## HOM-YANG-BAXTER EQUATION, HOM-LIE ALGEBRAS, AND QUASI-TRIANGULAR BIALGEBRAS (903)

Citations: | 9 - 5 self |

### BibTeX

@MISC{Yau903hom-yang-baxterequation,,

author = {Donald Yau},

title = {HOM-YANG-BAXTER EQUATION, HOM-LIE ALGEBRAS, AND QUASI-TRIANGULAR BIALGEBRAS},

year = {903}

}

### OpenURL

### Abstract

Abstract. We study a twisted version of the Yang-Baxter Equation, called the Hom-Yang-Baxter Equation (HYBE), which is motivated by Hom-Lie algebras. Three classes of solutions of the HYBE are constructed, one from Hom-Lie algebras and the others from Drinfeld’s (dual) quasitriangular bialgebras. Each solution of the HYBE can be extended to operators that satisfy the braid relations. Assuming an invertibility condition, these operators give a representation of the braid group. 1.

### Citations

1476 |
Exactly Solved Models in Statistical Mechanics
- Baxter
- 1982
(Show Context)
Citation Context ...relations. Assuming an invertibility condition, these operators give a representation of the braid group. 1. Introduction The Yang-Baxter Equation (YBE) originated in the work of Yang [24] and Baxter =-=[3, 4]-=- in statistical mechanics. Let V be a vector space, and let B: V ⊗V → V ⊗V be a linear automorphism. Then B is said to be an R-matrix if it satisfies the YBE: (IdV ⊗ B) ◦ (B ⊗ IdV ) ◦ (IdV ⊗ B) = (B ⊗... |

810 |
Quantum groups
- Drinfeld
- 1986
(Show Context)
Citation Context ...bras, which include many examples of quantum groups, and the braid group, among other topics. It is known that every (co)module M over a (dual) quasitriangular bialgebra H gives a solution of the YBE =-=[5, 6, 11]-=-. Also, every Lie algebra L gives a solution of the YBE [2]. Moreover, each solution of the YBE gives a representation of the braid group on n strands. We will study a twisted version of the YBE, whic... |

338 |
Faddeev, Quantization of Lie Groups and Lie Algebras
- Reshetikhin, D
- 1990
(Show Context)
Citation Context ... is well-known that B R is a solution of the YBE (1.0.1); see, e.g., [11, Proposition VIII.5.2]. This gives another systematic way to produce solutions of the YBE. Conversely, by the FRT construction =-=[20]-=-, every R-matrix for a finite dimensional vector space M has the form B R for someHOM-YANG-BAXTER EQUATION, HOM-LIE ALGEBRAS, AND QUASI-TRIANGULAR BIALGEBRAS 3 dual quasi-triangular bialgebra H and s... |

107 |
Foundations of Quantum Group Theory,” Cambridge Univ
- Majid
(Show Context)
Citation Context ...gebra, in which µ: H ⊗ H → H is the associative multiplication, η: k → H is the unit, ∆: H → H ⊗ H is the coassociative comultiplication, and ε: H → k is the counit. A quasi-triangular structure on H =-=[5, 6, 11, 15]-=- is an invertible element R ∈ H ⊗ H such that (τ ◦ ∆)(x) = R∆(x)R −1 □ □ (4.1.1) for x ∈ H, (∆ ⊗ IdH)(R) = R13R23, and (IdH ⊗ ∆)(R) = R13R12. (4.1.2) Here, if R = ∑ i si ⊗ ti, then R12 = ∑ si ⊗ ti ⊗ 1... |

92 |
Some exact results for the many-body problem in one dimension with repulsive deltafunction interactions
- Yang
- 1967
(Show Context)
Citation Context ...tisfy the braid relations. Assuming an invertibility condition, these operators give a representation of the braid group. 1. Introduction The Yang-Baxter Equation (YBE) originated in the work of Yang =-=[24]-=- and Baxter [3, 4] in statistical mechanics. Let V be a vector space, and let B: V ⊗V → V ⊗V be a linear automorphism. Then B is said to be an R-matrix if it satisfies the YBE: (IdV ⊗ B) ◦ (B ⊗ IdV ) ... |

43 |
Theory of braids, Ann
- Artin
(Show Context)
Citation Context ...sfy the braid relations, which can then be used to construct representations of the braid group. We extend this construction to solutions of the HYBE. Let n ≥ 3 and Bn be the braid group on n strands =-=[1]-=-. The braid group Bn has generators σi (1 ≤ i ≤ n − 1), which satisfy the determining braid relations: σiσj = σjσi if |i − j| > 1 and σiσi+1σi = σi+1σiσi+1. (1.3.1) The following result, which will be... |

