An analogue of the classical Yang-Baxter equation for general algebraic structures

1995 ◽  
Vol 119 (4) ◽  
pp. 327-346 ◽  
Author(s):  
Xiaoping Xu
Keyword(s):  
Baxter Equation ◽  
Algebraic Structures

scholarly journals Graded Medial n-Ary Algebras and Polyadic Tensor Categories

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1038
Author(s):  
Steven Duplij
Keyword(s):  
Tensor Product ◽  
Tensor Category ◽  
Baxter Equation ◽  
Algebraic Structures ◽  
Monoidal Categories ◽  
Graded Algebras ◽  
Tensor Categories ◽  
Lie Brackets

Algebraic structures in which the property of commutativity is substituted by the mediality property are introduced. We consider (associative) graded algebras and instead of almost commutativity (generalized commutativity or ε-commutativity), we introduce almost mediality (“commutativity-to-mediality” ansatz). Higher graded twisted products and “deforming” brackets (being the medial analog of Lie brackets) are defined. Toyoda’s theorem which connects (universal) medial algebras with abelian algebras is proven for the almost medial graded algebras introduced here. In a similar way we generalize tensor categories and braided tensor categories. A polyadic (non-strict) tensor category has an n-ary tensor product as an additional multiplication with n−1 associators of the arity 2n−1 satisfying a n2+1-gon relation, which is a polyadic analog of the pentagon axiom. Polyadic monoidal categories may contain several unit objects, and it is also possible that all objects are units. A new kind of polyadic categories (called groupal) is defined: they are close to monoidal categories but may not contain units: instead the querfunctor and (natural) functorial isomorphisms, the quertors, are considered (by analogy with the querelements in n-ary groups). The arity-nonreducible n-ary braiding is introduced and the equation for it is derived, which for n=2 coincides with the Yang–Baxter equation. Then, analogously to the first part of the paper, we introduce “medialing” instead of braiding and construct “medialed” polyadic tensor categories.

scholarly journals Pre-jordan Algebras

2013 ◽  
Vol 112 (1) ◽  
pp. 19 ◽  
Author(s):  
Dongping Hou ◽  
Xiang Ni ◽  
Chengming Bai
Keyword(s):  
Lie Algebras ◽  
Jordan Algebra ◽  
Symplectic Form ◽  
Jordan Algebras ◽  
Multiplication Operators ◽  
Baxter Equation ◽  
Algebraic Structures ◽  
Left Multiplication ◽  
Close Relationship ◽  
Dendriform Algebras

The purpose of this paper is to introduce and study a notion of pre-Jordan algebra. Pre-Jordan algebras are regarded as the underlying algebraic structures of the Jordan algebras with a nondegenerate symplectic form. They are the algebraic structures behind the Jordan Yang-Baxter equation and Rota-Baxter operators in terms of $\mathcal{O}$-operators of Jordan algebras introduced in this paper. Pre-Jordan algebras are analogues for Jordan algebras of pre-Lie algebras and fit into a bigger framework with a close relationship with dendriform algebras. The anticommutator of a pre-Jordan algebra is a Jordan algebra and the left multiplication operators give a representation of the Jordan algebra, which is the beauty of such a structure. Furthermore, we introduce a notion of $\mathcal{O}$-operator of a pre-Jordan algebra which gives an analogue of the classical Yang-Baxter equation in a pre-Jordan algebra.

A FURTHER STUDY ON NON-ABELIAN PHASE SPACES: LEFT-SYMMETRIC ALGEBRAIC APPROACH AND RELATED GEOMETRY

2006 ◽  
Vol 18 (05) ◽  
pp. 545-564 ◽  
Author(s):  
CHENGMING BAI
Keyword(s):  
Phase Space ◽  
Lie Algebra ◽  
Algebraic Approach ◽  
Baxter Equation ◽  
Algebraic Structures ◽  
Geometric Structures ◽  
Dual Spaces ◽  
Symmetric Algebras ◽  
Algebraic Properties

The notion of non-abelian phase space of a Lie algebra was first formulated and then discussed by Kuperschmidt. In this paper, we further study the non-abelian phase spaces in terms of left-symmetric algebras. We interpret the natural appearance of left-symmetric algebras from the intrinsic algebraic properties and the close relations with the classical Yang–Baxter equation. Furthermore, using the theory of left-symmetric algebras, we study some interesting geometric structures related to phase spaces. Moreover, we also discuss the generalized phase spaces with certain non-trivial algebraic structures on the dual spaces.

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