25,015 results on '"Mathematics::Quantum Algebra"'
Search Results
2. Geometric structure of affine Deligne-Lusztig varieties for GL3
- Author
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Ryosuke Shimada
- Subjects
Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Algebra and Number Theory ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) - Abstract
In this paper we study the geometric structure of affine Deligne-Lusztig varieties for $GL_3$ and $b$ basic. We completely determine the irreducible components of the affine Deligne-Lusztig variety. In particular, we classify the cases where all of the irreducible components are classical Deligne-Lusztig varieties times finite-dimensional affine spaces. If this is the case, then the irreducible components are pairwise disjoint.
- Published
- 2023
3. Representations of orientifold Khovanov–Lauda–Rouquier algebras and the Enomoto–Kashiwara algebra
- Author
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Przezdziecki, Tomasz
- Subjects
20C08 ,Mathematics::Quantum Algebra ,General Mathematics ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory - Abstract
We consider an "orientifold" generalization of Khovanov-Lauda-Rouquier algebras, depending on a quiver with an involution and a framing. Their representation theory is related, via a Schur-Weyl duality type functor, to Kac-Moody quantum symmetric pairs, and, via a categorification theorem, to highest weight modules over an algebra introduced by Enomoto and Kashiwara. Our first main result is a new shuffle realization of these highest weight modules and a combinatorial construction of their PBW and canonical bases in terms of Lyndon words. Our second main result is a classification of irreducible representations of orientifold KLR algebras and a computation of their global dimension in the case when the framing is trivial., 34 pages, some corrections in the proofs and improvements in the exposition
- Published
- 2023
4. Hochschild homology of twisted crossed products and twisted graded Hecke algebras
- Author
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Solleveld, Maarten
- Subjects
16E40, 16S35, 20C08 ,Rings and Algebras (math.RA) ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Mathematics - Rings and Algebras ,Geometry and Topology ,Representation Theory (math.RT) ,Mathematics::Algebraic Topology ,Mathematics ,Mathematics - Representation Theory ,Analysis - Abstract
Let A be a \C-algebra with an action of a finite group G, let $\natural$ be a 2-cocycle on $G$ and consider the twisted crossed product $A \rtimes \C [G,\natural]$. We determine the Hochschild homology of $A \rtimes \C [G,\natural]$ for two classes of algebras A: - rings of regular functions on nonsingular affine varieties, - graded Hecke algebras. The results are achieved via algebraic families of (virtual) representations and include a description of the Hochschild homology as module over the centre of $A \rtimes \C [G,\natural]$. This paper prepares for a computation of the Hochschild homology of the Hecke algebra of a reductive p-adic group.
- Published
- 2023
5. Extensions of Yang–Baxter sets
- Author
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Bardakov, Valeriy G. and Talalaev, Dmitry V.
- Subjects
Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Statistical and Nonlinear Physics ,Group Theory (math.GR) ,Mathematics - Group Theory ,Mathematical Physics - Abstract
The paper extends the notion of braided set and its close relative - the Yang-Baxter set - to the category of vector spaces and explore structure aspects of such a notion as morphisms and extensions. In this way we describe a family of solutions for the Yang-Baxter equation on the product of B and C if given B and C correspond to two linear (set-theoretic) solutions of the Yang-Baxter equation. One of the key observation is the relation of this question with the virtual pure braid group., Comment: 29 pages, 3 figures
- Published
- 2023
6. (Co)homology of crossed products by weak Hopf algebras
- Author
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Jorge A. Guccione, Juan J. Guccione, and Christian Valqui
- Subjects
Algebra and Number Theory ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics - K-Theory and Homology ,FOS: Mathematics ,K-Theory and Homology (math.KT) ,16E40, 16T05 - Abstract
We obtain a mixed complex simpler than the canonical one the computes the type cyclic homologies of a crossed product with invertible cocycle $A\times_{\rho}^f H$, of a weak module algebra $A$ by a weak Hopf algebra $H$. This complex is provided with a filtration. The spectral sequence of this filtration generalizes the spectral sequence obtained in \cite{CGG}. When $f$ takes its values in a separable subalgebra of $A$ that satisfies suitable conditions, the above mentioned mixed complex is provided with another filtration, whose spectral sequence generalize the Feigin-Tsygan spectral sequence., Comment: 22 pages. arXiv admin note: text overlap with arXiv:1811.02927
- Published
- 2023
7. Clifford-symmetric polynomials
- Author
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Lenzen, Fabian
- Subjects
Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,11E88, 16W55, 15A66 ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory - Abstract
Based on the NilHecke algebra $\mathsf{NH}_n$, the odd NilHecke algebra developed by Ellis, Khovanov and Lauda and Kang, Kashiwara and Tsuchioka's quiver Hecke superalgebra, we develop the Clifford Hecke superalgebra $\mathsf{NH}\mathfrak{C}_n$ as another super-algebraic analogue of $\mathsf{NH}_n$. We show that there is a notion of symmetric polynomials fitting in this picture, and we prove that these are generated by an appropriate analogue of elementary symmetric polynomials, whose properties we shall discuss in this text.