35 |
Deformations of Lie algebras using σ-derivations
- Hartwig, Larsson, et al.
(Show Context)
Citation Context ...y) and that the following Hom-Jacobi identity holds: [[x, y], α(z)] + [[z, x], α(y)] + [[y, z], α(x)]] = 0. (1.0.2) A Lie algebra is a Hom-Lie algebra with α = Id. Hom-Lie algebras were introduced in =-=[7]-=- (without multiplicativity) to describe the structures on certain q-deformations of the Witt and the Virasoro algebras. Earlier precursors of Hom-Lie algebras can be found in [10, 13]. Other classes o... |

33 | Two dual classes of bialgebras related to the concept of “quantum groups” and “quantum Lie algebra - Larson, Towber |

26 |
Representations, duals and quantum doubles of monoidal categories
- Majid
- 1991
(Show Context)
Citation Context ...le, and α: M → M be an H-module morphism. Then the map BR (1.1.2) is a solution of the HYBE (1.0.3) for (M, α). Dual to a quasi-triangular bialgebra is the notion of a dual quasi-triangular bialgebra =-=[8, 12, 14, 21]-=-. It consists of a bialgebra H and a dual quasi-triangular structure R ∈ Hom(H ⊗ H,k). The exact definition of a dual quasi-triangular bialgebra will be recalled in Section 5. Let (H, R) be a dual qua... |

20 |
Quantum groups and quantum determinants
- Hayashi
- 1992
(Show Context)
Citation Context ...le, and α: M → M be an H-module morphism. Then the map BR (1.1.2) is a solution of the HYBE (1.0.3) for (M, α). Dual to a quasi-triangular bialgebra is the notion of a dual quasi-triangular bialgebra =-=[8, 12, 14, 21]-=-. It consists of a bialgebra H and a dual quasi-triangular structure R ∈ Hom(H ⊗ H,k). The exact definition of a dual quasi-triangular bialgebra will be recalled in Section 5. Let (H, R) be a dual qua... |

19 |
On coquasitriangular Hopf algebras and the quantum Yang-Baxter equation, Algebra Berichte 67, Verlag Reinhard Fischer
- Schauenburg
- 1992
(Show Context)
Citation Context ...le, and α: M → M be an H-module morphism. Then the map BR (1.1.2) is a solution of the HYBE (1.0.3) for (M, α). Dual to a quasi-triangular bialgebra is the notion of a dual quasi-triangular bialgebra =-=[8, 12, 14, 21]-=-. It consists of a bialgebra H and a dual quasi-triangular structure R ∈ Hom(H ⊗ H,k). The exact definition of a dual quasi-triangular bialgebra will be recalled in Section 5. Let (H, R) be a dual qua... |

18 | Hom-algebra structures
- Makhlouf, Silvestrov
(Show Context)
Citation Context ... the structures on certain q-deformations of the Witt and the Virasoro algebras. Earlier precursors of Hom-Lie algebras can be found in [10, 13]. Other classes of Hom-Lie algebras were constructed in =-=[16, 26]-=-. We will describe some of these Hom-Lie algebras in Section 3. If one considers a Hom-Lie algebra as an α-twisted version of a Lie algebra, then there should be a corresponding twisted YBE. To state ... |

17 |
Notes on Formal Deformations of Hom-associative and HomLie Algebras, o appear
- Makhlouf, Silvestrov
(Show Context)
Citation Context ...ts universal enveloping Hom-associative algebra U(L) has the structure of a Hom-bialgebra. Besides the sources cited above, other papers that discuss Hom-Lie algebras and related Hom-algebras include =-=[7, 17, 18, 19, 26]-=-. □ Example 3.2 (Hom-Lie algebras as deformations of Lie algebras). Another systematic way to obtain Hom-Lie algebras is by deforming Lie algebras along endomorphisms. Let (L, [−, −]) be a Lie algebra... |

17 | Enveloping algebra of Hom-Lie algebras
- Yau
(Show Context)
Citation Context ...r, we point out that, similar to the universal enveloping algebra of a Lie algebra, there is a universal enveloping Hom-associative algebra functor U from Hom-Lie algebras to Hom-associative algebras =-=[25, 27]-=-. The ordinary enveloping algebra of a Lie algebra is a bialgebra. Similarly, one can define a Hom-bialgebra by dualizing and extending the definition of a Hom-associative algebra. It is shown in [27,... |

15 |
Higher-dimensional algebra VI
- Baez, Crans
(Show Context)
Citation Context ... group, among other topics. It is known that every (co)module M over a (dual) quasitriangular bialgebra H gives a solution of the YBE [5, 6, 11]. Also, every Lie algebra L gives a solution of the YBE =-=[2]-=-. Moreover, each solution of the YBE gives a representation of the braid group on n strands. We will study a twisted version of the YBE, which is motivated by Hom-Lie algebras. A Hom-Lie algebra L has... |

