## The Hom-Yang-Baxter equation and Hom-Lie algebras (2009)

Citations: | 5 - 3 self |

### BibTeX

@MISC{Yau09thehom-yang-baxter,

author = {Donald Yau},

title = {The Hom-Yang-Baxter equation and Hom-Lie algebras},

year = {2009}

}

### OpenURL

### Abstract

Abstract. Motivated by recent work on Hom-Lie algebras, a twisted version of the Yang-Baxter equation, called the Hom-Yang-Baxter equation (HYBE), was introduced by the author in [62]. In this paper, several more classes of solutions of the HYBE are constructed. Some of these solutions of the HYBE are closely related to the quantum enveloping algebra of sl(2), the Jones-Conway polynomial, and Yetter-Drinfel’d modules. We also construct a new infinite sequence of solutions of the HYBE from a given one. Along the way, we compute all the Lie algebra endomorphisms on the (1 + 1)-Poincaré algebra and sl(2). 1.

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Citation Context ...where V is a vector space and B: V ⊗2 → V ⊗2 is a bilinear automorphism. A solution of the YBE is also called an R-matrix (on V ). The YBE was first introduced in the context of statistical mechanics =-=[5, 6, 57]-=-. It plays an important role in many topics in mathematical physics, including quantum groups, quantum integrable systems [11], braided categories [30, 31, 32], the Zamolodchikov tetrahedron equation ... |

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Citation Context ... invariants of knots and links [56, 64], among others. Many R-matrices are known. In particular, various classes of quantum groups were introduced precisely for the purpose of constructing R-matrices =-=[8, 9, 10, 18, 34, 37, 44, 53]-=-. Among the many classes of R-matrices, a particularly interesting class comes from Lie algebras [4]. Thinking of Hom-Lie algebras as α-twisted analogues of Lie algebras, this raises the question: Wha... |

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Citation Context ...v1 in V1, then Φq,λ = B q −1 ,qλ. The general case of the R-matrix Bq,λ is a crucial ingredient in establishing the existence of the polynomial invariant of links known as the Jones-Conway polynomial =-=[7, 14, 28, 29]-=-, as discussed in [34, X.4 and XII.5]. The following result, which generalizes Theorem 1.1 to higher dimensions, describes all the linear maps that are compatible with Bq,λ and their induced solutions... |

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Citation Context ...ed in the context of statistical mechanics [5, 6, 57]. It plays an important role in many topics in mathematical physics, including quantum groups, quantum integrable systems [11], braided categories =-=[30, 31, 32]-=-, the Zamolodchikov tetrahedron equation in higher-dimensional categories [4, 33], and invariants of knots and links [56, 64], among others. Many R-matrices are known. In particular, various classes o... |

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Citation Context ...s. As the first example, consider the (unique up to isomorphism) two-dimensional simple Uq(sl(2))module V1. Here q ∈ C�{0, ±1} and Uq(sl(2)) is the quantum enveloping algebra of the Lie algebra sl(2) =-=[8, 9, 24, 34, 35, 45]-=-. We do not need to know the structure of Uq(sl(2)) for the discussion below. There is a basis {v0, v1} of V1 in which v0 is a highest weight vector. With respect to the basis {v0 ⊗ v0, v0 ⊗ v1, v1 ⊗ ... |

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Citation Context ...= λej ⊗ ei if i < j, ⎪⎩ λej ⊗ ei + λ(q − q−1 (1.1.2) )ei ⊗ ej if i > j. It is known that Bq,λ is an R-matrix on V [34, VIII.1.4]. The R-matrix Bq−1 ,q is known as the Jimbo operator of type A (1) N−1 =-=[24, 25, 26]-=-. Switching q and q−1 , the R-matrix B q,q −1 is used in constructing the quantum exterior algebra [23, section 3]. Moreover, the case N = 2 contains the R-matrix Φq,λ (1.0.7). In fact, if we switch t... |

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Citation Context ...v1 in V1, then Φq,λ = B q −1 ,qλ. The general case of the R-matrix Bq,λ is a crucial ingredient in establishing the existence of the polynomial invariant of links known as the Jones-Conway polynomial =-=[7, 14, 28, 29]-=-, as discussed in [34, X.4 and XII.5]. The following result, which generalizes Theorem 1.1 to higher dimensions, describes all the linear maps that are compatible with Bq,λ and their induced solutions... |

