added result concerning endowing (some) dual Hopf algebras with a symmetric special Frobenius algebra structure

]]>typo fixed

Fang

]]>Corrected the incorrect claim “Both Hopf algebras and Frobenius algebras are examples of bialgebras”

]]>added example of $H_8$ Kac-Paljutkin Hopf algebra

]]>Added thm that all finite-dimensional Hopf algebras can be given a Frobenius algebra structure.

]]>have hyperlinked *Matilde Marcolli*

removing old query box.

Mike, can you do something with these notes that I took at some point as a grad student? I don't know this stuff very well, which is why I don't incorporate them into the text, but at least I cleaned up the formatting a bit so that you can if you like it. —Toby

One can make a group into a Hopf algebra in at least $2$ very different ways. Both ways have a discrete version and a smooth version.

Given a (finite, discrete) group $G$ and a ground ring (field?) $K$, then the group ring $K[G]$ is a cocommutative Hopf algebra, with $M(g_0,g_1) = g_0 g_1$, $I = 1$, $D(g) = g \otimes g$, $E(g) = 1$, and the nifty Hopf antipodal operator $S(g) = g^{-1}$. Notice that the coalgebra operations $D,E$ depend only on $Set|G|$.

Given a (finite, discrete) group $G$ and a ground ring (field?) $K$, then the function ring $Fun(G,K)$ is a commutative Hopf algebra, with $M(f_0,f_1)(g) = f_0(g)f_1(g)$, $I(g) = 1$, $D(f)(g,h) = f(g h)$, $E(f) = f(1)$, and the nifty Hopf antipodal operator $S(f)(g) = f(g^{-1})$. Notice that the algebra operations $M,I$ depend only on $Set|G|$.

Given a (simply connected) Lie group $G$ and the complex (real?) field $K$, then the universal enveloping algebra $U(G)$ is a cocommutative Hopf algebra, with $M(\mathbf{g}_0,\mathbf{g}_1) = \mathbf{g}_0 \mathbf{g}_1$, $I = 1$, $D(\mathbf{g}) = \mathbf{g} \otimes 1 + 1 \otimes \mathbf{g}$, $E(\mathbf{g}) = 0$, and the nifty Hopf antipodal operator $S(\mathbf{g}) = -\mathbf{g}$. Notice that the coalgebra operation $D,E$ depend only on $K Vect|\mathfrak{g}|$.

Given a (compact) Lie group $G$ and the complex (real?) field $K$, then the algebraic function ring $Anal(G)$ is a cocommutative Hopf algebra, with $M(f_0,f_1)(g) = f_0(g) f_1(g)$, $I(g) = 1$, $D(f)(g,h) = f(g h)$, $E(f) = f(1)$, and the nifty Hopf antipodal operator $S(f)(g) = f(g^{-1})$. Notice that the algebra operations $M,I$ depend only on $Anal Man|G|$.

Anonymouse

]]>adding Noam Chomsky’s recent papers on Hopf algebras in generative linguistics:

Matilde Marcolli, Noam Chomsky, Robert Berwick,

*Mathematical Structure of Syntactic Merge*(arXiv:2305.18278)Matilde Marcolli, Robert Berwick, Noam Chomsky,

*Old and New Minimalism: a Hopf algebra comparison*(arXiv:2306.10270)

Anonymouse

]]>added pointer to:

- Christoph Schweigert,
*Hopf algebras, quantum groups and topological field theory*(2022) [pdf]

added pointer to:

- Nicolas Andruskiewitsch, Walter Ferrer Santos,
*The beginnings of the theory of Hopf algebras*, Acta Appl Math**108**(2009) 3-17 [arXiv:0901.2460]

All calculations with Sweedler notation would expand immensely if one does not give some thinking on its use in first couple of instances of usage. Here for example, the second line taken for $h_{(1)}\otimes g_{(1)}$ instead of $h\otimes g$ is multiplied (and summed over dummy indices) with $S g_{(2)} S h_{(2)}$ and then just the renaming of indices using coassociativity gives 3rd line. Just spend one hour practicing Sweedler-assisted computations (once in your lifetime) and you will automatically observe such things on the fly for the rest of your life. (Easily observed) rule of a thumb is that the notation is bilinear so you can “contract” it with following leg if the coassociativity holds. I think it is more useful to figure out this rule yourself (why the step is legal and not out of nowhere) when you see it for the first time than to read the verbose explanation. Every reference using Sweedler notation in practice will freely contract with the next “leg”.

]]>Okay, sure, anti-involution.

]]>Following the language of star algebra maybe it is better to say algebra anti-involution as the antipode is an antihomomorphism. I don’t care, just suggestion so maybe resulting in a bit more uniform convention. Set theoretically it is just an involution, but in algebra world there is a distinction.

]]>I have added mentioning (here) of the notion of *involutive Hopf algebra* (and will give this its own little page, for ease of hyperlinking).

Then in the proposition (here) that the antipoide is an anti-homomorphism I have added the statement that, hence, involutive Hopf algebras are star-algebras.

Also added hyperlinking to *anti-homomorphism*.

added pointer to:

- Douglas Ravenel, W. Stephen Wilson,
*The Hopf ring for complex cobordism*, Bull. Amer. Math. Soc. 80 (6) 1185 - 1189, November 1974 (doi:10.1016/0022-4049(77)90070-6, euclid, pdf)

added pointer to:

- Neil Strickland,
*Bott periodicity and Hopf rings*, 1992 (pdf)

added pointer to

- W. Stephen Wilson,
*Hopf rings in algebraic topology*, Expositiones Mathematicae, 18:369–388, 2000 (pdf)

(this used to be referenced only at *W. S. Wilson* and without the pdf link, so I completed the item and copied it to here)

Under “Examples” I added the line

]]>the ordinary homology of an H-space (for instance a based loop space) is a Hopf algebra via its Pontrjagin ring-structure

Okay, thanks. I thought I had added the “commutative” qualifier where necessary, but thanks for catching places where I missed it.

]]>This fact is observed in bigger generality by

- Benoit Fresse,
*Cogroups in algebras over an operad are free algebras*, Commen. Math. Helv.**73**:4, 1998, 637–676 doi

Here is the Israel Berstein’s (not Joseph Bernstein!) article showing that cogroups in associative algebras are extremely few (unlike Hopf algebras), and are basically free as algebras (the context is also algebraic topology):

- Israel Berstein, On cogroups in the category of graded algebras. Trans. Amer. Math. Soc. 115 (1965), 257–269 jstor

I changed the critical part in the entry into:

In particular, a co-group in the category of (unital) commutative rings is a commutative Hopf ring and a cogroup in the category of (unital) commutative $k$-algebras is a commutative Hopf $k$-algebra; a fact highlighted in homotopy theory by Haynes Miller (in view of his generalization to commutative Hopf algebroids as cogroupoids in commutative algebra) in the context of discussion of dual Steenrod algebras, see (Ravenel 86, appendix A) for review.

Discussion of commutative Hopf algebras as cogroups is in

- {#Ravenel86} Doug Ravenel, appendix A1 of
*Complex cobordism and stable homotopy groups of spheres*, Academic Press 1986