On algebras obtained by tensor product

Abstract

MSC:primary 18D50, 15A69, 17A30 secondary 17D25 Keywords:Tensor product of algebras over quadratic operads Let P be a quadratic operad with only one generating operation. We define an associated maximal operad P such that for any Palgebra A and P-algebra B, the algebra A ⊗ B is again a P-algebra for the classical tensor product.

Universal Deformation Formulas

“…Let A be an associative algebra and B = U(L). Then there is a one-to-one correspondence between B-module algebra structures on A and Lie algebra morphisms (17) α : L → Der(A) from L to the Lie algebra Der(A) of derivations of A.…”

“…Several other examples of diagonals ∇ : P → Q ⊗ R, sometimes very exotic, can be found in [17]. As far as B-module R-algebra structures are concerned, one has an obvious analog of Proposition 5.1, with the map (17) replaced by α : L → Der R (A), where Der R (A) denotes the Lie algebra of derivations of the R-algebra A.…”

“…Several other examples of diagonals ∇ : P → Q ⊗ R, sometimes very exotic, can be found in [17]. As far as B-module R-algebra structures are concerned, one has an obvious analog of Proposition 5.1, with the map (17) replaced by α : L → Der R (A), where Der R (A) denotes the Lie algebra of derivations of the R-algebra A. We use this randomly chosen type of algebras to illustrate Proposition 7.1 and demonstrate how our theory applies to algebras having other than binary operations.…”

Associative and Lie deformations of Poisson algebras

“…is equal to Poiss, that is, Poiss is self-dual. In [19] we defined, for a binary quadratic operad E, an associated quadratic operad E which gives a functor…”

On the non-Koszulity of ternary partially associative operad

“…In [8] we have defined, given a quadratic operad P, a quadratic operad P by the following caracteristic property: For every P-algebra A and every P-algebra B, the tensor product A ⊗ B is a P-algebra.…”