On the finite dimensional quantum group M_3 + (M_{2|1}(Lambda^2))_0

TL;DR

This paper explores a specific finite-dimensional non semi-simple quantum group algebra, detailing its structure, representations, and connections to quantum groups, modular representations, and potential applications in particle physics.

Contribution

It introduces and analyzes the properties of a new finite-dimensional quantum group algebra related to M_3 and Grassmann algebra, including its representations and physical implications.

Findings

H is a finite-dimensional non co-commutative Hopf algebra.

H relates to the quantum enveloping algebra of SLq(2) at a cubic root of unity.

The algebra's structure connects to modular representations of SL(2,F3) and potential modifications of the Standard Model.

Abstract

We describe a few properties of the non semi-simple associative algebra H = M_3 + (M_{2|1}(Lambda2))_0, where Lambda2 is the Grassmann algebra with two generators. We show that H is not only a finite dimensional algebra but also a (non co-commutative) Hopf algebra, hence a finite dimensional quantum group. By selecting a system of explicit generators, we show how it is related with the quantum enveloping algebra of SLq(2) when the parameter q is a cubic root of unity. We describe its indecomposable projective representations as well as the irreducible ones. We also comment about the relation between this object and the theory of modular representations of the group SL(2,F3), i.e. the binary tetrahedral group. Finally, we briefly discuss its relation with the Lorentz group and, as already suggested by A.Connes, make a few comments about the possible use of this algebra in a modification of the Standard Model of particle physics (the unitary group of the semi-simple algebra associated with H is U(3) x U(2) x U(1)).