[Paper Review] Hom-bialgebras and comodule Hom-algebras

Donald Yau|arXiv (Cornell University)|Oct 27, 2008

Advanced Topics in Algebra27 references15 citations

TL;DR

This paper introduces Hom-bialgebras and comodule Hom-algebras as Hom-type analogues of classical bialgebras and comodule algebras. It constructs a universal Hom-bialgebra $\mathbf{M}(2)$ representing $2\times2$ matrices over multiplicative Hom-associative algebras, and shows that the Hom-affine plane $\mathbf{A}_2$ is a non-trivial $\mathbf{M}(2)$-comodule Hom-algebra via a coaction defined on generators.

ABSTRACT

We study Hom-bialgebras and objects admitting coactions by Hom-bialgebras. In particular, we construct a Hom-bialgebra M representing the functor of 2x2-matrices on Hom-associative algebras. Then we construct a Hom-algebra analogue of the affine plane and show that it is a comodule Hom-algebra over M in a suitable sense. It is also shown that the enveloping Hom-associative algebra of a Hom-Lie algebra is naturally a Hom-bialgebra.

Motivation & Objective

• To develop the theory of Hom-bialgebras as a Hom-type generalization of classical bialgebras.
• To construct a universal Hom-bialgebra $\mathbf{M}(2)$ that represents the functor of $2\times2$ matrices over multiplicative Hom-associative algebras.
• To define and study comodule Hom-algebras over Hom-bialgebras, generalizing the classical notion of comodule algebras.
• To show that the Hom-affine plane $\mathbf{A}_2$ admits a non-trivial $\mathbf{M}(2)$-comodule structure.
• To establish that the enveloping Hom-associative algebra of a Hom-Lie algebra naturally carries a Hom-bialgebra structure.

Proposed method

• Uses the free functor from modules to multiplicative Hom-associative algebras to construct $\mathbf{M}(2)$ as the free algebra on generators $a,b,c,d$ with $\alpha$-twisted relations.
• Defines a comultiplication $\Delta$ on $\mathbf{M}(2)$ that represents matrix multiplication via the universal property.
• Equips $\mathbf{M}(2)$ with a comultiplication $\Delta$ satisfying $\alpha$-twisted coassociativity, making it a Hom-bialgebra.
• Constructs the Hom-affine plane $\mathbf{A}_2$ as the free multiplicative Hom-associative algebra on two generators $x,y$.
• Defines a coaction $\rho: \mathbf{A}_2 \to \u005cmathbf{M}(2) \otimes \u005cmathbf{A}_2$ by setting $\rho(x) = a\otimes x + b\otimes y$, $\rho(y) = c\otimes x + d\otimes y$, extended algebraically.
• Verifies the comodule condition using the identification $\u005cmathbf{M}(2)\otimes\u005cmathbf{M}(2)\otimes\u005cmathbf{A}_2 \cong F(\u005ck\langle a',a'',b',b'',c',c'',d',d'',x,y\rangle)$ and Hom-associativity in the target algebra.

Experimental results

Research questions

• RQ1 Can a universal Hom-bialgebra $\mathbf{M}(2)$ be constructed that represents the functor of $2\times2$ matrices over multiplicative Hom-associative algebras?
• RQ2 Does the Hom-affine plane $\mathbf{A}_2$ admit a non-trivial coaction making it a comodule Hom-algebra over $\mathbf{M}(2)$?
• RQ3 Is the enveloping Hom-associative algebra of a Hom-Lie algebra naturally a Hom-bialgebra?
• RQ4 How can the classical notion of comodule algebra be generalized to the Hom-algebra setting?
• RQ5 What is the role of $\alpha$-twisted coassociativity in defining Hom-bialgebras and their comodules?

Key findings

• The Hom-bialgebra $\mathbf{M}(2)$ exists and represents the functor sending a multiplicative Hom-associative algebra $A$ to the algebra $M_2(A)$ of $2\times2$ matrices over $A$, with a natural bijection $\mathbf{HA}(\u005cmathbf{M}(2), A) \cong M_2(A)$.
• The comultiplication $\Delta$ on $\mathbf{M}(2)$ universally represents matrix multiplication in $M_2(A)$, satisfying the $\alpha$-twisted coassociativity condition.
• The Hom-affine plane $\mathbf{A}_2$ is constructed as the free multiplicative Hom-associative algebra on two generators $x$ and $y$, and admits a non-trivial $\mathbf{M}(2)$-comodule structure via the coaction $\rho(x) = a\otimes x + b\otimes y$, $\rho(y) = c\otimes x + d\otimes y$.
• The coaction $\rho$ satisfies the comodule condition $(\Delta_M \otimes \alpha_A) \circ \rho = (\alpha_M \otimes \rho) \circ \rho$, verified using Hom-associativity in the tensor product algebra.
• The enveloping Hom-associative algebra $U_{HLie}(L)$ of a Hom-Lie algebra $L$ is naturally a Hom-bialgebra, extending the classical universal enveloping algebra construction.
• The construction relies on the adjunction $\mathbf{HA}(\u005cmathbf{A}_2, \u005cmathbf{M}(2)\otimes\u005cmathbf{A}_2) \cong \mathbf{Mod}(\u005ck\langle x,y\rangle, \u005cmathbf{M}(2)\otimes\u005cmathbf{A}_2)$, allowing the coaction to be defined on generators.