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Статті в журналах з теми "Twisted bialgebra"

Автор: Grafiati

Опубліковано: 17 лютого 2024

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1

Song, Guang’ai, and Xiaoqing Yue. "Dual Lie Bialgebra Structures of Twisted Schrödinger-Virasoro Type." Algebra Colloquium 25, no. 04 (December 2018): 627–52. http://dx.doi.org/10.1142/s1005386718000445.

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Анотація:

In this paper, the structures of dual Lie bialgebras of twisted Schrödinger-Virasoro type are investigated. By studying the maximal good subspaces, we determine the dual Lie coalgebras of the twisted Schrödinger-Virasoro algebras. Then based on this, we construct the dual Lie bialgebra structures of this type. As by-products, four new infinite dimensional Lie algebras are obtained.

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2

Su, Yihong, and Xue Chen. "Lie Bialgebras on the Rank Two Heisenberg–Virasoro Algebra." Mathematics 11, no. 4 (February 17, 2023): 1030. http://dx.doi.org/10.3390/math11041030.

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The rank two Heisenberg–Virasoro algebra can be viewed as a generalization of the twisted Heisenberg–Virasoro algebra. Lie bialgebras play an important role in searching for solutions of quantum Yang–Baxter equations. It is interesting to study the Lie bialgebra structures on the rank two Heisenberg–Virasoro algebra. Since the Lie brackets of rank two Heisenberg–Virasoro algebra are different from that of the twisted Heisenberg–Virasoro algebra and Virasoro-like algebras, and there are inner derivations (from itself to its tensor space) which are hidden more deeply in its interior algebraic structure, some new techniques and strategies are employed in this paper. It is proved that every Lie bialgebra structure on the rank two Heisenberg–Virasoro algebra is triangular coboundary.

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3

Fang, Xiao-Li, та Tae-Hwa Kim. "(𝜃,ω)-Twisted Radford’s Hom-biproduct and ϖ-Yetter–Drinfeld modules for Hom-Hopf algebras". Journal of Algebra and Its Applications 19, № 03 (березень 2020): 2050046. http://dx.doi.org/10.1142/s0219498820500462.

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Анотація:

To unify different definitions of smash Hom-products in a Hom-bialgebra [Formula: see text], we firstly introduce the notion of [Formula: see text]-twisted smash Hom-product [Formula: see text]. Secondly, we find necessary and sufficient conditions for the twisted smash Hom-product [Formula: see text] and the twisted smash Hom-coproduct [Formula: see text] to afford a Hom-bialgebra, which generalize the well-known Radford’s biproduct and the Hom-biproduct obtained in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math. 137 (2014) 43–65]. Furthermore, we introduce the notion of the category of [Formula: see text]-Yetter-Drinfeld modules which unifies the ones of Hom-Yetter Drinfeld category appeared in [H. Li and T. Ma, A construction of the Hom-Yetter–Drinfeld category, Colloq. Math. 137 (2014) 43–65] and [A. Makhlouf and F. Panaite, Twisting operators, twisted tensor products and smash products for Hom-associative algebras, J. Math. Glasgow 513–538 (2016) 58]. Finally, we prove that the [Formula: see text]-twisted Radford’s Hom-biproduct [Formula: see text] is a Hom-bialgebra if and only if [Formula: see text] is a Hom-bialgebra in the category of [Formula: see text]-Yetter–Drinfeld modules [Formula: see text], generalizing the well-known Majid’s conclusion.

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4

Ma, Tianshui, Linlin Liu, and Shaoxian Xu. "Twisted tensor biproduct monoidal Hom–Hopf algebras." Asian-European Journal of Mathematics 10, no. 01 (March 2017): 1750011. http://dx.doi.org/10.1142/s1793557117500115.