- Published
- 2023
8. Yoneda Ext-algebras of Takeuchi smash products
- Author
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Wu, Quanshui and Zhu, Ruipeng
- Subjects
Algebra and Number Theory ,Mathematics::Commutative Algebra ,Rings and Algebras (math.RA) ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Mathematics::Rings and Algebras ,FOS: Mathematics ,Mathematics - Rings and Algebras ,16E30, 16E45, 16S40, 16E65 - Abstract
We prove that the Yoneda Ext-algebra of a Takeuchi smash product is the graded Takeuchi smash product of the Yoneda Ext-algebras of the two algebras or modules involved. As an application, we prove that graded Takeuchi smash products preserve Artin-Schelter regularity, and describe the Nakayama automorphism of the product., 16 pages, comments welcome
- Published
- 2023
9. The relative Deligne tensor product over pointed braided fusion categories
- Author
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Thibault Decoppet
- Subjects
Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Mathematics::Rings and Algebras ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Category Theory (math.CT) ,Mathematics - Category Theory ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,18M15, 18M20, 18N25 (Primary), 16H05 (Secondary) ,Mathematics - Representation Theory - Abstract
We give a formula for the relative Deligne tensor product of two indecomposable finite semisimple module categories over a pointed braided fusion category over an algebraically closed field., Minor corrections
- Published
- 2023
10. The level two Zhu algebra for the Heisenberg vertex operator algebra
- Author
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Addabbo, Darlayne and Barron, Katrina
- Subjects
Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,7B68, 17B69, 17B81, 81R10, 81T40, 81T60 ,Quantum Algebra (math.QA) - Abstract
We determine the level two Zhu algebra for the Heisenberg vertex operator algebra $V$ for any choice of conformal element. We do this using only the following information for $V$: the internal structure of $V$; the level one Zhu algebra of $V$ already determined by the second author, along with Vander Werf and Yang; and the information the lower level Zhu algebras give regarding irreducible modules. We are able to carry out this calculation of the level two Zhu algebra for $V$ with this minimal information by employing the general results and techniques for determining generators and relations for higher level Zhu algebras for a vertex operator algebra, as developed previously by the authors in "On generators and relations for higher level Zhu algebras and applications", by Addabbo and Barron, J. Algebra, 2023. In particular, we show that the level $n$ Zhu algebras for the Heisenberg vertex operator algebra become noncommutative at level $n=2$. We also give a conjecture for the structure of the level $n$ Zhu algebra for the Heisenberg vertex operator algebra, for any $n >2$., 65 pages; minor typos corrected; some calculations shortened
- Published
- 2023
11. The McKay Conjecture and central isomorphic character triples
- Author
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Damiano Rossi and Rossi, D
- Subjects
Character triples ,Algebra and Number Theory ,Inductive McKay condition ,20C15 ,Group Theory (math.GR) ,McKay Conjecture ,Mathematics::Group Theory ,Mathematics::Algebraic Geometry ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Group Theory ,Mathematics - Representation Theory - Abstract
We refine the reduction theorem for the McKay Conjecture proved by Isaacs, Malle and Navarro. Assuming the inductive McKay condition, we obtain a strong version of the McKay Conjecture that gives central isomorphic character triples.Crown Copyright (c) 2022 Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
- Published
- 2023
12. A Theory of Orbit Braids
- Author
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Li, Hao, L��, Zhi, and Li, Fengling
- Subjects
Mathematics::Group Theory ,Mathematics - Geometric Topology ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Geometric Topology (math.GT) ,Group Theory (math.GR) ,Mathematics - Algebraic Topology ,Mathematics - Group Theory - Abstract
This paper upbuilds the theoretical framework of orbit braids in $M\times I$ by making use of the orbit configuration space $F_G(M,n)$, which enriches the theory of ordinary braids, where $M$ is a connected topological manifold of dimension at least 2 with an effective action of a finite group $G$ and the action of $G$ on $I$ is trivial. Main points of our work include as follows. We introduce the orbit braid group $\mathcal{B}_n^{orb}(M,G)$, and show that it is isomorphic to a group with an additional endowed operation (called the extended fundamental group of $F_G(M,n)$), formed by the homotopy classes of some paths (not necessarily closed paths) in $F_G(M,n)$, which is an essential extension for fundamental groups. The orbit braid group $\mathcal{B}_n^{orb}(M,G)$ is large enough to contain the fundamental group of $F_G(M,n)$ and other various braid groups as its subgroups. Around the central position of $\mathcal{B}_n^{orb}(M,G)$, we obtain five short exact sequences weaved in a commutative diagram. We also analyze the essential relations among various braid groups associated to those configuration spaces $F_G(M,n), F(M/G,n)$, and $F(M,n)$. We finally consider how to give the presentations of orbit braid groups in terms of orbit braids as generators. We carry out our work by choosing $M=\mathbb{C}$ with typical actions of $\mathbb{Z}_p$ and $(\mathbb{Z}_2)^2$. We obtain the presentations of the corresponding orbit braid groups, from which we see that the generalized braid group $Br(B_n)$ actually agrees with an orbit braid group and $Br(D_n)$ is a subgroup of another orbit braid group. In addition, the notion of extended fundamental groups is also defined in a general way in the category of topology and some characteristics extracted from the discussions of orbit braids are given., 30 pages, minor changes and corrections in section 4
- Published
- 2023
13. Constructing modular categories from orbifold data
- Author
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Mulevicius, Vincentas and Runkel, Ingo
- Subjects
High Energy Physics - Theory ,High Energy Physics - Theory (hep-th) ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Geometry and Topology ,Mathematics::Geometric Topology ,Mathematical Physics - Abstract
In Carqueville et al., arXiv:1809.01483, the notion of an orbifold datum $\mathbb{A}$ in a modular fusion category $\mathcal{C}$ was introduced as part of a generalised orbifold construction for Reshetikhin-Turaev TQFTs. In this paper, given a simple orbifold datum $\mathbb{A}$ in $\mathcal{C}$, we introduce a ribbon category $\mathcal{C}_{\mathbb{A}}$ and show that it is again a modular fusion category. The definition of $\mathcal{C}_{\mathbb{A}}$ is motivated by properties of Wilson lines in the generalised orbifold. We analyse two examples in detail: (i) when $\mathbb{A}$ is given by a simple commutative $\Delta$-separable Frobenius algebra $A$ in $\mathcal{C}$; (ii) when $\mathbb{A}$ is an orbifold datum in $\mathcal{C} = \operatorname{Vect}$, built from a spherical fusion category $\mathcal{S}$. We show that in case (i), $\mathcal{C}_{\mathbb{A}}$ is ribbon-equivalent to the category of local modules of $A$, and in case (ii), to the Drinfeld centre of $\mathcal{S}$. The category $\mathcal{C}_{\mathbb{A}}$ thus unifies these two constructions into a single algebraic setting., Comment: 58 pages
- Published
- 2023
14. Affine Kac-Moody Groups as Twisted Loop Groups obtained by Galois Descent Considerations
- Author
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Morita, Jun, Pianzola, Arturo, and Shibata, Taiki
- Subjects
Mathematics - Algebraic Geometry ,High Energy Physics::Theory ,Twisted Chevalley groups ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Affine Kac-Moody groups ,Group Theory (math.GR) ,Mathematics::Representation Theory ,20G44, 22E67 ,Mathematics - Group Theory ,Algebraic Geometry (math.AG) ,Loop groups - Abstract
We provide explicit generators and relations for the affine Kac-Moody groups, as well as a realization of them as (twisted) loop groups by means of Galois descent considerations., Comment: 39 pages; to appear in Mathematical Journal of Okayama University
