Smash biproducts of algebras and coalgebras

Let L and H be algebras and coalgebras (but not necessarily bialgebras, and consider two maps R : H ⊗L → L⊗H and W : L⊗H → H ⊗L. We introduce a product K = L W⊲⊳R H , and we give necessary and sufficient conditions for K to be a bialgebra. Our construction generalizes products introduced by Majid [13] and Radford [20]. Also some of the pointed Hopf algebras that were recently constructed by Beattie, Dǎscǎlescu and Grünenfelder [2] appear as special cases. 0 Introduction In a braided monoidal category, we can consider the smash product of two algebras A and B. The multiplication is then given by the formula mA#B = (mA ⊗mB) ◦ (IA ⊗RA,B ⊗ IB) (1) where R is the braiding on the category. In order to apply (1) to two particular algebras A and B, we do not need a braiding on the whole category, it suffices in fact to have a map R : B⊗A → A⊗B. This new algebra A#RB will be called a smash product, if it is associative with unit 1A#1B . We will work in the category of vector spaces over a field k, and give necessary and sufficient conditions for R to define a smash product. The smash product can be determined completely by a universal property, and it will also turn out that several variations of the smash product that appeared earlier in the literature are special cases. The construction can be dualized, leading to the definition of the smash coproduct of two coalgebras C and D (Section 3). The main result of this note (Section 4) is the fact that we can combine the two constructions, and this leads to the definition of the smash biproduct of two vector space Tempus visitor at UIA Research supported by the bilateral project “Hopf algebras and co-Galois theory” of the Flemish and Romanian governments. Research supported by the “FWO Flanders research network WO.011.96N”