scholarly journals Set-theoretic solutions of the Yang–Baxter equation, associated quadratic algebras and the minimality condition

2020 ◽  
Author(s):  
Ferran Cedó ◽  
Eric Jespers ◽  
Jan Okniński
Keyword(s):  
Baxter Equation ◽  
Minimality Condition ◽  
Quadratic Algebras

QUADRATIC ALGEBRAS OF SKEW TYPE SATISFYING THE CYCLIC CONDITION

2004 ◽  
Vol 14 (04) ◽  
pp. 479-498 ◽  
Author(s):  
ERIC JESPERS ◽  
JAN OKNIŃSKI
Keyword(s):  
Baxter Equation ◽  
Important Class ◽  
Polynomial Algebras ◽  
Quadratic Algebras ◽  
Cyclic Condition ◽  
Special Case ◽  
Special Classes

We consider algebras over a field with generators x1,x2,…,xn subject to [Formula: see text] square-free relations xixj=xkxl in which every product xpxq, p≠q, appears in one of the relations. The work of Gateva-Ivanova and Van den Bergh, motivated in particular by the study of set theoretic solutions of the Yang–Baxter equation, provided an important class of such algebras. In this special case the presentation satisfies the so-called cyclic condition that became an essential combinatorial tool in proving that these algebras share many strong ring theoretic properties of polynomial algebras in commuting variables. In this paper we describe the structure of algebras on four generators satisfying the cyclic condition. The emphasis is on some new unexpected features, not present in the motivating special classes.

Gröbner bases for quadratic algebras of skew type

2012 ◽  
Vol 55 (2) ◽  
pp. 387-401 ◽  
Author(s):  
Ferran Cedó ◽  
Jan Okniński
Keyword(s):  
Regular Language ◽  
Normal Forms ◽  
Free Monoid ◽  
Lexicographic Order ◽  
Semigroup Algebra ◽  
Baxter Equation ◽  
Finitely Generated ◽  
Quadratic Algebras ◽  
Defining Relations ◽  
Finite Module

AbstractNon-degenerate monoids of skew type are considered. This is a class of monoids S defined by n generators and $\binom{n}{2}$ quadratic relations of certain type, which includes the class of monoids yielding set-theoretic solutions of the quantum Yang–Baxter equation, also called binomial monoids (or monoids of I-type with square-free defining relations). It is shown that under any degree-lexicographic order on the associated free monoid FMn. of rank n the set of normal forms of elements of S is a regular language in FMn. As one of the key ingredients of the proof, it is shown that an identity of the form xN yN = yN xN holds in S. The latter is derived via an investigation of the structure of S viewed as a semigroup of matrices over a field. It also follows that the semigroup algebra K[S] is a finite module over a finitely generated commutative subalgebra of the form K[A] for a submonoid A of S.

scholarly journals Mapping TASEP back in time

2021 ◽  
Author(s):  
Leonid Petrov ◽  
Axel Saenz
Keyword(s):  
Markov Process ◽  
Continuous Time ◽  
Exclusion Process ◽  
Discrete Space ◽  
Baxter Equation ◽  
Simple Exclusion Process ◽  
Open Questions ◽  
Asymmetric Simple Exclusion ◽  
Simple Exclusion ◽  
Hammersley Process

AbstractWe obtain a new relation between the distributions $$\upmu _t$$ μ t at different times $$t\ge 0$$ t ≥ 0 of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions $$\upmu _t$$ μ t backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving $$\upmu _t$$ μ t which in turn brings new identities for expectations with respect to $$\upmu _t$$ μ t . The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang–Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions.

RATIONAL DEGENERATION OF ELLIPTIC QUADRATIC ALGEBRAS

1992 ◽  
Vol 07 (supp01b) ◽  
pp. 773-779 ◽  
Author(s):  
ALEKSANDR V. ODESSKI
Keyword(s):  
Quadratic Algebras

Ore extensions of quasitriangular Hopf group coalgebras

2014 ◽  
Vol 13 (06) ◽  
pp. 1450016 ◽  
Author(s):  
Daowei Lu ◽  
Dingguo Wang
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Baxter Equation ◽  
Ore Extension ◽  
Ore Extensions ◽  
Necessary And Sufficient

In this paper, we mainly consider some special Ore extension of quasitriangular Hopf group coalgebra, and give the necessary and sufficient conditions when the Ore extension of quasitriangular Hopf group coalgebras will preserve the same quasitriangular structure. Furthermore, in the two examples given at the end, we construct new solutions of Yang–Baxter equation of Hopf group coalgebras version.

Sign in / Sign up

Export Citation Format

Share Document