- Published
- 2023
15. Growth of nonsymmetric operads
- Author
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Qi, Zihao, Xu, Yongjun, Zhang, James J., and Zhao, Xiangui
- Subjects
Rings and Algebras (math.RA) ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,General Mathematics ,Mathematics::Rings and Algebras ,FOS: Mathematics ,16P90, 16Z10, 17A61, 13P10, 18M65 ,Category Theory (math.CT) ,Mathematics - Category Theory ,Mathematics - Rings and Algebras ,Mathematics::Representation Theory ,Mathematics::Algebraic Topology - Abstract
The paper concerns the Gelfand-Kirillov dimension and the generating series of nonsymmetric operads. An analogue of Bergman's gap theorem is proved, namely, no finitely generated locally finite nonsymmetric operad has Gelfand-Kirillov dimension strictly between $1$ and $2$. For every $r\in \{0\}\cup \{1\}\cup [2,\infty)$ or $r=\infty$, we construct a single-element generated nonsymmetric operad with Gelfand-Kirillov dimension $r$. We also provide counterexamples to two expectations of Khoroshkin and Piontkovski about the generating series of operads., 38 pages, 10 figures. The known typos have been corrected. A detailed proof to Lemma 7.2 has been added. The referee's comments have been incorporated
- Published
- 2023
16. Stokes Phenomenon and Reflection Equations
- Author
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Xu, Xiaomeng
- Subjects
High Energy Physics::Theory ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Mathematics::Number Theory ,Mathematics::Quantum Algebra ,FOS: Mathematics ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Representation Theory (math.RT) ,Mathematics - Representation Theory ,Mathematical Physics - Abstract
In this paper, we study the Stokes phenomenon of the cyclotomic Knizhnik-Zamolodchikov equation, and prove that its two types of Stokes matrices satisfy the Yang-Baxter and reflection equations respectively. We then discuss its isomonodromy deformation, and its relations with cyclotomic associators, twists, and quantum symmetric pairs., 16 pages
- Published
- 2022
17. Representations of some associative pseudoalgebras
- Author
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Zhixiang Wu
- Subjects
Mathematics::Operator Algebras ,Applied Mathematics ,General Mathematics ,Mathematics::Rings and Algebras ,Mathematics - Rings and Algebras ,17B05, 18D05, 18G60 ,Rings and Algebras (math.RA) ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory - Abstract
In this paper, we generalize Schur-Weyl duality and Morita Theorem on associative algebras to those on associative $H$-pseudoalgebras. Meanwhile, we get a plenty of associative $H$-pseudoalgebras over a cocommutative Hopf algebra $H$., All comments are welcome
- Published
- 2022
18. Cohomological obstructions and weak crossed products over weak Hopf algebras
- Author
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Ramón González Rodríguez and Ana Belén Rodríguez Raposo
- Subjects
1201.12 Álgebras no Asociativas ,Algebra and Number Theory ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,1201 Álgebra ,Mathematics - Rings and Algebras ,1201.05 Campos, Anillos, Álgebras - Abstract
Let $H$ be a cocommutative weak Hopf algebra and let $(B, \varphi_{B})$ a weak left $H$-module algebra. In this paper, for a twisted convolution invertible morphism $\sigma:H\otimes H\rightarrow B$ we define its obstruction $\theta_{\sigma}$ as a degree three Sweedler 3-cocycle with values in the center of $B$. We obtain that the class of this obstruction vanish in third Sweedler cohomology group $\mathcal{H}^3_{\varphi_{Z(B)}}(H, Z(B))$ if, and only if, there exists a twisted convolution invertible 2-cocycle $\alpha:H\otimes H\rightarrow B$ such that $H\otimes B$ can be endowed with a weak crossed product structure with $\alpha$ keeping a cohomological-like relation with $\sigma$. Then, as a consequence, the class of the obstruction of $\sigma$ vanish if, and only if, there exists a cleft extension of $B$ by $H$.
- Published
- 2022
19. Homogeneous quandles arising from automorphisms of symmetric groups
- Author
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Akihiro Higashitani and Hirotake Kurihara
- Subjects
Mathematics - Geometric Topology ,Algebra and Number Theory ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Geometric Topology (math.GT) ,Group Theory (math.GR) ,Primary, 20N02, Secondary: 20B05, 53C35 ,Mathematics - Group Theory ,Mathematics::Algebraic Topology ,Mathematics::Geometric Topology - Abstract
Quandle is an algebraic system with one binary operation, but it is quite different from a group. Quandle has its origin in the knot theory and good relationships with the theory of symmetric spaces, so it is well-studied from points of view of both areas. In the present paper, we investigate a special kind of quandles, called generalized Alexander quandles $Q(G,\psi)$, which is defined by a group $G$ together with its group automorphism $\psi$. We develop the quandle invariants for generalized Alexander quandles. As a result, we prove that there is a one-to-one correspondence between generalized Alexander quandles arising from symmetric groups $\Sf_n$ and the conjugacy classes of $\Sf_n$ for $3 \leq n \leq 30$ with $n \neq 6,15$, and the case $n=6$ is also discussed., Comment: 20 pages
- Published
- 2022
20. Hom-Yang-Baxter equations and Hom-Yang-Baxter systems
- Author
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Shengxiang Wang, Xiaohui Zhang, and Shuangjian Guo
- Subjects
Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Algebra and Number Theory ,Rings and Algebras (math.RA) ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Mathematics::Rings and Algebras ,FOS: Mathematics ,16T25, 17A30, 17B38 ,Physics::Accelerator Physics ,Mathematics - Rings and Algebras - Abstract
In this paper, we mainly present some new solutions of the Hom-Yang-Baxter equation from Hom-algebras, Hom-coalgebras and Hom-Lie algebras, respectively. Also, we prove that these solutions are all self-inverse and give some examples. Finally, we introduce the notion of Hom-Yang-Baxter systems and obtain two kinds of Hom-Yang-Baxter systems., Comment: 20
- Published
- 2022
21. Associator dependent algebras and Koszul duality
- Author
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Murray Bremner and Vladimir Dotsenko
- Subjects
Rings and Algebras (math.RA) ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Applied Mathematics ,Mathematics::Rings and Algebras ,Mathematics - K-Theory and Homology ,FOS: Mathematics ,K-Theory and Homology (math.KT) ,Mathematics - Rings and Algebras ,18M70 (Primary), 16R10, 17A30 (Secondary) ,Mathematics::Algebraic Topology - Abstract
We resolve a ten year old open question of Loday of describing Koszul operads that act on the algebra of octonions. In fact, we obtain the answer by solving a more general classification problem: we find all Koszul operads among those encoding associator dependent algebras., 21 pages
- Published
- 2022
22. Experiments on growth series of braid groups
- Author
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Jean Fromentin, Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville (LMPA), and Université du Littoral Côte d'Opale (ULCO)
- Subjects
Pure mathematics ,spherical growth series ,Geodesic ,Braid group ,68R15 Braid group ,Group Theory (math.GR) ,2020 Mathematics Subject Classification. Primary 20F36 ,01 natural sciences ,[MATH.MATH-GR]Mathematics [math]/Group Theory [math.GR] ,Mathematics::Group Theory ,Mathematics::Quantum Algebra ,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO] ,0103 physical sciences ,FOS: Mathematics ,Mathematics - Combinatorics ,0101 mathematics ,Mathematics ,algorithm ,Algebra and Number Theory ,Conjecture ,Series (mathematics) ,Secondary 20F69 ,010102 general mathematics ,Mathematics::Geometric Topology ,geodesic growth series ,Combinatorics (math.CO) ,010307 mathematical physics ,20F10 ,Mathematics - Group Theory - Abstract
We introduce an algorithmic framework to investigate spherical and geodesic growth series of braid groups relatively to the Artin's or Birman–Ko–Lee's generators. We present our experimentations in the case of three and four strands and conjecture rational expressions for the spherical growth series with respect to the Birman–Ko–Lee's generators.
