra $mathfrak{g}$ and the nilpotent element $F$ in an $mathfrak{sl}_2$-triple. We find a generating set of $mathcal{W}(mathfrak{g}, F)$ and compute the Poisson brackets between them. In the second part, which is the main part of the paper, we discuss supersymmetric classical W-algebras. We introduce two different constructions of a supersymmetric classical W-algebra $mathcal{W}(mathfrak{g}, f)$ associated with a Lie superalgebra $mathfrak{g}$ and an odd nilpotent element $f$ in a subalgebra isomorphic to $mathfrak{osp}(1|2)$. The first construction is via the SUSY classical BRST complex and the second is via the SUSY Drinfeld-Sokolov Hamiltonian reduction. We show that these two methods give rise to isomorphic SUSY Poisson vertex algebras. As a supersymmetric analogue of the first part, we compute explicit generators and Poisson brackets between the generators.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"70 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126242607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra $mathfrak{g}$ is simple if and only if the associated variety of its unique simple quotient is equal to $mathfrak{g}^*$. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.
{"title":"Simplicity of vacuum modules and associated varieties","authors":"T. Arakawa, Cuipo Jiang, Anne Moreau","doi":"10.5802/JEP.144","DOIUrl":"https://doi.org/10.5802/JEP.144","url":null,"abstract":"In this note, we prove that the universal affine vertex algebra associated with a simple Lie algebra $mathfrak{g}$ is simple if and only if the associated variety of its unique simple quotient is equal to $mathfrak{g}^*$. We also derive an analogous result for the quantized Drinfeld-Sokolov reduction applied to the universal affine vertex algebra.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131389677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ferdinand Georg Frobenius is generally considered the creator of character theory of finite groups. This achievement came from the study of the group determinant, which is the determinant of a matrix coming from the regular representation. In this paper, we generalize several of Frobenius' results about the group determinant and use them find a new formulation of supercharacter theory in terms of factorizations of the group determinant.
{"title":"Supercharacter theory via the group determinant","authors":"Shawn T. Burkett","doi":"10.1216/rmj.2021.51.447","DOIUrl":"https://doi.org/10.1216/rmj.2021.51.447","url":null,"abstract":"Ferdinand Georg Frobenius is generally considered the creator of character theory of finite groups. This achievement came from the study of the group determinant, which is the determinant of a matrix coming from the regular representation. In this paper, we generalize several of Frobenius' results about the group determinant and use them find a new formulation of supercharacter theory in terms of factorizations of the group determinant.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121357959","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present an overview of the close analogies between the character rings of finite groups and the fusion rings of rational conformal models, which follow from general principles related to orbifold deconstruction.
我们从轨道解构的一般原理出发，概述了有限群的特征环与有理共形模型的融合环之间的密切相似。
{"title":"Character rings and fusion algebras","authors":"P. Bantay","doi":"10.1090/CONM/768/15463","DOIUrl":"https://doi.org/10.1090/CONM/768/15463","url":null,"abstract":"We present an overview of the close analogies between the character rings of finite groups and the fusion rings of rational conformal models, which follow from general principles related to orbifold deconstruction.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133424344","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the dual ${rm G}^ast$ of a standard semisimple Poisson-Lie group ${rm G}$ from a perspective of cluster theory. We show that the coordinate ring $mathcal{O}({rm G}^ast)$ can be naturally embedded into a cluster Poisson algebra with a Weyl group action. We prove that $mathcal{O}({rm G}^ast)$ admits a natural basis which has positive integer structure coefficients and satisfies an invariance property with respect to a braid group action. We continue the study of the moduli space $mathscr{P}_{{rm G},mathbb{S}}$ of ${rm G}$-local systems introduced in cite{GS3}, and prove that the coordinate ring of $mathscr{P}_{{rm G}, mathbb{S}}$ coincides with its underlying cluster Poisson algebra.
{"title":"Duals of Semisimple Poisson–Lie Groups and Cluster Theory of Moduli Spaces of G-local Systems","authors":"Li-Chien Shen","doi":"10.1093/IMRN/RNAB094","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB094","url":null,"abstract":"We study the dual ${rm G}^ast$ of a standard semisimple Poisson-Lie group ${rm G}$ from a perspective of cluster theory. We show that the coordinate ring $mathcal{O}({rm G}^ast)$ can be naturally embedded into a cluster Poisson algebra with a Weyl group action. We prove that $mathcal{O}({rm G}^ast)$ admits a natural basis which has positive integer structure coefficients and satisfies an invariance property with respect to a braid group action. We continue the study of the moduli space $mathscr{P}_{{rm G},mathbb{S}}$ of ${rm G}$-local systems introduced in cite{GS3}, and prove that the coordinate ring of $mathscr{P}_{{rm G}, mathbb{S}}$ coincides with its underlying cluster Poisson algebra.","PeriodicalId":275006,"journal":{"name":"arXiv: Representation Theory","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133436638","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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