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Let [Formula: see text] be a monoidal Hom-bialgebra, [Formula: see text] a monoidal Hom-algebra and a monoidal Hom-coalgebra. Let [Formula: see text] and [Formula: see text] be two linear maps. First, we construct the [Formula: see text]-smash product monoidal Hom-algebra [Formula: see text] and [Formula: see text]-smash coproduct monoidal Hom-coalgebra [Formula: see text]. Second, the necessary and sufficient conditions for [Formula: see text] and [Formula: see text] to be a monoidal Hom-bialgebra are obtained, which generalizes the results in [8, 11]. Lastly, we give some examples and applications.

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5

Fa, Huanxia, Junbo Li, and Bin Xin. "Lie Super-bialgebra Structures on the Centerless Twisted N=2 Super-conformal Algebra." Algebra Colloquium 18, no. 03 (September 2011): 361–72. http://dx.doi.org/10.1142/s1005386711000253.

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6

Fa, Huanxia, Meijun Li, and Junbo Li. "The deformed twisted Heisenberg–Virasoro type Lie bialgebra." Communications in Algebra 48, no. 6 (February 9, 2020): 2713–22. http://dx.doi.org/10.1080/00927872.2020.1722824.

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7

Fa, Huanxia, Yanjie Li, and Junbo Li. "Schrödinger-Virasoro type Lie bialgebra: a twisted case." Frontiers of Mathematics in China 6, no. 4 (February 19, 2011): 641–57. http://dx.doi.org/10.1007/s11464-011-0105-1.

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8

Liu, Dong, Yufeng Pei, and Linsheng Zhu. "Lie bialgebra structures on the twisted Heisenberg–Virasoro algebra." Journal of Algebra 359 (June 2012): 35–48. http://dx.doi.org/10.1016/j.jalgebra.2012.03.009.

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9

Yang, Shilin, and Yongfeng Zhang. "Ore Extensions for the Sweedler’s Hopf Algebra H4." Mathematics 8, no. 8 (August 5, 2020): 1293. http://dx.doi.org/10.3390/math8081293.

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Анотація:

The aim of this paper is to classify all Hopf algebra structures on the quotient of Ore extensions H4[z;σ] of automorphism type for the Sweedler′s 4-dimensional Hopf algebra H4. Firstly, we calculate all equivalent classes of twisted homomorphisms (σ,J) for H4. Then we give the classification of all bialgebra (Hopf algebra) structures on the quotients of H4[z;σ] up to isomorphism.

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10

Liu, Ling, Abdenacer Makhlouf, Claudia Menini, and Florin Panaite. "-Rota–Baxter Operators, Infinitesimal Hom-bialgebras and the Associative (Bi)Hom-Yang–Baxter Equation." Canadian Mathematical Bulletin 62, no. 02 (January 7, 2019): 355–72. http://dx.doi.org/10.4153/cmb-2018-028-8.

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Анотація:

AbstractWe introduce the concept of a $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$ -Rota–Baxter operator, as a twisted version of a Rota–Baxter operator of weight zero. We show how to obtain a certain $\{\unicode[STIX]{x1D70E},\unicode[STIX]{x1D70F}\}$ -Rota–Baxter operator from a solution of the associative (Bi)Hom-Yang–Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.

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11

MAKHLOUF, ABDENACER, and SERGEI SILVESTROV. "HOM-ALGEBRAS AND HOM-COALGEBRAS." Journal of Algebra and Its Applications 09, no. 04 (August 2010): 553–89. http://dx.doi.org/10.1142/s0219498810004117.

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Анотація:

The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom–Lie algebras, we describe the notion and some properties of Hom-algebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and Hom–Hopf algebras, and prove some fundamental properties and give examples. Finally, we define the concept of Hom–Lie admissible Hom-coalgebra and provide their classification based on subgroups of the symmetric group.

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12

Kang, Chuangchuang, Guilai Liu, Zhuo Wang, and Shizhuo Yu. "Manin Triples and Bialgebras of Left-Alia Algebras Associated with Invariant Theory." Mathematics 12, no. 3 (January 26, 2024): 408. http://dx.doi.org/10.3390/math12030408.