- Published
- 2022
23. Rationality and C2-cofiniteness of certain diagonal coset vertex operator algebras
- Author
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Xingjun Lin
- Subjects
Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Mathematics::Representation Theory - Abstract
In this paper, it is shown that the diagonal coset vertex operator algebra $C(L_{\mathfrak{g}}(k+2,0),L_{\mathfrak{g}}(k,0)\otimes L_{\mathfrak{g}}(2,0))$ is rational and $C_2$-cofinite in case $\mathfrak{g}=so(2n), n\geq 3$ and $k$ is an admissible number for $\hat{\mathfrak{g}}$. It is also shown that the diagonal coset vertex operator algebra $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$ is rational and $C_2$-cofinite in case $k$ is an admissible number for $\hat{sl_2}$. Furthermore, irreducible modules of $C(L_{sl_2}(k+4,0),L_{sl_2}(k,0)\otimes L_{sl_2}(4,0))$ are classified in case $k$ is a positive odd integer., 28 pages
- Published
- 2022
24. Nonsymmetric Macdonald polynomials via integrable vertex models
- Author
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Michael Wheeler and Alexei Borodin
- Subjects
Vertex (graph theory) ,Path (topology) ,Integrable system ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Lattice (group) ,FOS: Physical sciences ,Mathematical Physics (math-ph) ,Eigenfunction ,01 natural sciences ,Combinatorics ,Macdonald polynomials ,Mathematics::Quantum Algebra ,Vertex model ,FOS: Mathematics ,Bijection ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Representation Theory (math.RT) ,0101 mathematics ,Mathematical Physics ,Mathematics - Representation Theory ,Mathematics - Abstract
Starting from an integrable rank-$n$ vertex model, we construct an explicit family of partition functions indexed by compositions $\mu = (\mu_1,\dots,\mu_n)$. Using the Yang-Baxter algebra of the model and a certain rotation operation that acts on our partition functions, we show that they are eigenfunctions of the Cherednik-Dunkl operators $Y_i$ for all $1 \leq i \leq n$, and are thus equal to nonsymmetric Macdonald polynomials $E_{\mu}$. Our partition functions have the combinatorial interpretation of ensembles of coloured lattice paths which traverse a cylinder. Applying a simple bijection to such path ensembles, we show how to recover the well-known combinatorial formula for $E_{\mu}$ due to Haglund-Haiman-Loehr., Comment: 36 pages
- Published
- 2022
25. Multiparameter quantum groups at roots of unity
- Author
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Garc��a, Gast��n Andr��s and Gavarini, Fabio
- Subjects
Algebra and Number Theory ,17B37 (primary), 16T05, 16T20 (secondary) ,17B37 ,16T05 ,Settore MAT/02 - Algebra ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Settore MAT/03 - Geometria ,Geometry and Topology ,17B37, 16T05, 16T20 ,16T20 ,Mathematical Physics - Abstract
We address the study of multiparameter quamtum groups (=MpQG's) at roots of unity, namely quantum universal enveloping algebras $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) $ depending on a matrix of parameters $ \boldsymbol{\rm q} = {\big( q_{ij} \big)}_{i, j \in I} \, $. This is performed via the construction of quantum root vectors and suitable "integral forms" of $ U_{\boldsymbol{\rm q}}(\mathfrak{g}) \, $, a \textsl{restricted one} - generated by quantum divided powers and quantum binomial coefficients - and an \textsl{unrestricted\/} one - where quantum root vectors are suitably renormalized. The specializations at roots of unity of either forms are the "MpQG's at roots of unity" we look for. In particular, we study special subalgebras and quotients of our MpQG's at roots of unity - namely, the multiparameter version of small quantum groups - and suitable associated quantum Frobenius morphisms, that link the MpQG's at roots of 1 with MpQG's at 1, the latter being classical Hopf algebras bearing a well precise Poisson-geometrical content. A key point in the discussion - often at the core of our strategy - is that every MpQG is actually a 2-cocycle deformation of the algebra structure of (a lift of) the "canonical" one-parameter quantum group by Jimbo-Lusztig, so that we can often rely on already established results available for the latter. On the other hand, depending on the chosen multiparameter $ \boldsymbol{\rm q} $ our quantum groups yield (through the choice of integral forms and their specialization) different semiclassical structures, namely different Lie coalgebra structures and Poisson structures on the Lie algebra and algebraic group underlying the canonical one-parameter quantum group., 71 pages. Final version, shortened and polished after the referee's comments - the mathematical content, though, is unchanged with respect to the previous version. To appear in "Journal of Noncommutative Geometry"
- Published
- 2022
26. A quantitative Birman–Menasco finiteness theorem and its application to crossing number
- Author
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Ito, Tetsuya
- Subjects
Mathematics - Geometric Topology ,Mathematics::Group Theory ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Geometric Topology (math.GT) ,57K10 (Primary) ,Geometry and Topology ,Mathematics::Geometric Topology - Abstract
Birman-Menasco proved that there are finitely many knots having a given genus and braid index. We give a quantitative version of Birman-Menasco finiteness theorem, an estimate of the crossing number of knots in terms of genus and braid index. This has various applications of crossing numbers, such as, the crossing number of connected sum or satellites., Comment: 11 pages, 5 figures; v3. error in Proposition 2 is corrected. Main applications (Corollary 1-- 6) are not changed. v2. Corollary 6, an estimate for the crossing number of satellite knot, is added
- Published
- 2022
27. The compact presentation for the alternating central extension of the q-Onsager algebra
- Author
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Terwilliger, Paul
- Subjects
Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Mathematics - Combinatorics ,Quantum Algebra (math.QA) ,Combinatorics (math.CO) ,17B37 - Abstract
The $q$-Onsager algebra $O_q$ is defined by two generators and two relations, called the $q$-Dolan/Grady relations. We investigate the alternating central extension $\mathcal O_q$ of $O_q$. The algebra $\mathcal O_q$ was introduced by Baseilhac and Koizumi, who called it the current algebra of $O_q$. Recently Baseilhac and Shigechi gave a presentation of $\mathcal O_q$ by generators and relations. The presentation is attractive, but the multitude of generators and relations makes the presentation unwieldy. In this paper we obtain a presentation of $\mathcal O_q$ that involves a subset of the original set of generators and a very manageable set of relations. We call this presentation the compact presentation of $\mathcal O_q$. This presentation resembles the compact presentation of the alternating central extension for the positive part of $U_q(\widehat{\mathfrak{sl}}_2)$., Comment: 26 pages. arXiv admin note: text overlap with arXiv:2103.03028
- Published
- 2023
28. Quasitriangular operator algebras
- Author
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Massoud Amini, Mehdi Moradi, and Ismaeil Mousavi
- Subjects
Mathematics::K-Theory and Homology ,Mathematics::Operator Algebras ,Primary: 46K50, Secondary: 46L07 ,Mathematics::Quantum Algebra ,Applied Mathematics ,FOS: Mathematics ,Mathematics - Operator Algebras ,Operator Algebras (math.OA) ,Analysis - Abstract
We give characterizations of quasitriangular operator algebras along the line of Voiculescu's characterization of quasidiagonal $C^*$-algebras.