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Citation Context ...s. As the first example, consider the (unique up to isomorphism) two-dimensional simple Uq(sl(2))module V1. Here q ∈ C�{0, ±1} and Uq(sl(2)) is the quantum enveloping algebra of the Lie algebra sl(2) =-=[8, 9, 24, 34, 35, 45]-=-. We do not need to know the structure of Uq(sl(2)) for the discussion below. There is a basis {v0, v1} of V1 in which v0 is a highest weight vector. With respect to the basis {v0 ⊗ v0, v0 ⊗ v1, v1 ⊗ ... |

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Citation Context ...where V is a vector space and B: V ⊗2 → V ⊗2 is a bilinear automorphism. A solution of the YBE is also called an R-matrix (on V ). The YBE was first introduced in the context of statistical mechanics =-=[5, 6, 57]-=-. It plays an important role in many topics in mathematical physics, including quantum groups, quantum integrable systems [11], braided categories [30, 31, 32], the Zamolodchikov tetrahedron equation ... |

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Citation Context ...infel’d modules and are valid over any field k of characteristic 0. We will produce a family of solutions of the HYBE from each Yetter-Drinfel’d module. A Yetter-Drinfel’d module V over a bialgebra H =-=[51, 65]-=- consists of (i) a left H-module structure on V and (ii) a left H-comodule structure on V (written ρ(v) = ∑ v−1 ⊗v0) such that the Yetter-Drinfel’d condition ∑ x1v−1 ⊗ x2 · v0 = ∑ (x1 · v)−1x2 ⊗ (x1 ·... |

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Citation Context ...he YBE was first introduced in the context of statistical mechanics [5, 6, 57]. It plays an important role in many topics in mathematical physics, including quantum groups, quantum integrable systems =-=[11]-=-, braided categories [30, 31, 32], the Zamolodchikov tetrahedron equation in higher-dimensional categories [4, 33], and invariants of knots and links [56, 64], among others. Many R-matrices are known.... |

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Citation Context ...v1 in V1, then Φq,λ = B q −1 ,qλ. The general case of the R-matrix Bq,λ is a crucial ingredient in establishing the existence of the polynomial invariant of links known as the Jones-Conway polynomial =-=[7, 14, 28, 29]-=-, as discussed in [34, X.4 and XII.5]. The following result, which generalizes Theorem 1.1 to higher dimensions, describes all the linear maps that are compatible with Bq,λ and their induced solutions... |

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Citation Context ...he element representing crossing the ith and the (i + 1)st strands with the former going under the latter. The elements σi (1 ≤ i ≤ n − 1) generate Bn and satisfy the defining braid relations (1.0.5) =-=[1, 2]-=-. If both α and B are invertible, then clearly so are the Bi (1.0.4). In this case, there is a unique group morphism ρ B n : Bn → Aut(V ⊗n ) satisfying ρ B n (σi) = Bi. This generalizes the usual brai... |

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Citation Context ... this paper is to construct some concrete classes of solutions of the Hom-YangBaxter equation (HYBE), which was introduced by the author in [62]. Let us first recall some motivations for the HYBE. In =-=[16]-=- a generalization of Lie algebras, called Hom-Lie algebras, were introduced in which the Jacobi identity is twisted by a linear self-map. More precisely, a Hom-Lie algebra L = (L, [−, −], α) consists ... |

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Citation Context ... invariants of knots and links [56, 64], among others. Many R-matrices are known. In particular, various classes of quantum groups were introduced precisely for the purpose of constructing R-matrices =-=[8, 9, 10, 18, 34, 37, 44, 53]-=-. Among the many classes of R-matrices, a particularly interesting class comes from Lie algebras [4]. Thinking of Hom-Lie algebras as α-twisted analogues of Lie algebras, this raises the question: Wha... |

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Citation Context ...ed in the context of statistical mechanics [5, 6, 57]. It plays an important role in many topics in mathematical physics, including quantum groups, quantum integrable systems [11], braided categories =-=[30, 31, 32]-=-, the Zamolodchikov tetrahedron equation in higher-dimensional categories [4, 33], and invariants of knots and links [56, 64], among others. Many R-matrices are known. In particular, various classes o... |

