{"id":879,"date":"2020-02-05T09:58:44","date_gmt":"2020-02-05T09:58:44","guid":{"rendered":"http:\/\/mat.uab.cat\/web\/perera\/?p=879"},"modified":"2020-03-11T10:04:06","modified_gmt":"2020-03-11T10:04:06","slug":"seminar-ring-theory-2","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/perera\/2020\/02\/05\/seminar-ring-theory-2\/","title":{"rendered":"Seminar (Ring Theory)"},"content":{"rendered":"

Eric Jespers<\/a> (Vrije Universiteit Brussel)<\/p>\n

Associative structures associated to set-theoretic solutions of the Yang–Baxter equation<\/em><\/p>\n

Abstract: Let $(X,r)$ be a set-theoretic solution of the YBE, that is $X$ is a set and $r\\colon X\\times X \\to X\\times X$ satisfies
\n$$(r \\times\u00a0 \\mathrm{id})\\circ (\\mathrm{id} \\times\u00a0 r)\\circ (r \\times\u00a0 \\mathrm{id}) = (\\mathrm{id} \\times\u00a0 r)\\circ (r \\times\u00a0 \\mathrm{id})\\circ (\\mathrm{id} \\times r)$$ on $X^{3}$. Write $r(x,y)=(\\lambda_x (y), \\rho_y (x))$, for $x,y\\in X$. Gateva-Ivanova and Majid showed that the study of such solutions is determined by solutions $(M,r_M)$, where
\n\\[M=M(X,r) =\\langle x\\in X\\mid xy=\\lambda_x(y) \\rho_y(x), \\text{ for all } x,y\\in X \\rangle\\]
\nis the structure monoid of\u00a0 $(X,r)$, and $r_M$ restricts to $r$ on $X^2<\/sup>$. For left non-degenerate solutions, i.e. all $\\sigma_x$ are bijective, it has been shown that $M(X,r)$ is a regular submonoid of $A(X,r)\\rtimes \\mathcal{G}(X,r)$, where $\\mathcal{G}(X,r)=\\langle\u00a0 \\lambda_x\\mid x\\in X\\rangle$ is the permutation group of $(X,r)$, and
\n\\[A(X,r) =\\langle x\\in X \\mid x\\lambda_{x}(y)\u00a0 =\\lambda_{x}(y) \\lambda_{\\sigma_{x}(y)}(\\rho_{y}(x) \\rangle\\]
\nis the derived monoid of $(X,r)$. It also is the structure monoid of the rack solution $(X,r’)$ with
\n\\[r'(x,y)=(y,\\lambda_y\\rho_{\\lambda^{-1}_x(y)}(x)).\\]
\nThis solution “encodes”\u00a0 the relations determined by the map $r^{2} \\colon X^{2} \\to X^{2}$. The elements of $A=A(X,r)$ are normal, i.e.\u00a0 $aA=Aa$ for all $a\\in A$. It is this “richer structure” that has been exploited by several authors to obtain information on the structure monoid $M(X,r)$ and the structure algebra $kM(X,r)$.<\/p>\n

In this talk\u00a0 we report on some\u00a0 recent investigations for arbitrary solutions, i.e. not necessarily left non-degenerate nor bijective.
\nThis is joint work with F. Ced\\’o and C. Verwimp.\u00a0 We prove that there is a\u00a0 unique $1$-cocycle $M(X,r)\\to A(X,r)$ and we determine when this mapping is injective, surjective, respectively bijective. One then obtains a monoid homomorphism $M(X,r) \\to A(X,r)\\rtimes \\langle \\sigma_x\u00a0 \\mid x\\in X\\rangle$. This mapping is injective when all $\\sigma_x$ are injective. Further we determine the left cancellative congruence $\\eta$ on $M(X,r)$ and show that $(X,r)$ induces a set-theoretic solution in $M(X,r)\/\\eta$ provided $(X,r)$ is left non-degenerate.<\/p>\n","protected":false},"excerpt":{"rendered":"

Eric Jespers (Vrije Universiteit Brussel) Associative structures associated to set-theoretic solutions of the Yang–Baxter equation Abstract: Let $(X,r)$ be a set-theoretic solution of the YBE, that is $X$ is a set and $r\\colon X\\times X \\to X\\times X$ satisfies $$(r \\times\u00a0 \\mathrm{id})\\circ (\\mathrm{id} \\times\u00a0 r)\\circ (r \\times\u00a0 \\mathrm{id}) = (\\mathrm{id} \\times\u00a0 r)\\circ (r \\times\u00a0 \\mathrm{id})\\circ … <\/p>\n

Continua llegint \u00abSeminar (Ring Theory)\u00bb<\/span><\/a><\/p>\n","protected":false},"author":22,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6,3],"tags":[],"class_list":["post-879","post","type-post","status-publish","format-standard","hentry","category-ring-theory","category-seminars"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/perera\/wp-json\/wp\/v2\/posts\/879","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.uab.cat\/web\/perera\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.uab.cat\/web\/perera\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/perera\/wp-json\/wp\/v2\/users\/22"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/perera\/wp-json\/wp\/v2\/comments?post=879"}],"version-history":[{"count":6,"href":"https:\/\/mat.uab.cat\/web\/perera\/wp-json\/wp\/v2\/posts\/879\/revisions"}],"predecessor-version":[{"id":885,"href":"https:\/\/mat.uab.cat\/web\/perera\/wp-json\/wp\/v2\/posts\/879\/revisions\/885"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/perera\/wp-json\/wp\/v2\/media?parent=879"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/perera\/wp-json\/wp\/v2\/categories?post=879"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/perera\/wp-json\/wp\/v2\/tags?post=879"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}