Realization of bicovariant differential calculus on the Lie algebra type noncommutative spaces

Abstract

Abstract. This paper investigates bicovariant differential calculus on noncommutative spaces of the Lie algebra type. For a given Lie algebra g0 we construct a Lie superalgebra g = g0 ⊕ g1 containing noncommutative coordinates and one-forms. We show that g can be extended by a set of generators TAB whose action on the enveloping algebra U (g) gives the commutation relations between monomials in U (g0) and one-forms. Realizations of noncommutative coordinates, one-forms and the generators TAB as formal power series in a semicompleted Weyl superalgebra are found. In the special case dim(g0) = dim (g1) we also find a realization of the exterior derivative on U (g0). The realizations of these geometric objects yield a bicovariant differential calculus on U (g0) as a deformation of the standard calculus on the Euclidean space.

Generalization of Weyl realization to a class of Lie superalgebras

“…This proves the conjecture stated in Ref. [29]. We point out that the proof presented here applies only to Lie superalgebras g = g 0 ⊕ g 1 with [g 1 , g 1 ] = {0}.…”

“…This establishes the conjecture stated in Ref. [29]. The Appendix provides alternative methods for solving the functional equation for f (t).…”

“…[21], and by now has been investigated by many authors from different points of view in Refs. [22][23][24][25][26][27][28][29]. A bicovariant differential calculus on Lie algebra type NC spaces was investigated in Ref.…”

“…A bicovariant differential calculus on Lie algebra type NC spaces was investigated in Ref. [29]. In this paper the authors use realizations of Lie superalgebras by formal power series in the Weyl superalgebra (Clifford-Weyl algebra) in order to construct geometric objects as deformations of the corresponding notions on the Euclidean space.…”

Derivation based differential calculi for noncommutative algebras deforming a class of three dimensional spaces

“…Such derivations satisfy the standard Leibniz rule, so our approach E-mail address: marmo@na.infn.it, patrizia.vitale@na.infn.it, azampini@na.infn.it. Date: 16 May 2018. differs from the one developped in [37,38], where a quantum phase space on a Lie algebra type non commutative space is defined by deforming the coproduct on a suitable Hopf algebra.…”