30 |
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Citation Context ...lication. In the special case that H is a finite dimensional Hopf algebra, the category of Yetter-Drinfel’d modules over H is equivalent to the category of left modules over the Drinfel’d double of H =-=[9, 43]-=-. Using the Yetter-Drinfel’d condition (1.5.5), a direct computation shows that each Yetter-Drinfel’d module V over a bialgebra H has an associated R-matrix [36, 50] given by B(v ⊗ w) = ∑ v−1 · w ⊗ v0... |

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Citation Context ...he element representing crossing the ith and the (i + 1)st strands with the former going under the latter. The elements σi (1 ≤ i ≤ n − 1) generate Bn and satisfy the defining braid relations (1.0.5) =-=[1, 2]-=-. If both α and B are invertible, then clearly so are the Bi (1.0.4). In this case, there is a unique group morphism ρ B n : Bn → Aut(V ⊗n ) satisfying ρ B n (σi) = Bi. This generalizes the usual brai... |

28 |
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Citation Context ...infel’d modules and are valid over any field k of characteristic 0. We will produce a family of solutions of the HYBE from each Yetter-Drinfel’d module. A Yetter-Drinfel’d module V over a bialgebra H =-=[51, 65]-=- consists of (i) a left H-module structure on V and (ii) a left H-comodule structure on V (written ρ(v) = ∑ v−1 ⊗v0) such that the Yetter-Drinfel’d condition ∑ x1v−1 ⊗ x2 · v0 = ∑ (x1 · v)−1x2 ⊗ (x1 ·... |

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Citation Context ...l(γ) is also equal to the number of pairs i < j for which γ(i) > γ(j). A decomposition γ = γ1 · · · γk with each γi ∈ Σn is called reduced if l(γ) = l(γ1)+···+l(γk). By a well-known result of Iwahori =-=[21]-=-, there is a well-defined map θ: Σn → Bn given by θ(γ) = σi1 · · · σi l(γ) , where γ = τi1 · · · τi l(γ) is any reduced decomposition of γ in terms of transpositions. Now let B: V ⊗2 → V ⊗2 be a solut... |

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Citation Context ... invariants of knots and links [56, 64], among others. Many R-matrices are known. In particular, various classes of quantum groups were introduced precisely for the purpose of constructing R-matrices =-=[8, 9, 10, 18, 34, 37, 44, 53]-=-. Among the many classes of R-matrices, a particularly interesting class comes from Lie algebras [4]. Thinking of Hom-Lie algebras as α-twisted analogues of Lie algebras, this raises the question: Wha... |

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Citation Context ...s. As the first example, consider the (unique up to isomorphism) two-dimensional simple Uq(sl(2))module V1. Here q ∈ C�{0, ±1} and Uq(sl(2)) is the quantum enveloping algebra of the Lie algebra sl(2) =-=[8, 9, 24, 34, 35, 45]-=-. We do not need to know the structure of Uq(sl(2)) for the discussion below. There is a basis {v0, v1} of V1 in which v0 is a highest weight vector. With respect to the basis {v0 ⊗ v0, v0 ⊗ v1, v1 ⊗ ... |

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Citation Context ...introduced in [16] to describe the structures on some q-deformations of the Witt and the Virasoro algebras. They are also closely related to discrete and deformed vector fields, differential calculus =-=[16, 39, 40, 52, 54]-=-, and number theory [38]. Earlier precursors of Hom-Lie algebras can be found in [19, 42]. Hom-Lie algebras and related algebraic structures have been further studied in [3, 12, 13, 27, 41, 46, 47, 48... |

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Citation Context ...ules over the Drinfel’d double of H [9, 43]. Using the Yetter-Drinfel’d condition (1.5.5), a direct computation shows that each Yetter-Drinfel’d module V over a bialgebra H has an associated R-matrix =-=[36, 50]-=- given by B(v ⊗ w) = ∑ v−1 · w ⊗ v0 (1.5.6) for v, w ∈ V . The following result shows that this operator B is also a solution of the HYBE for (V, α), provided that α is compatible with the H-(co)modul... |

