Derivations of Tensor Product of Algebras

We establish several results regarding the algebra of derivations of tensor product of two algebras, and its connection to finite order automorphisms. These results generalize some well-know theorems in the literature. Dedicated to Professor Bruce Allison on the occasion of his sixtieth birthday 0. Introduction In 1969, R. E. Block [B] showed that the algebra of derivations of tensor product of two algebras (satisfying certain finite dimensionality conditions) can be expressed in terms of the algebra of derivations and the centroid of each of the involved algebras. In 1986, G. Benkart and R. V. Moody [BM] used this (with a new proof) to establish some very interesting results concerning the finite order automorphisms and its connection to the algebra of derivations of tensor product of two algebras. They applied their results to determine the algebra of derivations of several important classes if infinite dimensional Lie algebras, including twisted and untwisted affine Kac–Moody Lie algebras [K], Virasoro algebras, and some subclasses of extended affine Lie algebras (see [AABGP],[BGK] and [N2] for extended affine Lie algebras). All algebras we consider will be over a field k. We denote by D(A) and C(A) the algebra of derivations and centroid of an algebra A, respectively. Let A be a perfect algebra and S be a commutative associative unital algebra. It is proved in [B, Theorem 7.1] and [BM, Theorem 1.1] that if A is finite dimensional then D(A ⊗ S) = D(A)⊗ S ⊕ C(A)⊗ D(S). (1) Therefore any derivation d ∈ D(A ⊗ S) can be represented as d = ∑