(α,β)-derivations on quantum function algebras

Abstract

(α,β)-derivations, introduced by Cohn, are defined on several noncommutative function algebras including the Hopf function algebras of the quantum groups SLq(n), n=2,3, where their interpretation is most apparent. In analogy with algebraic groups, left-invariant (α,β)-derivations on the Hopf function algebra of SLq(n), n=2,3, generate its quantum universal enveloping algebra Uq(sl(n)). Derivations and their algebras are also found on the non-Hopf, function algebras of two noncommutative varieties, which are constructed similarly to SLq(2) and SLq(3). General (α,β) derivations (analogs of arbitrary vector fields) are found for SLq(2) and the quantum plane. The analog of the tangent bundle TSLq(2) is defined, and its relation to derivations is discussed. Some examples of (α,β) derivations on commutative algebras are also given. Generalities about (α,β) derivations and their algebras are discussed.