Caracterisation des derivations interieures de l'algebre de weyl et de l'algebre d’heisenberg quantiques

Abstract

The inner automorphisms of an arbitrary group G are characterized by the fact that they extend to any group which contains G as subgroup [17] . Generally we show in [7] that an analog of this result for rings is not true . However, if A is a ring embedded in another ring B then any inner derivation (resp. automorphism) if of A extends to a derivation (resp. automorphism) of B. Furthemore, for all ideal I of A, ϕ induce a derivation (resp. automorphism) of the quotient algebra A/I so that . extends to all algebra including AS/I as subalgebra. We wonder whether these properties actually characterize inner derivations (resp. automorphisms) of A. We show that every derivation of the quantized Weyl algebra A 1(k,q) is a sum of an inner derivation and a derivation well specified, where the inner derivations and automorphisms are characterized by the preceding properties. Also, we give a complete description of derivations of Heisenberg algebra and a characterization of its inner derivations.