- Published
- 2023
29. Schur-Weyl duality for quantum toroidal superalgebras
- Author
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Lu, Kang
- Subjects
Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory - Abstract
We establish the Schur-Weyl type duality between double affine Hecke algebras and quantum toroidal superalgebras, generalizing the well known result of Vasserot-Varagnolo [VV96] to the super case., Comment: 17 pages; v3 a new Proposition 2.7 to fix a gap in the proof of Proposition 2.9; to appear in Journal of Pure and Applied Algebra
- Published
- 2023
30. Gaudin Algebras, RSK and Calogero-Moser cells in type A
- Author
-
Brochier, Adrien, Gordon, Iain, and White, Noah
- Subjects
Mathematics - Algebraic Geometry ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,General Mathematics ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Algebraic Geometry (math.AG) ,Mathematics - Representation Theory - Abstract
We study the spectrum of a family of algebras, the inhomogeneous Gaudin algebras, acting on the $n$-fold tensor representation $\mathbb{C}[x_1, \ldots, x_r]^{\otimes n}$ of the Lie algebra $\mathfrak{gl}_r$. We use the work of Halacheva-Kamnitzer-Rybnikov-Weekes to demonstrate that the Robinson-Schensted-Knuth correspondence describes the behaviour of the spectrum as we move along special paths in the family. We apply the work of Mukhin-Tarasov-Varchenko, which proves that the rational Calogero-Moser phase space can be realised as a part of this spectrum, to relate this to behaviour at $t=0$ of rational Cherednik algebras of $\mathfrak{S}_n$. As a result, we confirm for symmetric groups a conjecture of Bonnaf\'e-Rouquier which proposes an equality between the Calogero-Moser cells they defined and the well-known Kazhdan-Lusztig cells., Comment: 24 pages
- Published
- 2023
31. Tangles in affine Hecke algebras
- Author
-
Morton, Hugh
- Subjects
Mathematics::Group Theory ,Algebra and Number Theory ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,57K10, 20C08 (Primary) 57K14, 20F36 (Secondary) ,Mathematics::Geometric Topology - Abstract
The affine Hecke algebra $\dot H_n$ of type $A$ is often presented as a quotient of the braid algebra of $n$-braids in the annulus. This leads to diagrammatic representations in terms of braids in the annulus, subject to a quadratic relation for the simple Artin braids, as in the description by Graham and Lehrer in \cite{GL03}. I show here that the use of more general framed oriented $n$-tangle diagrams in the annulus, subject to the Homfly skein relations, produces an algebra which is isomorphic to $\dot H_n$ with an extended ring of coefficients. This setting allows the use of some attractive diagrams for elements of $\dot H_n$, using closed curves as well as braids, and gives neat pictures for its central elements., 10 pages, several embedded diagrams
- Published
- 2023
32. Set-theoretic Yang-Baxter (co)homology theory of involutive non-degenerate solutions
- Author
-
Przytycki, Józef H., Vojtěchovský, Petr, and Yang, Seung Yeop
- Subjects
Mathematics - Geometric Topology ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Algebra and Number Theory ,Mathematics::Quantum Algebra ,16T25, 20N05, 57M27 ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Geometric Topology (math.GT) ,Mathematics - Algebraic Topology ,Mathematics::Geometric Topology - Abstract
W. Rump showed that there exists a one-to-one correspondence between involutive right non-degenerate solutions of the Yang-Baxter equation and Rump right quasigroups. J. S. Carter, M. Elhamdadi, and M. Saito, meanwhile, introduced a homology theory of set-theoretic solutions of the Yang-Baxter equation in order to define cocycle invariants of classical knots. In this paper, we introduce the normalized homology theory of an involutive right non-degenerate solution of the Yang-Baxter equation and prove that the set-theoretic Yang-Baxter homology of certain solutions can be split into the normalized and degenerated parts., Comment: 14 pages, 6 figures
- Published
- 2023
33. Lie algebras of curves and loop-bundles on surfaces
- Author
-
Alonso, Juan, Paternain, Miguel, Peraza, Javier, and Reisenberger, Michael
- Subjects
Mathematics - Geometric Topology ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,55P35 (Primary), 17B62 (Secondary) ,FOS: Mathematics ,Geometric Topology (math.GT) ,Group Theory (math.GR) ,Geometry and Topology ,Mathematics::Geometric Topology ,Mathematics::Algebraic Topology ,Mathematics - Group Theory - Abstract
W. Goldman and V. Turaev defined a Lie bialgebra structure on the $\mathbb Z$-module generated by free homotopy classes of loops of an oriented surface (i.e. the conjugacy classes of its fundamental group). We develop a generalization of this construction replacing homotopies by thin homotopies, based on the combinatorial approach given by M. Chas. We use it to give a geometric proof of a characterization of simple curves in terms of the Goldman-Turaev bracket, which was conjectured by Chas., 40 pages, 5 figures
- Published
- 2023
34. Anisotropic Spin Generalization of Elliptic Macdonald–Ruijsenaars Operators and R-Matrix Identities
- Author
-
M. Matushko and Andrei Zotov
- Subjects
High Energy Physics - Theory ,Nuclear and High Energy Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,High Energy Physics - Theory (hep-th) ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Exactly Solvable and Integrable Systems (nlin.SI) ,Mathematical Physics - Abstract
We propose commuting set of matrix-valued difference operators in terms of the elliptic Baxter-Belavin $R$-matrix in the fundamental representation of ${\rm GL}_M$. In the scalar case $M=1$ these operators are the elliptic Macdonald-Ruijsenaars operators, while in the general case they can be viewed as anisotropic versions of the quantum spin Ruijsenaars Hamiltonians. We show that commutativity of the operators for any $M$ is equivalent to a set of $R$-matrix identities. The proof of identities is based on the properties of elliptic $R$-matrix including the quantum and the associative Yang-Baxter equations. As an application of our results, we introduce elliptic generalization of q-deformed Haldane-Shastry model., Comment: 38 pages, minor corrections
- Published
- 2023
35. Affine Pieri rule for periodic Macdonald spherical functions and fusion rings
- Author
-
J. F. van Diejen, Ignacio Zurrián, and E. Emsiz
- Subjects
