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𝒪-operators on associative algebras and associative Yang–Baxter equations

Chengming Bai, Li Guo, Xiang Ni

2012
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Pacific Journal of Mathematics
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An ᏻ-operator on an associative algebra is a generalization of a Rota-Baxter operator that plays an important role in the Hopf algebra approach of Connes and Kreimer to the renormalization of quantum field theory. It is also the associative analog of an ᏻ-operator on a Lie algebra in the study of the classical Yang-Baxter equation. We introduce the concept of an extended ᏻ-operator on an associative algebra whose Lie algebra analog has been applied to generalized Lax pairs and PostLie algebras.
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... We study algebraic structures coming from extended ᏻ-operators. Continuing the work of Aguiar deriving Rota-Baxter operators from the associative Yang-Baxter equation, we show that its solutions correspond to extended ᏻ-operators through a duality. We also establish a relationship of extended ᏻ-operators with the generalized associative Yang-Baxter equation. C. 257 258 CHENGMING BAI, LI GUO AND XIANG NI numerous studies to better understand the role played by the Rota-Baxter identity in quantum field theory renormalization, as well as in applying the idea of renormalization to study divergency in mathematics [Ebrahimi-Fard et al. 2004; Ebrahimi-Fard et al. 2006; Guo and Zhang 2008; Manchon and Paycha 2010] . In this paper we consider a generalization of the Rota-Baxter operator in the relative context called the ᏻ-operator. It came from another connection between Rota-Baxter operators (on Lie algebras) and mathematical physics. In special cases, the Rota-Baxter identity for Lie algebras coincides with the operator form of the classical Yang-Baxter equation, named after the well-known physicists Yang [1967] and Baxter [1972]. The connection has its origin in the work of Semenov-Tyan-Shanskiȋ [1983] and its extension led to the concept of ᏻ-operators [Bai 2007; Bordemann 1990; Kupershmidt 1999] . The relation defining an ᏻ-operator was also called the Schouten curvature by Kosmann-Schwarzbach and Magri [1988] , and is the algebraic version of the contravariant analog of the Cartan curvature of the Lie algebra-valued one-form on a Lie group. Back to associative algebras, the first connection between Rota-Baxter operators and an associative analog of the classical Yang-Baxter equation was made by Aguiar [2000a; 2000b] , who showed that a solution of the associative Yang-Baxter equation (AYBE) gives rise to a Rota-Baxter operator of weight zero. Our study of this connection in this paper was motivated by the ᏻ-operator approach to the classical Yang-Baxter equation, but we go beyond what was known in the Lie algebra case. On one hand, we generalize the concept of a Rota-Baxter operator to that of an ᏻ-operator (of any weight) 1 and further to extended ᏻ-operators. On the other hand, we investigate the operator properties of the associative Yang-Baxter equation motivated by the study in the Lie algebra case. Through this approach, we show that the operator property of solutions of the associative Yang-Baxter equation is to a large extent characterized by ᏻ-operators. This generalization in the associative context, motivated by Lie algebra studies, has in turn motivated us to establish a similar generalization for Lie algebras and to apply it to generalized Lax pairs, classical Yang-Baxter equations and PostLie algebras [Bai et al. 2010b; 2011; Vallette 2007] . Our approach connects (extended) ᏻ-operators to solutions of the AYBE and its generalizations, and therefore [Bai 2010] to the construction of antisymmetric infinitesimal bialgebras and their related Frobenius algebras. The latter plays an important role in topological field theory [Runkel et al. 2007 ]. In particular, we are able to reverse the connection made by Aguiar and derive, from a Rota-Baxter 1 In the weight zero case, this has been considered by Uchino [2008] under the name "generalized Rota-Baxter operator". In the general case, the term "relative Rota-Baxter operator" is also used [Bai et al. 2010a ].

doi:10.2140/pjm.2012.256.257
fatcat:cjfxllz6evbohcybbqujmtyl5a