Quadratic algebras and idempotent braided sets
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914937681477632 |
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| author | Gateva-Ivanova, Tatiana Majid, Shahn |
| author_facet | Gateva-Ivanova, Tatiana Majid, Shahn |
| contents | We study the Yang-Baxter algebras $A(K,X,r)$ associated to finite set-theoretic solutions $(X,r)$ of the braid relations. We introduce an equivalent set of quadratic relations $\Re\subseteq G$, where $G$ is the reduced Gröbner basis of $(\Re)$. We show that if $(X,r)$ is left-nondegenerate and idempotent then $\Re= G$ and the Yang-Baxter algebra is PBW. We use graphical methods to study the global dimension of PBW algebras in the $n$-generated case and apply this to Yang-Baxter algebras in the left-nondegenerate idempotent case. We study the $d$-Veronese subalgebras for a class of quadratic algebras and use this to show that for $(X,r)$ left-nondegenerate idempotent, the $d$-Veronese subalgebra $A(K,X,r)^{(d)}$ can be identified with $A(K,X,r^{(d)})$, where $(X,r^{(d)})$ are all left-nondegenerate idempotent solutions. We determined the Segre product in the left-nondegenerate idempotent setting. Our results apply to a previously studied class of `permutation idempotent' solutions, where we show that all their Yang-Baxter algebras for a given cardinality of $X$ are isomorphic and are isomorphic to their $d$-Veronese subalgebras. In the linearised setting, we construct the Koszul dual of the Yang-Baxter algebra and the Nichols-Woronowicz algebra in the idempotent case, showing that the latter is quadratic. We also construct noncommutative differentials on some of these quadratic algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_02939 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quadratic algebras and idempotent braided sets Gateva-Ivanova, Tatiana Majid, Shahn Quantum Algebra Mathematical Physics Rings and Algebras We study the Yang-Baxter algebras $A(K,X,r)$ associated to finite set-theoretic solutions $(X,r)$ of the braid relations. We introduce an equivalent set of quadratic relations $\Re\subseteq G$, where $G$ is the reduced Gröbner basis of $(\Re)$. We show that if $(X,r)$ is left-nondegenerate and idempotent then $\Re= G$ and the Yang-Baxter algebra is PBW. We use graphical methods to study the global dimension of PBW algebras in the $n$-generated case and apply this to Yang-Baxter algebras in the left-nondegenerate idempotent case. We study the $d$-Veronese subalgebras for a class of quadratic algebras and use this to show that for $(X,r)$ left-nondegenerate idempotent, the $d$-Veronese subalgebra $A(K,X,r)^{(d)}$ can be identified with $A(K,X,r^{(d)})$, where $(X,r^{(d)})$ are all left-nondegenerate idempotent solutions. We determined the Segre product in the left-nondegenerate idempotent setting. Our results apply to a previously studied class of `permutation idempotent' solutions, where we show that all their Yang-Baxter algebras for a given cardinality of $X$ are isomorphic and are isomorphic to their $d$-Veronese subalgebras. In the linearised setting, we construct the Koszul dual of the Yang-Baxter algebra and the Nichols-Woronowicz algebra in the idempotent case, showing that the latter is quadratic. We also construct noncommutative differentials on some of these quadratic algebras. |
| title | Quadratic algebras and idempotent braided sets |
| topic | Quantum Algebra Mathematical Physics Rings and Algebras |
| url | https://arxiv.org/abs/2409.02939 |