Talk:Semisimple representation

To respond to the edit summary “the more general semisimple objects don't really belong in the first sentence, ” by {{ping|MarkH21}}, first I don’t disagree. But it’s still important, I think, to mention semisimple module as well as semisimple operator are examples of semisimple representations in the lead; perhaps in the second paragraph. (Incidentally, the opposite is also possible in the sense: given a set of operators, one can consider the algebra generated by it and then a semisimple representation is also a semisimple module over that algebra.) I wonder what the best way is. —— Taku (talk) 21:13, 15 November 2019 (UTC)

:{{ping|TakuyaMurata}} Sure, at the very least we can create a section in the article body and then summarize that later in the lead. — MarkH21 (talk) 10:58, 16 November 2019 (UTC)

::The thing is this article is about a specific type of a representation (as opposed to a representation in general) and so it seems strange to have a section that discusses the definition of a representation in a somehow generalized sense. Anyway, for now, I will add a few sentences to the "equivalent characterization" section; note the proof is abstract in that it doesn't care a representation is of a group, an algebra or a single operator. -- Taku (talk) 23:53, 17 November 2019 (UTC)

:::I think that having different sections for different kinds of representations is fine. For instance, adding to the existing sections on groups and Lie algebras. — MarkH21 (talk) 23:56, 17 November 2019 (UTC)

::::No, I am *against* such a structure. Because this article is about a rep not an algebraic object, the emphasis should be on the structure of a rep. For example, the article needs (and I am planning to add later) a discussion on multiplicity of simple reps; that's independent of whether one is considering a rep of a group or an algebra. (In fact, with due care, one can more ore less identify a rep of a Lie group and a rep of a Lie algebra.) -- Taku (talk) 01:02, 18 November 2019 (UTC)

:::::Ah sorry, I misunderstood what you meant. But representations of algebras and operators are representations; should this just be about group representations? It wouldn’t be terrible to include a generalization section even if it was focused on group representations, particularly since “semisimple representation” may refer to them in literature (I also don’t mean to include semisimple objects in general). — MarkH21 (talk) 01:23, 18 November 2019 (UTC)

::::::I am not sure if I follow. I am against structuring articles according to types of algebraic objects. In physics, *as far as I understand*, it's common to treat the same rep as a that of a Lie group and that of a Lie algebra. In other words, the focus is on a rep and you stay flexible about whether the action is through an algebra or a group. This is why it's a bad idea to have an article structure based on algebraic types (am I making sense?) -- Taku (talk) 22:56, 18 November 2019 (UTC)

::::::I am for having a separate section on the infinite-dimensional rep case, since in that case, one usually has the *completion* of a semisimple representation, as opposed to a semisimple representation, as you have pointed out, (and so one needs an extra care). -- Taku (talk) 23:27, 18 November 2019 (UTC)

::::::On the last point: another option is to have a section of a regular representation (which is semisimple rep or the completion of a semisimple rep in the interesting case). Maybe we don’t need to give a detailed discussion of an infinite-dimensional semisimple rep. —- Taku (talk) 03:25, 19 November 2019 (UTC)