On distributions on a topological group - there is a theory due to Bruhat, where test functions are the Schwarz-Bruhat functions. But I think what is meant here is the appropriate group algebra concept, which can be illustrated by convolution of measures (? appropriate hypotheses). Since the point made is about variance, it shouldn't matter so much which convolution algebra is taken.
Charles Matthews 11:07, 2 Nov 2003 (UTC)
- can form two different Hopf algebras over it. The first is the algebra of continuous functions from G to K whose product is the pointwise product. ε acting on a function gives its value at the identity and (Δ f)(x,y)=f(xy) for all x and y in G. (Sf)(x)=f(x-1).
The formula (Δ f)(x,y)=f(xy) does not work formally, since (Δ f) is supposed to be an element of the tensor product of C(G,K) and C(G,K). So apparently some map from C(GxG,K) to the tensor product of C(G,K) and C(G,K) is silently being used. I can see that it works for finite discrete G, but in general I don't know what map to use. Or are we using some "continuous" tensor product here? AxelBoldt 20:22, 10 Aug 2004 (UTC)
I removed the following list of examples from the main page, as I think they are not Hopf algebras as defined in this article. I believe they are possibly examples of "locally compact quantum groups", some sort of topological version of Hopf algebras. AxelBoldt 20:58, 3 Sep 2004 (UTC)
- Given a topological group G, we can form two different Hopf algebras over it. The first is the algebra of continuous functions from G to K whose product is the pointwise product. ε acting on a function gives its value at the identity and (Δ f)(x,y)=f(xy) for all x and y in G. (Sf)(x)=f(x-1). The coaction of this Hopf algebra upon noncommutative spaces is as a left (right) comodule. The other Hopf algebra we can construct is the convolution product algebra of distributions over G (i.e. its group ring). This time, the action of this Hopf algebra upon noncommutative spaces is as a left (right) module.
- If, in addition, G is a Lie group, it has a Lie algebra g. Its universal enveloping algebra can be turned into a Hopf algebra by εx=0, Δx=x⊗1+1⊗x and Sx=-x for all elements of the Lie algebra. There's an injective homomorphism from this Hopf algebra to the Hopf algebra of convolutions over G such that the image of this homomorphism is the subalgebra generated by the Dirac delta distribution and its derivatives over the identity of G.
- Given a Lie superalgebra L, we can turn it into a Hopf algebra as follows: Let A be the unital associative algebra generated by the elements of L and an element g subject to g2=1, gxg=(-1)xx for pure elements x of L, cx+dy (as a Lie superalgebra linear combination)=cx+dy (as a linear combination in A) and [x,y]=xy-(-1)xyyx for pure elements x, y in L. It's not quite the universal enveloping algebra although there is a canonical injective embedding of the universal enveloping algebra within A. Now, let ε(g)=1 and ε(x)=0 and and for pure elements x in L.
I always thought one can define a Hopf algebra over an arbitrary integral domain, not neccesary a field. At the very least, Hopf algebras over integral domains are being studied. Elenthel 22:24, 6 June 2006 (UTC)
Unique. The set Hom(H,H) is a monoid w.r.t. the convolution product. In this monoid, the antipode S : H --> H is inverse to the identity morphism 1 : H --> H. —Preceding unsigned comment added by 220.127.116.11 (talk) 02:13, 25 May 2009 (UTC)
This topic is shows up in quantum field theory in ways that I don't understand at all (Template:Arxiv from here). Maybe some of that can be useful to the article. 18.104.22.168 (talk) 10:29, 14 March 2011 (UTC)
Having only come across Hopf algebras in algebraic topology, the antipode was something new to me from this page. I think it would be helpful to others like me to add the fact that if a bialgebra is graded (with non-negative grades only) and the zero grade part of it is isomorphic to the underlying field, then an antipode always exists, so such a bialgebra can always be made into a Hopf algebra. There is a proof of this at why graded bi algebras have antipodes.
Now, in the algebraic topology case that the bialgebra is the cohomology ring of a path-connected H-space, we also have that for an element h of grade n>0, the grade n component of (Δh) is . Am I right in thinking that this is not in general true for an arbitrary Hopf algebra ? Rfs2 (talk) 10:37, 21 July 2011 (UTC)