Actions of Commutative Hopf Algebras

Crossed Products and Inner Actions of Hopf Algebras

“…(5) follows from (l)- (4) (1) for all h,l G H. Applying id g £ to both sides, we obtain t(h, I) = e(hl)l, so that t is trivial. This yields the second assertion.…”

“…Since 4.5(2) is true for (ipi,r), t(H x H) Ç Z(A), and the right side of 4.5(1) for (ipu,cr) becomes (1) so that this yields 4.5(1) for (ipu,o). Conversely, 4.5(1) for (ipu,<^) yields 4.5(1) for (ipf,r), using the invertibility of u.…”

Identities of graded algebras and codimension growth

“…. , wt(b jq+2 · · · b (j+1)q ) are distinct non-identity elements of H since they are equal to the respective elements in (1 are distinct non-identity elements of H for h from (2). Now take 0 = c…”

“…The situation is quite different in the case of finite groups. It is well known that A = ⊕ g∈G A g is a PI-algebra if and only if A e satisfies a non-trivial polynomial identity provided that |G| < ∞ [1]. Moreover, even if G is infinite but the number of elements g ∈ G such that A g = 0 is finite, any non-trivial identity of A e implies a non-trivial identity on whole of A (see [3]).…”

“…For instance, if A e is nilpotent and G finite, then it is not hard to show that A is also nilpotent. If A e satisfies a standard identity of degree m, then A satisfies a power of the standard identity of degree dm where d = |G| [1]. If A e satisfies an identity of degree d, then A has a non-trivial identity of degree bounded by a function of d and |G| (see [2], [3]).…”