Pure mathematics ,05E05, 17B67, 33D52, 33D80, 81T40 ,General Mathematics ,FOS: Physical sciences ,Field (mathematics) ,Genus (mathematics) ,Mathematics::Quantum Algebra ,Lie algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Wess-Zumino-Witten fusion rings ,Quantum Algebra (math.QA) ,Representation Theory (math.RT) ,Mathematical Physics ,Mathematics ,Ring (mathematics) ,Mathematics::Combinatorics ,Zero (complex analysis) ,Affine Lie algebras ,Basis (universal algebra) ,Mathematical Physics (math-ph) ,Affine Lie algebra ,Macdonald spherical functions ,Affine transformation ,Affine Hecke algebras ,Mathematics - Representation Theory - Abstract
Let $\hat{\mathfrak{g}}$ be an untwisted affine Lie algebra or the twisted counterpart thereof (which excludes the affine Lie algebras of type $\widehat{BC}_n=A^{(2)}_{2n}$). We present an affine Pieri rule for a basis of periodic Macdonald spherical functions associated with $\hat{\mathfrak{g}}$. In type $\hat{A}_{n-1}=A^{(1)}_{n-1}$ the formula in question reproduces an affine Pieri rule for cylindric Hall-Littlewood polynomials due to Korff, which at $t=0$ specializes in turn to a well-known Pieri formula in the fusion ring of genus zero $\widehat{\mathfrak{sl}}(n)_c$-Wess-Zumino-Witten conformal field theories., 25 pages
- Published
- 2023
36. Comparison of quantizations of symmetric spaces: cyclotomic Knizhnik–Zamolodchikov equations and Letzter–Kolb coideals
- Author
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Kenny De Commer, Sergey Neshveyev, Lars Tuset, Makoto Yamashita, Algebra and Analysis, Mathematics, Mathematics-TW, and Topological Algebra, Functional Analysis and Category Theory
- Subjects
Statistics and Probability ,Algebra and Number Theory ,Mathematics::Rings and Algebras ,Quantum groups ,deformation quantization ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Discrete Mathematics and Combinatorics ,Quantum Algebra (math.QA) ,Geometry and Topology ,Representation Theory (math.RT) ,tensor categories ,Mathematical Physics ,Analysis ,Mathematics - Representation Theory - Abstract
We establish an equivalence between two approaches to quantization of irreducible symmetric spaces of compact type within the framework of quasi-coactions, one based on the Enriquez-Etingof cyclotomic Knizhnik-Zamolodchikov (KZ) equations and the other on the Letzter-Kolb coideals. This equivalence can be upgraded to that of ribbon braided quasi-coactions, and then the associated reflection operators (K-matrices) become a tangible invariant of the quantization. As an application we obtain a Kohno-Drinfeld type theorem on type B braid group representations defined by the monodromy of KZ-equations and by the Balagovi\'c-Kolb universal K-matrices. The cases of Hermitian and non-Hermitian symmetric spaces are significantly different. In particular, in the latter case a quasi-coaction is essentially unique, while in the former we show that there is a one-parameter family of mutually nonequivalent quasi-coactions., Comment: v2: minor changes; v1: 62 pages
- Published
- 2023
37. General comodule-contramodule correspondence
- Author
-
Katerina Hristova, John Jones, and Dmitriy Rumynin
- Subjects
General Mathematics ,Mathematics::Rings and Algebras ,Mathematics - Category Theory ,Mathematics::Algebraic Topology ,Primary 18D20. Secondary 18N40, 16T15, 55U40 ,Computational Theory and Mathematics ,Mathematics::K-Theory and Homology ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Algebraic Topology (math.AT) ,Quantum Algebra (math.QA) ,Category Theory (math.CT) ,Mathematics - Algebraic Topology ,Representation Theory (math.RT) ,Statistics, Probability and Uncertainty ,Mathematics - Representation Theory - Abstract
This paper is a fundamental study of comodules and contramodules over a comonoid in a symmetric closed monoidal category. We study both algebraic and homotopical aspects of them. Algebraically, we enrich the comodule and contramodule categories over the original category, construct enriched functors between them and enriched adjunctions between the functors. Homotopically, for simplicial sets and topological spaces, we investigate the categories of comodules and contramodules and the relations between them., Comment: Version 2: major revision to make the paper easier to read. Version 3: another major revision: we refer to various old results on the functorial semantics that makes paper shorter and more accessible. Version 4: yet another major revision: we replace a biclosed monoidal category with a symmetric closed monoidal category, simplifying many proofs. Version 5: minor edits, final journal version
- Published
- 2023
38. Non-linear Hopf Manifolds are Locally Conformally Kähler
- Author
-
Liviu Ornea and Misha Verbitsky
- Subjects
Mathematics - Differential Geometry ,Differential Geometry (math.DG) ,Mathematics::Complex Variables ,Mathematics - Complex Variables ,Mathematics::Quantum Algebra ,Mathematics::Rings and Algebras ,FOS: Mathematics ,Mathematics::Differential Geometry ,Geometry and Topology ,Complex Variables (math.CV) ,Mathematics::Symplectic Geometry - Abstract
A Hopf manifold is a quotient of $C^n\backslash 0$ by the cyclic group generated by a holomorphic contraction. Hopf manifolds are diffeomorphic to $S^1\times S^{2n-1}$ and hence do not admit Kahler metrics. It is known that Hopf manifolds defined by linear contractions (called linear Hopf manifolds) have locally conformally Kahler (LCK) metrics. In this paper we prove that the Hopf manifolds defined by non-linear holomorphic contractions admit holomorphic embeddings into linear Hopf manifolds, and, moreover they admit LCK metrics., 11 pages, Latex (no change in the article, but the title in the submission was wrong)
- Published
- 2023
39. Quantum Borcherds-Bozec algebras via semi-derived Ringel-Hall algebras
- Author
-
Lu, Ming
- Subjects
High Energy Physics::Theory ,Mathematics::Quantum Algebra ,Applied Mathematics ,General Mathematics ,Mathematics - Quantum Algebra ,Mathematics::Rings and Algebras ,FOS: Mathematics ,Quantum Algebra (math.QA) ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory - Abstract
We use semi-derived Ringel-Hall algebras of quivers with loops to realize the whole quantum Borcherds-Bozec algebras and quantum generalized Kac-Moody algebras., Comment: v2, 12 pages, minor edits, accepted by Proc. AMS
- Published
- 2023
40. A parafermionic hypergeometric function and supersymmetric 6 -symbols
- Author
-
Elena Apresyan, Gor Sarkissian, and Vyacheslav P. Spiridonov
