[Paper Review] Weighted infinitesimal unitary bialgebras on matrix algebras and weighted associative Yang-Baxter equations

Yi Zhang, Xing Gao+1 more|arXiv (Cornell University)|Nov 2, 2018

Advanced Topics in Algebra16 references6 citations

TL;DR

This paper constructs weighted infinitesimal unitary bialgebras on matrix algebras via a coproduct derived from solutions to the weighted associative Yang-Baxter equation (AYBE). It establishes a bijection between solutions of the AYBE of weight $\lambda$ and Rota-Baxter operators of weight $-\lambda$, and uses this to build weighted quasitriangular infinitesimal unitary bialgebras, which further induce dendriform algebra structures on matrix algebras.

ABSTRACT

We equip a matrix algebra with a weighted infinitesimal unitary bialgebraic structure, via a construction of a suitable coproduct. Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on a matrix algebra. By exploring the relationship between weighted infinitesimal bialgebras and pre-Lie algebras, we construct a pre-Lie algebraic structure and then a new Lie algebraic structure on a matrix algebra. We also introduce the weighted associative Yang-Baxter equations (AYBEs) and obtain the relationship between solutions of weighted AYBEs and weighted infinitesimal unitary bialgebras. We give a bijection between the solutions of the associative Yang-Baxter equation of weight $λ$ and Rota-Baxter operators of weight $-λ$ on matrix algebras. As a consequence, weighted quasitriangular infinitesimal unitary bialgebras are constructed, which generalize the results studied by Aguiar. Finally, We show that any weighted quasitriangular infinitesimal unitary bialgebra can be made into a dendriform algebra.

Motivation & Objective

• To construct weighted infinitesimal unitary bialgebras on matrix algebras using a coproduct derived from solutions to the weighted associative Yang-Baxter equation (AYBE).
• To establish a bijection between solutions of the weighted AYBE of weight $\lambda$ and Rota-Baxter operators of weight $-\lambda$ on matrix algebras.
• To generalize Aguiar's quasitriangular infinitesimal bialgebra framework by introducing weighted quasitriangular $\epsilon$-unitary bialgebras.
• To show that any weighted quasitriangular $\epsilon$-unitary bialgebra can be used to construct a dendriform algebra structure on the underlying algebra.
• To explore the interplay between pre-Lie algebras, weighted infinitesimal bialgebras, and the algebraic structures induced by Rota-Baxter operators on matrix algebras.

Proposed method

• Define a coproduct $\Delta_r$ on a matrix algebra $A$ using a solution $r \in A \otimes A$ of the weighted AYBE of weight $\lambda$, satisfying $\Delta_r(ab) = a \cdot \Delta_r(b) + \Delta_r(a) \cdot b + \lambda(a \otimes b)$.
• Construct an infinitesimal unitary Hopf algebra on a matrix algebra by equipping the weighted infinitesimal bialgebra with an antipode, under Aguiar's framework.
• Establish a bijection between solutions of the weighted AYBE of weight $\lambda$ and Rota-Baxter operators of weight $-\lambda$ via the formula $r = \sum u_i \otimes v_i$ and the operator $P(a) = \sum u_i a v_i$.
• Define a pre-Lie algebra structure on the matrix algebra using the coproduct and the weighted bialgebra structure, with the product $a \ast b = \sum u_i a v_i b$.
• Construct a dendriform algebra structure on $A$ using the quasitriangular $\epsilon$-unitary bialgebra via operations $a \succ b = \sum u_i a v_i b$ and $a \prec b = \sum a u_i b v_i - \lambda ab$.
• Verify that the resulting operations satisfy the dendriform algebra axioms by leveraging the weighted AYBE and properties of Rota-Baxter algebras.

Experimental results

Research questions

• RQ1 How can a matrix algebra be equipped with a weighted infinitesimal unitary bialgebra structure via a suitable coproduct?
• RQ2 What is the precise relationship between solutions of the weighted associative Yang-Baxter equation of weight $\lambda$ and Rota-Baxter operators of weight $-\lambda$ on matrix algebras?
• RQ3 Can weighted quasitriangular $\epsilon$-unitary bialgebras be constructed from solutions of the weighted AYBE, and what algebraic structures do they induce?
• RQ4 How do weighted infinitesimal bialgebras relate to pre-Lie and dendriform algebra structures on matrix algebras?
• RQ5 What conditions ensure that a weighted quasitriangular $\epsilon$-unitary bialgebra gives rise to a dendriform algebra?

Key findings

• A bijection is established between solutions of the weighted associative Yang-Baxter equation of weight $\lambda$ and Rota-Baxter operators of weight $-\lambda$ on matrix algebras.
• The construction of a weighted infinitesimal unitary bialgebra on a matrix algebra is achieved via a coproduct $\Delta_r$ defined by $\Delta_r(ab) = a \cdot \Delta_r(b) + \Delta_r(a) \cdot b + \lambda(a \otimes b)$, where $r$ is a solution of the weighted AYBE.
• An infinitesimal unitary Hopf algebra structure is realized on the matrix algebra by equipping the weighted infinitesimal bialgebra with an antipode, extending Aguiar's framework.
• A pre-Lie algebra structure is constructed on the matrix algebra using the coproduct and the weighted bialgebra axioms, with the product $a \ast b = \sum u_i a v_i b$.
• A dendriform algebra structure is induced on the matrix algebra from a weighted quasitriangular $\epsilon$-unitary bialgebra via the operations $a \succ b = \sum u_i a v_i b$ and $a \prec b = \sum a u_i b v_i - \lambda ab$, satisfying the dendriform axioms.
• The triple $(A, \succ, \prec)$ forms a dendriform algebra, as proven by verifying the axioms using the weighted AYBE and the Rota-Baxter algebra framework.