18 | Hom-algebra structures
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Citation Context ... calculus [16, 39, 40, 52, 54], and number theory [38]. Earlier precursors of Hom-Lie algebras can be found in [19, 42]. Hom-Lie algebras and related algebraic structures have been further studied in =-=[3, 12, 13, 27, 41, 46, 47, 48, 49, 58, 59, 60, 61, 62, 63]-=-. Recall that the Yang-Baxter equation (YBE) states (IdV ⊗ B) ◦ (B ⊗ IdV ) ◦ (IdV ⊗ B) = (B ⊗ IdV ) ◦ (IdV ⊗ B) ◦ (B ⊗ IdV ), (1.0.2) where V is a vector space and B: V ⊗2 → V ⊗2 is a bilinear automor... |

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Citation Context ... calculus [16, 39, 40, 52, 54], and number theory [38]. Earlier precursors of Hom-Lie algebras can be found in [19, 42]. Hom-Lie algebras and related algebraic structures have been further studied in =-=[3, 12, 13, 27, 41, 46, 47, 48, 49, 58, 59, 60, 61, 62, 63]-=-. Recall that the Yang-Baxter equation (YBE) states (IdV ⊗ B) ◦ (B ⊗ IdV ) ◦ (IdV ⊗ B) = (B ⊗ IdV ) ◦ (IdV ⊗ B) ◦ (B ⊗ IdV ), (1.0.2) where V is a vector space and B: V ⊗2 → V ⊗2 is a bilinear automor... |

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Citation Context ... calculus [16, 39, 40, 52, 54], and number theory [38]. Earlier precursors of Hom-Lie algebras can be found in [19, 42]. Hom-Lie algebras and related algebraic structures have been further studied in =-=[3, 12, 13, 27, 41, 46, 47, 48, 49, 58, 59, 60, 61, 62, 63]-=-. Recall that the Yang-Baxter equation (YBE) states (IdV ⊗ B) ◦ (B ⊗ IdV ) ◦ (IdV ⊗ B) = (B ⊗ IdV ) ◦ (IdV ⊗ B) ◦ (B ⊗ IdV ), (1.0.2) where V is a vector space and B: V ⊗2 → V ⊗2 is a bilinear automor... |

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Citation Context ...ebras. They are also closely related to discrete and deformed vector fields, differential calculus [16, 39, 40, 52, 54], and number theory [38]. Earlier precursors of Hom-Lie algebras can be found in =-=[19, 42]-=-. Hom-Lie algebras and related algebraic structures have been further studied in [3, 12, 13, 27, 41, 46, 47, 48, 49, 58, 59, 60, 61, 62, 63]. Recall that the Yang-Baxter equation (YBE) states (IdV ⊗ B... |

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Citation Context ...ebras. They are also closely related to discrete and deformed vector fields, differential calculus [16, 39, 40, 52, 54], and number theory [38]. Earlier precursors of Hom-Lie algebras can be found in =-=[19, 42]-=-. Hom-Lie algebras and related algebraic structures have been further studied in [3, 12, 13, 27, 41, 46, 47, 48, 49, 58, 59, 60, 61, 62, 63]. Recall that the Yang-Baxter equation (YBE) states (IdV ⊗ B... |

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Citation Context ...ules over the Drinfel’d double of H [9, 43]. Using the Yetter-Drinfel’d condition (1.5.5), a direct computation shows that each Yetter-Drinfel’d module V over a bialgebra H has an associated R-matrix =-=[36, 50]-=- given by B(v ⊗ w) = ∑ v−1 · w ⊗ v0 (1.5.6) for v, w ∈ V . The following result shows that this operator B is also a solution of the HYBE for (V, α), provided that α is compatible with the H-(co)modul... |

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Citation Context ...math-ph] 19 May 2009 Abstract. Motivated by recent work on Hom-Lie algebras, a twisted version of the Yang-Baxter equation, called the Hom-Yang-Baxter equation (HYBE), was introduced by the author in =-=[62]-=-. In this paper, several more classes of solutions of the HYBE are constructed. Some of these solutions of the HYBE are closely related to the quantum enveloping algebra of sl(2), the Jones-Conway pol... |

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