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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
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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
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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
864:. But that's different -- no one is claiming that cofree coalgebras are Hopf algebras (although I guess they could be if all the right definitions are made, it being essentially just the dual to the tensor algebra). In the meanwhile, I am cleaning up the confusion that was in tensor algebra.
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where the sum is not a sum over grading, but simply a sum over other elements in the algebra. Its exactly like structure constants in a Lie algebra, but going the other way. I vaguely recall that there are some
Steeenrod-y things that have this kind of more complex structure.
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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.
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I don't understand how theta can be unique in the braided monoidal categories section. Isn't the whole point of a braiding (rather than a symmetry) that theta(a, b, c, d) may not be equal to the inverse of theta(a, c, b, d)?
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540:. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra $ \Hc$ which is commutative as an algebra. It is the dual Hopf algebra of the
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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.
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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: -->
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x of L, cx+dy (as a Lie superalgebra linear combination)=cx+dy (as a linear combination in A) and =xy-(-1)yx for pure elements x, y in L. It's not quite the
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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). The
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801:, it is mentioned that this coproduct does not turn the Tensor algebra into a Hopf algebra. One of the two articles has to be wrong. Which one is it? --
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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).
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Renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the
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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
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although there is a canonical injective embedding of the universal enveloping algebra within A. Now, let ε(g)=1 and ε(x)=0 and
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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
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Unique. The set Hom(H,H) is a monoid w.r.t. the convolution product. In this monoid, the antipode S : H --: -->
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I don't see anywhere where it says that, but the article on tensor algebra is seriously messed up.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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general I don't know what map to use. Or are we using some "continuous" tensor product here?
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made is about variance, it shouldn't matter so much which convolution algebra is taken.
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This topic is shows up in quantum field theory in ways that I don't understand at all (
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Is the antipode for the given bialgebra exactly unique or only up to iso? --
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The article no longer mentions a field, and gives a general definition.
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G, we can form two different Hopf algebras over it. The first is the
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is given as an example of a Hopf algebra. However, in the article
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In general, Hopf algebra are not graded. In general the
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In this article, the Tensor
Algebra with the coproduct
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