[Paper Review] Quantum groups of GL(2) representation type

Colin Mrozinski|arXiv (Cornell University)|Jan 17, 2012

Algebraic structures and combinatorial models19 references3 citations

TL;DR

This paper classifies cosemisimple Hopf algebras over an algebraically closed field of characteristic zero whose corepresentation semi-ring is isomorphic to that of $GL(2)$. Using Tannaka-Krein duality and a novel family of Hopf algebras $\mathcal{G}(A,B)$ defined by matrix relations involving bilinear forms, the authors establish monoidal equivalence to quantum $GL_q(2)$ and provide a complete isomorphism classification based on matrix invariants and genericity conditions on $q$. The key contribution is a structural and categorical classification of $GL(2)$-type quantum groups via universal algebras parametrized by invertible matrices.

ABSTRACT

We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear form. A detailed study of these Hopf algebras gives us an isomorphic classification and the description of their corepresentation categories.

Motivation & Objective

• To classify all cosemisimple Hopf algebras over an algebraically closed field of characteristic zero whose corepresentation semi-ring is isomorphic to that of $GL(2)$.
• To generalize the quantum similitude group construction for non-degenerate bilinear forms to a new family of Hopf algebras $\mathcal{G}(A,B)$.
• To establish a monoidal equivalence between the comodule categories of $\mathcal{G}(A,B)$ and $\mathcal{O}(GL_q(2))$ under specific matrix conditions.
• To provide a complete isomorphism classification of such Hopf algebras via matrix invariants and genericity of $q$.
• To extend the classification to the compact case and analyze bi-Galois objects and cohomological invariants.

Proposed method

• Define the universal Hopf algebra $\mathcal{G}(A,B)$ with generators $x_{ij}, d, d^{-1}$ and relations $x^t A x = A d$, $x B x^t = B d$, $d d^{-1} = 1$, encoding symmetries of bilinear forms.
• Equip $\mathcal{G}(A,B)$ with a Hopf algebra structure via comultiplication $\Delta(x_{ij}) = \sum_k x_{ik} \otimes x_{kj}$, counit $\varepsilon(x_{ij}) = \delta_{ij}$, and antipode $S(x) = d^{-1} A^{-1} x^t A$.
• Use Tannaka-Krein duality and Schauenburg's theorem on Hopf bi-Galois objects to prove monoidal equivalence between $\mathrm{Comod}(\mathcal{G}(A,B))$ and $\mathrm{Comod}(\mathcal{O}(GL_q(2)))$ under the condition $B^t A^t B A = \lambda I_n$ and $q$ satisfying a quadratic equation.
• Construct a connected cogroupoid linking $\mathcal{G}(A,B)$ and $\mathcal{O}(GL_q(2))$ to prove the equivalence, relying on the diamond lemma to establish a basis of reduced monomials and non-degeneracy of $d$.
• Apply the classification to derive isomorphism conditions: $\mathcal{G}(A,B) \cong \mathcal{G}(C,D)$ iff $n=m$ and $C,D$ are related to $A,B$ via conjugation by a matrix $P$ or its inverse transpose.
• Analyze $\mathcal{G}(A,B)$-Galois objects, their group, and lazy cohomology, and extend results to the compact case.

Experimental results

Research questions

• RQ1 Which cosemisimple Hopf algebras over an algebraically closed field of characteristic zero have a corepresentation semi-ring isomorphic to that of $GL(2)$?
• RQ2 Under what conditions is the comodule category of $\mathcal{G}(A,B)$ monoidally equivalent to that of $\mathcal{O}(GL_q(2))$?
• RQ3 When are two Hopf algebras $\mathcal{G}(A,B)$ and $\mathcal{G}(C,D)$ isomorphic, given the matrix constraints $B^t A^t B A = \lambda I_n$?
• RQ4 What is the role of the parameter $q$ in the classification, and how does its genericity affect the structure of $\mathcal{G}(A,B)$?
• RQ5 How do the Hopf bi-Galois objects and lazy cohomology of $\mathcal{G}(A,B)$ relate to its monoidal and structural properties?

Key findings

• The comodule category of $\mathcal{G}(A,B)$ is monoidally equivalent to that of $\mathcal{O}(GL_q(2))$ if $B^t A^t B A = \lambda I_n$ and $q$ satisfies $q^2 - \sqrt{\lambda^{-1}} \operatorname{tr}(AB^t) q + 1 = 0$.
• All cosemisimple Hopf algebras with corepresentation semi-ring isomorphic to that of $GL(2)$ are isomorphic to $\mathcal{G}(A,B)$ for some $A,B \in GL_n(k)$ satisfying $B^t A^t B A = \lambda I_n$ and $q$ generic (not a root of unity or $\pm 1$).
• Two Hopf algebras $\mathcal{G}(A,B)$ and $\mathcal{G}(C,D)$ are isomorphic if and only if $n = m$ and there exists $P \in GL_n(k)$ such that either $(C,D) = (P^t A P, P^{-1} B P^{-1t})$ or $(C,D) = (P^t B^{-1} P, P^{-1} A^{-1} P^{-1t})$.
• The algebra $\mathcal{G}(A,B)$ admits a basis of reduced monomials, and $d$ is not a zero divisor, ensuring the algebra is non-zero and well-behaved under localization.
• The Hopf algebra $\mathcal{G}(A,B)$ is shown to be non-degenerate and well-structured via the diamond lemma, confirming the existence of a PBW-type basis.
• The classification extends to the compact case, and the group of $\mathcal{G}(A,B)$-bi-Galois objects and its lazy cohomology group are fully described.