[Paper Review] Differential calculus and connections on a quantum plane at a cubic root of unity

Robert Coquereaux, Ariel García+1 more|arXiv (Cornell University)|Jul 12, 1998

Algebraic structures and combinatorial models13 references3 citations

TL;DR

This paper constructs a differential calculus on the quantum plane at a cubic root of unity by realizing the algebra of N×N matrices as a reduced quantum plane acted upon by a finite-dimensional quantum group H (a quotient of U_q(sl(2,C)) at q³=1, dim H=27 for N=3). It introduces connections as 1-forms in this calculus, studies their transformation properties under H, and shows how tensoring with spacetime forms yields generalized connections covariant under both Lorentz and quantum group symmetries, providing a framework for noncommutative geometry with finite quantum group invariance.

ABSTRACT

We consider the algebra of N x N matrices as a reduced quantum plane on which a finite-dimensional quantum group H acts. This quantum group is a quotient of U_q(sl(2,C)), q being an N-th root of unity. Most of the time we shall take N=3; in that case \dim(H) = 27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess-Zumino complex. The quantum group H also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of H. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finite-dimensional quantum group.

Motivation & Objective

• To develop a differential calculus on the quantum plane at a cubic root of unity using matrix algebras as a reduced quantum plane.
• To study the action of a finite-dimensional quantum group H (a quotient of U_q(sl(2,C)) at q³=1) on this differential algebra.
• To define and analyze connections in the sense of noncommutative geometry as elements of the differential calculus.
• To construct generalized connections by tensoring the differential calculus with spacetime forms, ensuring covariance under both Lorentz and quantum group symmetries.
• To decompose the differential algebra into irreducible representations of the quantum group H, using representation theory.

Proposed method

• Realize the algebra of N×N matrices as a reduced quantum plane on which a quantum group H acts, with H being a quotient of U_q(sl(2,C)) at q being an Nth root of unity.
• Construct a differential calculus as a quotient of the Wess-Zumino complex, ensuring compatibility with the quantum group action.
• Define connections as 1-forms within this differential calculus, consistent with the axioms of noncommutative geometry.
• Decompose the differential algebra into irreducible representations of H using character theory and representation decomposition techniques.
• Tensor the differential calculus with differential forms on spacetime to generate generalized connections with joint covariance under Lorentz and quantum group symmetries.
• Use the representation theory of H to analyze the structure and transformation properties of the resulting connections.

Experimental results

Research questions

• RQ1 How can a differential calculus be consistently defined on the quantum plane at a cubic root of unity, particularly when the underlying algebra is realized as a matrix algebra?
• RQ2 What is the structure of the differential algebra under the action of the finite-dimensional quantum group H, and how does it decompose into irreducible representations?
• RQ3 How do connections in the sense of noncommutative geometry behave when they are elements of this differential calculus?
• RQ4 What are the transformation properties of generalized connections formed by tensoring the differential calculus with spacetime forms under both Lorentz and quantum group symmetries?
• RQ5 What role does the quantum group H (a quotient of U_q(sl(2,C)) at q³=1) play in organizing the differential structure and connection theory?

Key findings

• The differential calculus on the quantum plane is constructed as a quotient of the Wess-Zumino complex, ensuring compatibility with the quantum group action and providing a well-defined noncommutative differential structure.
• The quantum group H, of dimension 27 when N=3, acts on the differential algebra, and its decomposition into irreducible representations is explicitly analyzed using representation theory.
• Connections defined as 1-forms in the differential calculus inherit transformation properties under the action of H, enabling a noncommutative geometric framework with finite quantum group symmetry.
• Tensoring the differential calculus with spacetime differential forms yields generalized connections that are covariant under both the Lorentz group and the finite quantum group H.
• The construction realizes a unified gauge-theoretic framework where quantum group symmetry coexists with spacetime Lorentz covariance, offering a novel approach to gauge theories in noncommutative geometry.
• The final version of the paper, published in Rev. Math. Phys. 12 (2000), 227–285, includes corrections and a refined analysis of the representation decomposition and connection properties.