- Subjects
High Energy Physics - Theory ,Nuclear and High Energy Physics ,High Energy Physics::Theory ,High Energy Physics - Theory (hep-th) ,Mathematics::Quantum Algebra ,Mathematics::Classical Analysis and ODEs ,FOS: Physical sciences ,Computer Science::Symbolic Computation - Abstract
We study properties of a parafermionic generalization of the hyperbolic hypergeometric function appearing as the most important part in the fusion matrix for Liouville field theory and the Racah-Wigner symbols for the Faddeev modular double. We show that this generalized hypergeometric function is a limiting form of the rarefied elliptic hypergeometric function $V^{(r)}$ and derive its transformation properties and a mixed difference-recurrence equation satisfied by it. At the intermediate level we describe symmetries of a more general rarefied hyperbolic hypergeometric function. An important $r=2$ case corresponds to the supersymmetric hypergeometric function given by the integral appearing in the fusion matrix of $N=1$ super Liouville field theory and the Racah-Wigner symbols of the quantum algebra ${\rm U}_q({\rm osp}(1|2))$. We indicate relations to the standard Regge symmetry and prove some previous conjectures for the supersymmetric Racah-Wigner symbols by establishing their different parametrizations., 29 pages
- Published
- 2023
41. On pentagon identity in Ding-Iohara-Miki algebra
- Author
-
Zenkevich, Yegor
- Subjects
High Energy Physics - Theory ,Nuclear and High Energy Physics ,High Energy Physics - Theory (hep-th) ,Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,FOS: Physical sciences - Abstract
We notice that the famous pentagon identity for quantum dilogarithm functions and the five-term relation for certain operators related to Macdonald polynomials discovered by Garsia and Mellit can both be understood as specific cases of a general "master pentagon identity" for group-like elements in the Ding-Iohara-Miki (or quantum toroidal, or elliptic Hall) algebra. We perform some checks of this remarkable identity and discuss its implications., 10 pages
- Published
- 2023
42. Invariants of Multi-linkoids
- Author
-
Boštjan Gabrovšek and Neslihan Gügümcü
- Subjects
Mathematics - Geometric Topology ,Mathematics::Quantum Algebra ,General Mathematics ,FOS: Mathematics ,Geometric Topology (math.GT) ,Mathematics::Geometric Topology - Abstract
In this paper, we extend the definition of a knotoid that was introduced by Turaev, to multi-linkoids that consist of a number of knot and knotoid components. We study invariants of multi-linkoids that lie in a closed orientable surface, namely the Kauffman bracket polynomial, ordered bracket polynomial, the Kauffman skein module, and the $T$-invariant in relation with generalized $\Theta$-graphs., Comment: 15 pages
- Published
- 2023
43. Set-valued tableaux rule for Lascoux polynomials
- Author
-
Yu, Tianyi
- Subjects
Mathematics::Algebraic Geometry ,Mathematics::Combinatorics ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Mathematics - Combinatorics ,Combinatorics (math.CO) ,Mathematics::Representation Theory - Abstract
Lascoux polynomials generalize Grassmannian stable Grothendieck polynomials and may be viewed as K-theoretic analogs of key polynomials. The latter two polynomials have combinatorial formulas involving tableaux: Lascoux and Sch\"{u}tzenberger gave a combinatorial formula for key polynomials using right keys; Buch gave a set-valued tableau formula for Grassmannian stable Grothendieck polynomials. We establish a novel combinatorial rule for Lascoux polynomials involving right keys and set-valued tableaux. Our rule recovers the tableaux formulas of key polynomials and Grassmannian stable Grothendieck polynomials. To prove our rule, we construct a new abstract Kashiwara crystal structure on set-valued tableaux. This construction answers an open problem of Monical, Pechenik and Scrimshaw in the context of abstract Kashiwara crystal.
- Published
- 2023
44. Centers of Hecke algebras of complex reflection groups
- Author
-
Eirini Chavli and Götz Pfeiffer
- Subjects
Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics::Number Theory ,FOS: Mathematics ,Group Theory (math.GR) ,20C08, 20F55 ,Geometry and Topology ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Group Theory ,Mathematics - Representation Theory - Abstract
We provide a dual version of the Geck--Rouquier Theorem on the center of an Iwahori--Hecke algebra, which also covers the complex case. For the eight complex reflection groups of rank $2$, for which the symmetrising trace conjecture is known to be true, we provide a new faithful matrix model for their Hecke algebra $H$. These models enable concrete calculations inside $H$. For each of the eight groups, we compute an explicit integral basis of the center of $H$., Comment: Final version; 14 pages. Appears in: Beitr\"age zur Algebra und Geometrie
- Published
- 2023
45. A scanning algorithm for odd Khovanov homology
- Author
-
Dirk Schütz
- Subjects
Mathematics - Geometric Topology ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Geometric Topology (math.GT) ,Geometry and Topology ,57K18 (primary) 57K10 (secondary) ,Mathematics::Symplectic Geometry ,Mathematics::Geometric Topology - Abstract
We adapt Bar-Natan's scanning algorithm for fast computations in (even) Khovanov homology to odd Khovanov homology. We use a mapping cone construction instead of a tensor product, which allows us to deal efficiently with the more complicated sign assignments in the odd theory. The algorithm has been implemented in a computer program. We also use the algorithm to determine the odd Khovanov homology of 3-strand torus links., Comment: 30 pages, 9 figures. For program file, see https://www.maths.dur.ac.uk/~dma0ds/KnotJob.zip
- Published
- 2022
46. On the h-adic Quantum Vertex Algebras Associated with Hecke Symmetries
- Author
-
Slaven Kožić
- Subjects
Mathematics::Quantum Algebra ,Mathematics - Quantum Algebra ,FOS: Mathematics ,Quantum Algebra (math.QA) ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Hecke symmetry ,Involutive symmetry ,Quantum determinant ,Quantum vertex algebra ,Yangian ,φ-Coordinated module ,Mathematics::Representation Theory ,17B37, 17B69, 81R50 ,Mathematical Physics - Abstract
