Cyclic Paragrassmann Representations for Covariant Quantum Algebras

This report is devoted to the consideration from the algebraic point of view the paragrassmann algebras with one and many paragrassmann generators θ 2, θ 2 p+1 = 0. We construct the paragrassmann versions of the Heisenberg algebra. For the special case, this algebra is nothing but the algebra for coordinates and derivatives considered in the context of covariant differential calculus on quantum hyperplane. The parameter of deformation q in our case is (p+1)-root of unity. Our construction is nondegenerate only for even p. Taking bilinear combinations of paragrassmann derivatives and coordinates we realize generators for the covariant quantum algebras as tensor products of (p + 1) × (p + 1) matrices. There is now the extensive literature about finite dimensional cyclicrepresentations for quantum algebras with q being a root of unity (see e.g. [2],[24]). It is rather interesting to relate our paragrassmann representations with representationsexplored in [2],[24]. At the end of our talk we discuss the paragrassmann extensions of the Virasoro algebra. This report is largely based on the papers [25–27].