Classical Yang–Baxter Equation and the A∞-Constraint

Polishchuk, Alexander

doi:10.1006/aima.2001.2047

Cited by 82 publications

(186 citation statements)

References 20 publications

Supporting

Mentioning

178

Contrasting

Unclassified

Order By: Relevance

“…It is worth mentioning that the elliptic functions appearing in this paper can be interpreted as discrete versions of the solutions of the associative Yang-Baxter equation that was introduced in a recent paper of Polishchuk [15].…”

Section: Introductionmentioning

confidence: 99%

Elliptic functions and equations of modular curves

Borisov

Gunnells²,

Popescu³

2001

Mathematische Annalen

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 99%

Elliptic functions and equations of modular curves

Borisov

Gunnells²,

Popescu³

2001

Mathematische Annalen

View full text Add to dashboard Cite

show abstract

“…This analogy, in fact, underlies the results of papers [22] and [18,19,28]. In this sense the unitarity condition (1.2) is analogue of (A.11).…”

Section: Discussionmentioning

confidence: 60%

“…Associative Yang-Baxter equation was originally introduced in [1] for constant R-matrices and then generalized by Polishchuk [22] to the form:…”

Section: Brief Reviewmentioning

confidence: 99%

See 1 more Smart Citation

Associative Yang-Baxter equation for quantum (semi-)dynamical R-matrices

Сечин

Zotov

2016

Journal of Mathematical Physics

View full text Add to dashboard Cite

In this paper we propose versions of the associative Yang-Baxter equation and higher order R-matrix identities which can be applied to quantum dynamical R-matrices. As is known quantum non-dynamical R-matrices of Baxter-Belavin type satisfy this equation. Together with unitarity condition and skew-symmetry it provides the quantum YangBaxter equation and a set of identities useful for different applications in integrable systems. The dynamical R-matrices satisfy the Gervais-Neveu-Felder (or dynamical Yang-Baxter) equation. Relation between the dynamical and non-dynamical cases is described by the IRF-Vertex transformation. An alternative approach to quantum (semi-)dynamical Rmatrices and related quantum algebras was suggested by Arutyunov, Chekhov and Frolov (ACF) in their study of the quantum Ruijsenaars-Schneider model. The purpose of this paper is twofold. First, we prove that the ACF elliptic R-matrix satisfies the associative Yang-Baxter equation with shifted spectral parameters. Second, we directly prove a simple relation of the IRF-Vertex type between the Baxter-Belavin and the ACF elliptic R-matrices predicted previously by Avan and Rollet. It provides the higher order R-matrix identities and an explanation of the obtained equations through those for non-dynamical R-matrices. As a by-product we also get an interpretation of the intertwining transformation as matrix extension of scalar theta function likewise R-matrix is interpreted as matrix extension of the Kronecker function. Relations to the Gervais-Neveu-Felder equation and identities for the Felder's elliptic R-matrix are also discussed.

show abstract

“…Assume first that n = 2. Then m 3 (α 1 , α 2 , β) is the unique value of the well-defined triple Massey product in D b (C) (see [9], 1.1). Using the standard recipe for its calculation (see [2], IV.2) we immediately get that m 3 (α 1 , α 2 , β) = id.…”

Section: Lemma 32 Let Cmentioning

confidence: 99%

Extensions of homogeneous coordinate rings to $A_{\infty}$-algebras

Polishchuk

2003

Homology, Homotopy and Applications

Self Cite

View full text Add to dashboard Cite

Abstract. We study A ∞ -structures extending the natural algebra structure on the cohomology of ⊕ n∈Z L n , where L is a very ample line bundle on a projective d-dimensional variety X such that H i (X, L n ) = 0 for 0 < i < d and all n ∈ Z. We prove that there exists a unique such nontrivial A ∞ -structure up to a strict A ∞ -isomorphism (i.e., an A ∞ -isomorphism with the identity as the first structure map) and rescaling. In the case when X is a curve we also compute the group of strict A ∞ -automorphisms of this A ∞ -structure.

show abstract

Classical Yang–Baxter Equation and the A∞-Constraint

Cited by 82 publications

References 20 publications

Elliptic functions and equations of modular curves

Elliptic functions and equations of modular curves

Associative Yang-Baxter equation for quantum (semi-)dynamical R-matrices

Extensions of homogeneous coordinate rings to $A_{\infty}$-algebras

Contact Info

Product

Resources

About