Talk:Bialgebra

message: In the "Comultiplication and unit" diagram, shouldn't eta and eta x eta trade places?  I mean, eta x eta : K x K --> B x B should be on the left, and eta : K --> B should be on the right (without changing the direction of the arrows), so that the digram states Delta(eta(k)) = eta(k) x eta(k) for all k in K.
Damiano, no username, 5 January 2007
Agreed...if also the top thing says K x K = K instead of K = K x K.
SeaRisk 03:32, 21 August 2007 (UTC)
thread_title: Misplaced arrows

message: Hey there, you crazy kids; this hasn't got any introduction! It's pretty important to have one because they're good at sliding into the topics. Uxorion 14:04, 14 August 2007 (UTC)
thread_title:  This article has no introduction! 

message: I'm puzzled by where it says "Unit and counit (antipode)" -- I've never heard "antipode" used in this way. Isn't the antipode a thing that a bialgebra might or might not have (and if it does, it's a Hopf algebra)?
SeaRisk 02:17, 21 August 2007 (UTC)
thread_title:  Antipode? 

message: Is it necessary that K is a field? Can one take a ring (maybe commutative with unit)?
--Udoh (talk) 09:29, 19 May 2009 (UTC)
thread_title:  Generalization 

message: The coassociativity and counit section appears to be reusing a diagram from another page. Here, the object (bialgebra) in question is uniformly denoted throughout the article by B, but in the diagram, it is denoted by C, leading to the awkward phrasing "Coassociativy and counit are expressed by the commutativity of the following two diagrams with B in place of C." For clarity, a diagram using the same notation as the rest of the article should be used. (As an aside, I am very new to editing wikipedia, and do not trust myself to make the appropriate change effectively. My apologies if I neglected some normal piece of protocol in my post.)
131.220.135.75 (talk) 16:06, 10 September 2015 (UTC)
thread_title: Bad Notation

message: If G is a finite group or monoid then functions from G to R give a bialgebra. If G is an arbitrary group then it seems to me that much of the example explained in the page as it stood before I just inserted the word "finite" is vague, to say the least. If G is not finite then the functions from G to the reals are not spanned by the "delta functions" sending g in G to 1 and everything else to 0; the space is much bigger, and hence the definition in the text makes no sense. One can instead use finitely-supported functions and as far as I can see this would be a fine example of a bialgebra for an arbitrary group or monoid (although I did not check this yet; I'm about to formalise this example in Lean so we'll find out soon enough!). However much of the section seems to be talking about an example coming from probability theory, and as it stands the explanation is far too vague to make any kind of sense: one would presumably have to look at continuous or measurable functions, and demand at the very least a topology or sigma algebra structure on G, and then restrict to functions from G to R which are continuous or measurable or something: but I am not an expert in this area and don't have a clue how to fix up this probability example. If one restricts to G finite all these problems go away, so inserting the word "finite" as I have done at least makes the first example make sense, and makes the probability example more likely to make sense. I would suggest that if someone wants to revert my edit and remove this word "finite" then they should probably split this example into two examples: first for a general group or monoid where one uses finitely supported functions, and secondly the probability example, where the correct assumptions on G and on the functions from G to the reals are explicitly written down in order to make the example have some mathematical meaning beyond the vague ideas currently on the page (which I don't understand and am thus in no position to fix :-( ) KevinBuzzard (talk) 21:36, 18 December 2023 (UTC)
thread_title:  Example coming from group theory and probability theory