Let (H,R) be a quasitriangular weak Hopf algebra, and A a quantum commutative weak H-module algebra. We establish the relationship of homological dimensions between weak smash product algebra A#H and A under some conditions. As an application, we consider the case of twisted weak Hopf algebra. Mathematics Subject Classification (2010): 16T05

We introduce a more general version of the so-called L-R-smash product and study its relations with other kinds of crossed products (two-sided smash and crossed product and diagonal crossed product).

We introduce a more general version of the so-called L-R-smash product and study its relations with other kinds of crossed products (two-sided smash and crossed product and diagonal crossed product). We also give an interpretation of the L-R-smash product in terms of an L-R-twisting datum.

Suppose that a group G acts by automorphisms on a (restricted) Lie algebra L over a field K of positive characteristic. This gives rise to smash products U(L)#K[G] and u(L)#K[G] . We find necessary and sufficient conditions for these smash products to satisfy a nontrivial polynomial identity.

Let k be a commutative ring. E∞ k-algebras are associative and commutative k-algebras up to homotopy, as codified in the action of an E∞ operad; A∞ k-algebras are obtained by ignoring permutations. Using a particularly well-behaved E∞ algebra, we explain an associative and commutative operadic tensor product that effectively hides the operad: an A∞ algebra or E∞ algebra A is defined in terms of...

We construct a smash product operation on secondary homotopy groups yielding the structure of a lax symmetric monoidal functor. Applications on cup-one products, Toda brackets and Whitehead products are considered.