We study the quantum vertex algebraic framework for the Yangians of RTT-type and the braided Yangians associated with Hecke symmetries, introduced by Gurevich and Saponov. First, we construct several families of modules for the aforementioned Yangian-like algebras which, in the RTT-type case, lead to a certain $h$-adic quantum vertex algebra $\mathcal{V}_c (R)$ via the Etingof-Kazhdan construction, while, in the braided case, they produce ($\phi$-coordinated) $\mathcal{V}_c (R)$-modules. Next, we show that the coefficients of suitably defined quantum determinant can be used to obtain central elements of $\mathcal{V}_c (R)$, as well as the invariants of such ($\phi$-coordinated) $\mathcal{V}_c (R)$-modules. Finally, we investigate a certain algebra which is closely connected with the representation theory of $\mathcal{V}_c (R)$., Comment: 23 pages, comments are welcome
- Published
- 2022
47. On the convergence of the Kac-Moody correction factor
- Author
-
Chen, Yanze
- Subjects
High Energy Physics::Theory ,Mathematics::Quantum Algebra ,General Mathematics ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory - Abstract
The Kac-Moody correction factor, first studied by Macdonald in the affine case, corrects the failure of an identity found by Macdonald in finite-dimensional root systems in 1972. Subsequntly this factor appeared in several formulas in the affine or Kac-Moody analogue of $p$-adic spherical theory for reductive groups. In this article we view the inverse of this correction factor as a function, prove the convergency and holomorphy of this function on a certain domain., Comment: 22 pages
- Published
- 2022
48. Snowflake modules and Enright functor for Kac–Moody superalgebras
- Author
-
Gorelik, Maria and Serganova, Vera
- Subjects
Algebra and Number Theory ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,17B67 - Abstract
We introduce a class of modules over Kac-Moody superalgebras; we call these modules snowflake. These modules are characterized by invariance property of their characters with respect to a certain subgroup of the Weyl group. Examples of snowflake modules appear as admissible modules in representation theory of affine vertex algebras and in classification of bounded weight modules. Using these modules we prove Arakawa's Theorem for the Lie superalgebra osp(1|2n).
- Published
- 2022
49. Stated skein algebras of surfaces
- Author
-
Costantino, Francesco, Le, Thang T. Q., Institut de Mathématiques de Toulouse UMR5219 (IMT), Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Université Toulouse - Jean Jaurès (UT2J)-Université Toulouse III - Paul Sabatier (UT3), Université Fédérale Toulouse Midi-Pyrénées-Centre National de la Recherche Scientifique (CNRS), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National des Sciences Appliquées - Toulouse (INSA Toulouse), Institut National des Sciences Appliquées (INSA)-Université de Toulouse (UT)-Institut National des Sciences Appliquées (INSA)-Université Toulouse - Jean Jaurès (UT2J), Université de Toulouse (UT)-Université Toulouse III - Paul Sabatier (UT3), Université de Toulouse (UT)-Centre National de la Recherche Scientifique (CNRS), ANR-16-CE40-0017,Quantact,Topologie quantique et géométrie de contact(2016), and ANR-11-LABX-0040,CIMI,Centre International de Mathématiques et d'Informatique (de Toulouse)(2011)
- Subjects
Mathematics - Geometric Topology ,[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT] ,Mathematics::Category Theory ,Mathematics::Quantum Algebra ,Applied Mathematics ,General Mathematics ,FOS: Mathematics ,Geometric Topology (math.GT) ,Mathematics::Geometric Topology ,Primary 57N10, secondary 57M25 - Abstract
We study the algebraic and geometric properties of stated skein algebras of surfaces with punctured boundary. We prove that the skein algebra of the bigon is isomorphic to the quantum group ${\mathcal O}_{q^2}(\mathrm{SL}(2))$ providing a topological interpretation for its structure morphisms. We also show that its stated skein algebra lifts in a suitable sense the Reshetikhin-Turaev functor and in particular we recover the dual $R$-matrix for ${\mathcal O}_{q^2}(\mathrm{SL}(2))$ in a topological way. We deduce that the skein algebra of a surface with $n$ boundary components is an algebra-comodule over ${\mathcal O}_{q^2}(\mathrm{SL}(2))^{\otimes{n}}$ and prove that cutting along an ideal arc corresponds to Hochshild cohomology of bicomodules. We give a topological interpretation of braided tensor product of stated skein algebras of surfaces as "glueing on a triangle"; then we recover topologically some braided bialgebras in the category of ${\mathcal O}_{q^2}(\mathrm{SL}(2))$-comodules, among which the "transmutation" of ${\mathcal O}_{q^2}(\mathrm{SL}(2))$. We also provide an operadic interpretation of stated skein algebras as an example of a "geometric non symmetric modular operad". In the last part of the paper we define a reduced version of stated skein algebras and prove that it allows to recover Bonahon-Wong's quantum trace map and interpret skein algebras in the classical limit when $q\to 1$ as regular functions over a suitable version of moduli spaces of twisted bundles., Comment: 74 pages, 33 figures. In version 2 : strengthened Theorem 4.17. To be published in the Journal of the European Mathematical Society
- Published
- 2022
50. Cocycle deformations for Hom-Hopf algebras
- Author
-
J.N. Alonso Álvarez, J.M. Fernández Vilaboa, and R. González Rodríguez
- Subjects
1201.12 Álgebras no Asociativas ,Algebra and Number Theory ,Mathematics::Quantum Algebra ,Mathematics::Category Theory ,Mathematics::Rings and Algebras ,Physics::Accelerator Physics - Abstract
Financiado para publicación en acceso aberto: Universidade de Vigo/CISUG In this paper we introduce the Hom-analogue of the definition of 2-cocycle for Hopf algebras, called Hom-2-cocycle, and study its properties in order to give a theory of multiplication alteration by Hom-2-cocycles for Hom-Hopf algebras. We show that, just like in the classical setting, if H is a Hom-Hopf algebra with associated endomorphism α and σ is a convolution invertible Hom-2-cocycle, it is possible to define a new product in H to get a new Hom-Hopf algebra if . Moreover we introduce the notions of matched pair and skew pairing in the Hom case and, by the close connection between Hom-2-cocycles and Hom-skew pairings, we show that a special case of Hom-matched pair can be obtained as a deformation of a Hom-Hopf algebra by a Hom-2-cocycle built by a Hom-skew pairing. Xunta de Galicia | Ref. ED431C 2019/10 Agencia Estatal de Investigación | Ref. PID2020-115155GB-I00
- Published
- 2022
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