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Анотація:

A left-Alia algebra is a vector space together with a bilinear map satisfying the symmetric Jacobi identity. Motivated by invariant theory, we first construct a class of left-Alia algebras induced by twisted derivations. Then, we introduce the notions of Manin triples and bialgebras of left-Alia algebras. Via specific matched pairs of left-Alia algebras, we figure out the equivalence between Manin triples and bialgebras of left-Alia algebras.

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13

Patras, F., and C. Reutenauer. "On Descent Algebras and Twisted Bialgebras." Moscow Mathematical Journal 4, no. 1 (2004): 199–216. http://dx.doi.org/10.17323/1609-4514-2004-4-1-199-216.

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14

Giaquinto, Anthony, and James J. Zhang. "Bialgebra actions, twists, and universal deformation formulas." Journal of Pure and Applied Algebra 128, no. 2 (June 1998): 133–51. http://dx.doi.org/10.1016/s0022-4049(97)00041-8.

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15

Zhang, Xiaohui, Wei Wang, and Xiaofan Zhao. "Drinfeld twists for monoidal Hom-bialgebras." Colloquium Mathematicum 156, no. 2 (2019): 199–228. http://dx.doi.org/10.4064/cm7359-4-2018.

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16

Ocal, Pablo S., and Amrei Oswald. "A dichotomy between twisted tensor products of bialgebras and Frobenius algebras." Journal of Algebra 644 (April 2024): 351–80. http://dx.doi.org/10.1016/j.jalgebra.2023.12.039.

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17

Abe, Yasuhiro. "Holonomies of gauge fields in twistor space 1: Bialgebra, supersymmetry, and gluon amplitudes." Nuclear Physics B 825, no. 1-2 (January 2010): 242–67. http://dx.doi.org/10.1016/j.nuclphysb.2009.09.026.

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18

Abedin, Raschid, and Stepan Maximov. "Classification of classical twists of the standard Lie bialgebra structure on a loop algebra." Journal of Geometry and Physics 164 (June 2021): 104149. http://dx.doi.org/10.1016/j.geomphys.2021.104149.

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19

Halbout, Gilles. "Formality theorem for Lie bialgebras and quantization of twists and coboundary r-matrices." Advances in Mathematics 207, no. 2 (December 2006): 617–33. http://dx.doi.org/10.1016/j.aim.2005.12.006.

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20

Song, Guang’ai, Yucai Su, and Xiaoqing Yue. "Dual Lie bialgebra structures of the twisted Heisenberg–Virasoro type." Journal of Algebra and Its Applications, November 15, 2022. http://dx.doi.org/10.1142/s0219498824500300.

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Анотація:

In this paper, by studying the maximal good subspaces, we determine the dual Lie coalgebras of the centerless twisted Heisenberg–Virasoro algebra. Based on this, we construct the dual Lie bialgebras structures of the twisted Heisenberg–Virasoro type. As by-products, four new infinite dimensional Lie algebras are obtained.

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21

Abedin, Raschid, and Igor Burban. "Algebraic Geometry of Lie Bialgebras Defined by Solutions of the Classical Yang–Baxter Equation." Communications in Mathematical Physics, September 4, 2021. http://dx.doi.org/10.1007/s00220-021-04188-7.

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AbstractThis paper is devoted to algebro-geometric study of infinite dimensional Lie bialgebras, which arise from solutions of the classical Yang–Baxter equation. We regard trigonometric solutions of this equation as twists of the standard Lie bialgebra cobracket on an appropriate affine Lie algebra and work out the corresponding theory of Manin triples, putting it into an algebro-geometric context. As a consequence of this approach, we prove that any trigonometric solution of the classical Yang–Baxter equation arises from an appropriate algebro-geometric datum. The developed theory is illustrated by some concrete examples.

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22

Chen, Hank. "Drinfel’d double symmetry of the 4d Kitaev model." Journal of High Energy Physics 2023, no. 9 (September 21, 2023). http://dx.doi.org/10.1007/jhep09(2023)141.

