2002/07/31 by Michel Dubois-Violette, Todor Popov
Mathematics · Physics and Astronomy · #math.QA #hep-th #math-ph #math.MP
published as Lett.Math.Phys. 61 (2002) 159-170 · 14 pages
arxiv created 2002/09/10 · arxiv updated 2009/11/30
After some generalities on homogeneous algebras, we give a formula connecting the Poincaré series of a homogeneous algebra with the homology of the corresponding Koszul complex generalizing thereby a standard result for quadratic algebras. We then investigate two particular types of cubic algebras: The first one called the parafermionic (parabosonic) algebra is the algebra generated by the creation operators of the universal fermionic (bosonic) parastatics with D degrees of freedom while the second is the plactic algebra that is the algebra of the plactic monoid with entries in \1,2,..., D\. In the case D=2 we describe the relations with the cubic Artin-Schelter algebras. It is pointed out that the natural action of GL(2) on the parafermionic algebra for D=2 extends as an action of the quantum group GLp,q(2) on the generic cubic Artin-Schelter regular algebra of type S1; p and q being related to the Artin-Schelter parameters. It is claimed that this has a counterpart for any integer D≥ 2.