[Paper Review] Automorphisms of Regular Algebras

Todor Popov|ArXiv.org|Jan 11, 2006

Algebraic structures and combinatorial models7 references3 citations

TL;DR

This paper extends Manin's quantum matrix group construction to non-quadratic regular Artin-Schelter algebras of dimension 3, constructing a Hopf algebra of automorphisms for both quadratic and cubic cases. For cubic algebras, it yields new quantum groups not covered in prior classifications, with the quantum group's Gelfand-Kirillov dimension being 7 for certain types like $S_2$, contrary to the expected 9.

ABSTRACT

Manin associated to a quadratic algebra (quantum space) the quantum matrix group of its automorphisms. This Talk aims to demonstrate that Manin's construction can be extended for quantum spaces which are non-quadratic homogeneous algebras. Here given a regular Artin-Schelter algebra of dimension 3 we construct the quantum group of its symmetries, i.e., the Hopf algebra of its automorphisms. For quadratic Artin-Schelter algebras these quantum groups are contained in the the classification of the GL(3) quantum matrix groups due to Ewen and Ogievetsky. For cubic Artin-Schelter algebras we obtain new quantum groups which are automorphisms of cubic quantum spaces.

Motivation & Objective

• To extend Manin’s construction of quantum matrix groups from quadratic to non-quadratic (specifically cubic) regular Artin-Schelter algebras of dimension 3.
• To demonstrate that the bialgebra of endomorphisms of a regular algebra can be extended to a Hopf algebra by adding inverse relations, thus defining a quantum group of symmetries.
• To classify and construct new quantum groups for cubic regular algebras, which are not included in the Ewen-Ogievetsky classification of $GL(3)$-type quantum groups.
• To compute the Gelfand-Kirillov dimension of the automorphism quantum group and show it can be 7 instead of the expected 9 for some cubic algebras.

Proposed method

• Define the endomorphism bialgebra $e( ext{A}) = ext{A}^! ullet ext{A}$ using the tensor product of the algebra and its dual, with relations derived from the annihilator of the defining relations.
• Construct the bialgebra $e( ext{A})$ as a quotient of $A(E^* igotimes E, r igoplus ar{r})$ by identifying generators from $ ext{A}$ and $ ext{A}^!$ via a metric $g^{ij} = ar{ ho}^{ij}$.
• Introduce the quasi-determinant ${ m D} = { m D}( ext{A})$ as a quasi-central element satisfying ${ m D} u_j^i = h_{(j)}^{(i)} u_j^i { m D}$, with $h_{(j)}^{(i)}$ determined by the invariant tensor $ ilde{ ho}$.
• Define the antipode $S(u_j^i) = { m D}^{-1} { m S}_L(u_j^i) = { m S}_R(u_j^i) { m D}^{-1}$, extending the bialgebra to a Hopf algebra $H( ext{A})$ by inverting ${ m D}$.
• Verify the antipode axioms using the Cramer adjoint maps ${ m S}_L$ and ${ m S}_R$, ensuring $m ig( ext{Id} igotimes S ig) igcirc riangle = m ig( S igotimes ext{Id} ig) igcirc riangle = ho igcirc ho$.
• Use the cyclicity of the invariant tensor $ ho$ to derive the structure of $h_{(j)}^{(i)}$, which determines the quasi-central action of ${ m D}$.

Experimental results

Research questions

• RQ1 Can Manin’s quantum matrix group construction be generalized beyond quadratic algebras to include non-quadratic regular algebras?
• RQ2 What is the structure of the automorphism quantum group for a 3-dimensional cubic regular Artin-Schelter algebra?
• RQ3 How does the Gelfand-Kirillov dimension of the automorphism quantum group compare to the classical case $gk ext{-} ext{dim} = 9$ for 3D algebras?
• RQ4 What role does the quasi-determinant ${ m D}$ play in defining the antipode and ensuring the Hopf algebra structure?
• RQ5 Are the quantum groups arising from cubic regular algebras new, or do they appear in existing classifications such as Ewen and Ogievetsky’s $GL(3)$-type quantum groups?

Key findings

• The construction yields a Hopf algebra $H( ext{A})$ of automorphisms for any generic 3-dimensional regular Artin-Schelter algebra, extending Manin’s framework beyond quadratic cases.
• For cubic regular algebras, the resulting quantum groups are new and not contained in the Ewen-Ogievetsky classification of $GL(3)$-type quantum matrix groups.
• The Gelfand-Kirillov dimension of $H( ext{A})$ is 7 for certain cubic algebras of type $S_2$, contradicting the expectation of $gk ext{-} ext{dim} = 9$.
• The quasi-determinant ${ m D}$ is quasi-central, satisfying ${ m D} u_j^i = h_{(j)}^{(i)} u_j^i { m D}$, with $h_{(j)}^{(i)} = h^{j-i}$ and $h = -1$ for $S_2$ type algebras.
• The antipode is explicitly constructed via $S(u_j^i) = { m D}^{-1} { m S}_L(u_j^i)$, and satisfies the antipode axioms, confirming that $H( ext{A})$ is a Hopf algebra.
• The inverse of the quasi-determinant satisfies $S({ m D}) = { m D}^{-1}$, consistent with the Hopf algebra axioms and the counit property $ ho({ m D}) = 1$.