15 | Hom-Lie admissible Hom-coalgebras and Hom-Hopf algebras, Published as Chapter 17, pp 189-206
- Makhlouf, Silvestrov
- 2008
(Show Context)
Citation Context ...ts universal enveloping Hom-associative algebra U(L) has the structure of a Hom-bialgebra. Besides the sources cited above, other papers that discuss Hom-Lie algebras and related Hom-algebras include =-=[7, 17, 18, 19, 26]-=-. □ Example 3.2 (Hom-Lie algebras as deformations of Lie algebras). Another systematic way to obtain Hom-Lie algebras is by deforming Lie algebras along endomorphisms. Let (L, [−, −]) be a Lie algebra... |

13 |
Characterizations of quantum Witt algebra
- Liu
- 1992
(Show Context)
Citation Context ...as were introduced in [7] (without multiplicativity) to describe the structures on certain q-deformations of the Witt and the Virasoro algebras. Earlier precursors of Hom-Lie algebras can be found in =-=[10, 13]-=-. Other classes of Hom-Lie algebras were constructed in [16, 26]. We will describe some of these Hom-Lie algebras in Section 3. If one considers a Hom-Lie algebra as an α-twisted version of a Lie alge... |

12 |
q-Witt algebras, q-Lie algebras, q-holomorph structure and representations
- Hu
- 1999
(Show Context)
Citation Context ...as were introduced in [7] (without multiplicativity) to describe the structures on certain q-deformations of the Witt and the Virasoro algebras. Earlier precursors of Hom-Lie algebras can be found in =-=[10, 13]-=-. Other classes of Hom-Lie algebras were constructed in [16, 26]. We will describe some of these Hom-Lie algebras in Section 3. If one considers a Hom-Lie algebra as an α-twisted version of a Lie alge... |

11 |
Hom-algebras and homology
- Yau
- 2009
(Show Context)
Citation Context ... the structures on certain q-deformations of the Witt and the Virasoro algebras. Earlier precursors of Hom-Lie algebras can be found in [10, 13]. Other classes of Hom-Lie algebras were constructed in =-=[16, 26]-=-. We will describe some of these Hom-Lie algebras in Section 3. If one considers a Hom-Lie algebra as an α-twisted version of a Lie algebra, then there should be a corresponding twisted YBE. To state ... |

8 |
Solutions of the braid equation related to Hopf algebras
- Woronowicz
- 1991
(Show Context)
Citation Context ...[5, 6, 11]. This gives an efficient and systematic way to produce solutions of the YBE. Particular examples of solutions of the YBE arising this way include the Woronowicz operators on a Hopf algebra =-=[23]-=-, as shown in [9]. The following generalization will be proved in Section 4. Theorem 1.2. Let (H, R) be a quasi-triangular bialgebra, M be an H-module, and α: M → M be an H-module morphism. Then the m... |

4 |
Hom-bialgebras and comodule algebras, arXiv:0810.4866
- Yau
(Show Context)
Citation Context ...r, we point out that, similar to the universal enveloping algebra of a Lie algebra, there is a universal enveloping Hom-associative algebra functor U from Hom-Lie algebras to Hom-associative algebras =-=[25, 27]-=-. The ordinary enveloping algebra of a Lie algebra is a bialgebra. Similarly, one can define a Hom-bialgebra by dualizing and extending the definition of a Hom-associative algebra. It is shown in [27,... |

3 |
Partition function fo the eight-vertex lattice model
- Baxter
- 1972
(Show Context)
Citation Context ...relations. Assuming an invertibility condition, these operators give a representation of the braid group. 1. Introduction The Yang-Baxter Equation (YBE) originated in the work of Yang [24] and Baxter =-=[3, 4]-=- in statistical mechanics. Let V be a vector space, and let B: V ⊗V → V ⊗V be a linear automorphism. Then B is said to be an R-matrix if it satisfies the YBE: (IdV ⊗ B) ◦ (B ⊗ IdV ) ◦ (IdV ⊗ B) = (B ⊗... |

2 | On almost cocommutative Hopf algebras, Alg. i Analiz 1:2 - Drinfel’d - 1989 |

2 |
On solutions to the braid equation identified by Woronowicz
- Hennings
- 1993
(Show Context)
Citation Context ...ives an efficient and systematic way to produce solutions of the YBE. Particular examples of solutions of the YBE arising this way include the Woronowicz operators on a Hopf algebra [23], as shown in =-=[9]-=-. The following generalization will be proved in Section 4. Theorem 1.2. Let (H, R) be a quasi-triangular bialgebra, M be an H-module, and α: M → M be an H-module morphism. Then the map BR (1.1.2) is ... |