On some Garsideness properties of structure groups of set-theoretic solutions of the Yang-Baxter equation

Fabienne Chouraqui

doi:10.1080/00927872.2023.2180181

On some Garsideness properties of structure groups of set-theoretic solutions of the Yang-Baxter equation

Fabienne Chouraqui

Department of Mathematics, Physics and Computer Science (Oranim Campus)

Research output: Contribution to journal › Article › peer-review

Abstract

There exists a multiplicative homomorphism from the braid group (Formula presented.) on k + 1 strands to the Temperley–Lieb algebra TL_k. Moreover, the homomorphic images in TL_k of the simple elements form a basis for the vector space underlying TL_k. In analogy with the case of B_k, there exists a multiplicative homomorphism from the structure group G(X, r) of a non-degenerate, involutive set-theoretic solution (X, r), with (Formula presented.), to an algebra, which extends to a homomorphism of algebras. We construct a finite basis of the underlying vector space of the image of G(X, r) using the Garsideness properties of G(X, r).

Original language	English
Pages (from-to)	3221-3231
Number of pages	11
Journal	Communications in Algebra
Volume	51
Issue number	8
DOIs	https://doi.org/10.1080/00927872.2023.2180181
State	Published - 2023

Bibliographical note

Publisher Copyright:
© 2023 Taylor & Francis Group, LLC.

Keywords

Coxeter-like quotient groups
Garside groups and monoids
set-theoretic solutions of the quantum Yang-Baxter equation

ASJC Scopus subject areas

Algebra and Number Theory

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10.1080/00927872.2023.2180181

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abstract = "There exists a multiplicative homomorphism from the braid group (Formula presented.) on k + 1 strands to the Temperley–Lieb algebra TLk. Moreover, the homomorphic images in TLk of the simple elements form a basis for the vector space underlying TLk. In analogy with the case of Bk, there exists a multiplicative homomorphism from the structure group G(X, r) of a non-degenerate, involutive set-theoretic solution (X, r), with (Formula presented.), to an algebra, which extends to a homomorphism of algebras. We construct a finite basis of the underlying vector space of the image of G(X, r) using the Garsideness properties of G(X, r).",

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AB - There exists a multiplicative homomorphism from the braid group (Formula presented.) on k + 1 strands to the Temperley–Lieb algebra TLk. Moreover, the homomorphic images in TLk of the simple elements form a basis for the vector space underlying TLk. In analogy with the case of Bk, there exists a multiplicative homomorphism from the structure group G(X, r) of a non-degenerate, involutive set-theoretic solution (X, r), with (Formula presented.), to an algebra, which extends to a homomorphism of algebras. We construct a finite basis of the underlying vector space of the image of G(X, r) using the Garsideness properties of G(X, r).

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KW - Garside groups and monoids

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JO - Communications in Algebra

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ER -

On some Garsideness properties of structure groups of set-theoretic solutions of the Yang-Baxter equation

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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