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Abstract Following the general theory of categorified quantum groups developed by the author previously, we construct the Drinfel’d double 2-bialgebra associated to a finite group N = G0. For N = ℤ2, we explicitly compute the braided 2-categories of 2-representations of certain version of this Drinfel’d double 2-bialgebra, and prove that they characterize precisely the 4d toric code and its spin-ℤ2 variant. This result relates the two descriptions (categorical vs. field theoretical) of 4d gapped topological phases in existing literature and displays an instance of higher Tannakian duality for braided 2-categories. In particular, we show that particular twists of the underlying Drinfel’d double 2-bialgebra is responsible for much of the higher-structural properties that arise in 4d topological orders.

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23

"Polyadic Hopf Algebras and Quantum Groups." 2, no. 2 (2021). http://dx.doi.org/10.26565/2312-4334-2021-2-01.

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This article continues the study of concrete algebra-like structures in our polyadic approach, where the arities of all operations are initially taken as arbitrary, but the relations between them, the arity shapes, are to be found from some natural conditions (“arity freedom principle”). In this way, generalized associative algebras, coassociative coalgebras, bialgebras and Hopf algebras are defined and investigated. They have many unusual features in comparison with the binary case. For instance, both the algebra and its underlying field can be zeroless and nonunital, the existence of the unit and counit is not obligatory, and the dimension of the algebra is not arbitrary, but “quantized”. The polyadic convolution product and bialgebra can be defined, and when the algebra and coalgebra have unequal arities, the polyadic version of the antipode, the querantipode, has different properties. As a possible application to quantum group theory, we introduce the polyadic version of braidings, almost co-commutativity, quasitriangularity and the equations for the R-matrix (which can be treated as a polyadic analog of the Yang-Baxter equation). We propose another concept of deformation which is governed not by the twist map, but by the medial map, where only the latter is unique in the polyadic case. We present the corresponding braidings, almost co-mediality and M-matrix, for which the compatibility equations are found.

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24

Doikou, Anastasia, Alexandros Ghionis, and Bart Vlaar. "Quasi-bialgebras from set-theoretic type solutions of the Yang–Baxter equation." Letters in Mathematical Physics 112, no. 4 (August 2022). http://dx.doi.org/10.1007/s11005-022-01572-9.

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AbstractWe examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang–Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists, we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fact quasi-triangular quasi-bialgebras. Specific illustrative examples compatible with our generic findings are worked out. In the q-deformed case of set-theoretic solutions, we also construct admissible Drinfeld twists similar to the set-theoretic ones, subject to certain extra constraints dictated by the q-deformation. These findings greatly generalize recent relevant results on set-theoretic solutions and their q-deformed analogues.

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25

Borowiec, Andrzej, Jerzy Kowalski-Glikman та Josua Unger. "3-dimensional Λ-BMS symmetry and its deformations". Journal of High Energy Physics 2021, № 11 (листопад 2021). http://dx.doi.org/10.1007/jhep11(2021)103.

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Abstract In this paper we study quantum group deformations of the infinite dimensional symmetry algebra of asymptotically AdS spacetimes in three dimensions. Building on previous results in the finite dimensional subalgebras we classify all possible Lie bialgebra structures and for selected examples we explicitely construct the related Hopf algebras. Using cohomological arguments we show that this construction can always be performed by a so-called twist deformation. The resulting structures can be compared to the well-known κ-Poincaré Hopf algebras constructed on the finite dimensional Poincaré or (anti) de Sitter algebra. The dual κ Minkowski spacetime is supposed to describe a specific non-commutative geometry. Importantly, we find that some incarnations of the κ-Poincaré can not be extended consistently to the infinite dimensional algebras. Furthermore, certain deformations can have potential physical applications if subalgebras are considered. Since the conserved charges associated with asymptotic symmetries in 3-dimensional form a centrally extended algebra we also discuss briefly deformations of such algebras. The presence of the full symmetry algebra might have observable consequences that could be used to rule out these deformations.

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