[Paper Review] Kaplansky's Construction Type and Classification of Weak bialgebras and Weak Hopf algebras

Zoheir Chebel, Abdenacer Makhlouf|arXiv (Cornell University)|Jan 13, 2010

Algebraic structures and combinatorial models19 references3 citations

TL;DR

This paper introduces Kaplansky-type constructions to generate weak bialgebras and weak Hopf algebras from regular algebras or bialgebras, providing a classification of all 2- and 3-dimensional weak bialgebras and weak Hopf algebras up to isomorphism. It identifies their stabilizer subgroups under the action of the general linear group, revealing distinct automorphism groups for different isomorphism classes, including groups of order 2 and 6, and families parameterized by complex parameters.

ABSTRACT

In this paper, we study weak bialgebras and weak Hopf algebras. These algebras form a class wider than bialgebras respectively Hopf algebras. The main results of this paper are Kaplansky's constructions type which lead to weak bialgebras or weak Hopf algebras starting from a regular algebra or a bialgebra. Also we provide a classification of 2-dimensional and 3-dimensional weak bialgebras and weak Hopf algebras. We determine then the stabilizer group and the representative of these classes, the action being that of the linear group.

Motivation & Objective

• To extend Kaplansky’s construction method for bialgebras to weak bialgebras and weak Hopf algebras.
• To classify all isomorphism classes of finite-dimensional weak bialgebras and weak Hopf algebras for dimensions 2 and 3.
• To compute the stabilizer subgroups of these algebras under the action of the general linear group.
• To establish that the set of weak bialgebras and weak Hopf algebras forms an algebraic variety fibred by linear group actions, with orbits corresponding to isomorphism classes.

Proposed method

• Adapts Kaplansky’s construction technique to generate weak bialgebras and weak Hopf algebras from arbitrary algebras or bialgebras.
• Uses Sweedler’s notation for comultiplication and defines compatibility conditions via three key identities: comultiplication as an algebra morphism, weak unit preservation, and weak counit multiplicativity.
• Applies the action of the general linear group GL(V) on the space of weak bialgebra structures to define isomorphism classes via orbits.
• Derives explicit matrix conditions (equations 4.1–4.3) for isomorphism between two weak bialgebras in terms of structure constants and linear transformation matrices.
• Performs case-by-case classification for 2- and 3-dimensional algebras by solving the system of equations under basis-specific assumptions.
• Computes automorphism groups by solving the isomorphism conditions with the identity transformation, yielding discrete and continuous groups depending on the algebraic structure.

Experimental results

Research questions

• RQ1 How can Kaplansky’s construction method be generalized to produce weak bialgebras and weak Hopf algebras from regular algebras or bialgebras?
• RQ2 What are the complete isomorphism classes of 2- and 3-dimensional weak bialgebras and weak Hopf algebras over an algebraically closed field of characteristic zero?
• RQ3 What are the stabilizer subgroups of these algebras under the action of the general linear group GL(V)?
• RQ4 How do the automorphism groups of weak bialgebras and weak Hopf algebras vary across different isomorphism classes in low dimensions?
• RQ5 Can the space of weak bialgebras and weak Hopf algebras be described as an algebraic variety fibred by GL(V)-orbits?

Key findings

• There are exactly 11 isomorphism classes of 3-dimensional weak bialgebras, with 10 having automorphism groups of order 6 and one (class 18) having a continuous automorphism group parameterized by complex numbers.
• The automorphism group of the 2-dimensional weak bialgebras is of order 2, generated by a single matrix with entries (1,1; 0,-1).
• For 3-dimensional weak bialgebras, classes (1)–(11) have automorphism groups isomorphic to the symmetric group S3, generated by two specific 3×3 matrices.
• Class (18) has an automorphism group isomorphic to a one-parameter family of matrices involving complex parameters r and e with 4e - r² ≠ 0.
• Class (20) has an automorphism group parameterized by r and e with 4e + r² ≠ 0, also forming a one-parameter family.
• All 3-dimensional weak Hopf algebras have automorphism groups of order 6, isomorphic to S3, matching the group structure of classes (1)–(11) of weak bialgebras.