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Yang–Baxter equationThe Yang–Baxter equation is an equation which was first introduced in the field of statistical mechanics. It takes its name from independent work of C. N. Yang from 1968, and R. J. Baxter from 1982. Additional recommended knowledge
Parameterdependent YangBaxter equationLet A be a unital associative algebra. The parameterdependent Yang–Baxter equation is an equation for R(u), a parameterdependent invertible element of the tensor product (here, u is the parameter, which usually ranges over all real numbers in the case of an additive parameter, or over all positive real numbers in the case of a multiplicative parameter). The Yang–Baxter equation is for all values of u and v, in the case of an additive parameter, and for all values of u and v, in the case of a multiplicative parameter, where R_{12}(w) = φ_{12}(R(w)), R_{13}(w) = φ_{13}(R(w)), and R_{23}(w) = φ_{23}(R(w)), for all values of the parameter w, and , , and , are algebra morphisms determined by
Parameterindependent Yang–Baxter equationLet A be a unital associative algebra. The parameterindependent Yang–Baxter equation is an equation for R, an invertible element of the tensor product . The YangBaxter equation is where R_{12} = φ_{12}(R), R_{13} = φ_{13}(R), and R_{23} = φ_{23}(R). Let V be a module of A. Let be the linear map satisfying for all , then a representation of the braid group, B_{n}, can be constructed on by for , where on . This representation can be used to determine quasiinvariants of braids, knots and links. See also
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Yang–Baxter_equation". A list of authors is available in Wikipedia. 