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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 22nd 2010
    • (edited Oct 30th 2013)

    Chris Schommer-Pries posted a question/suggestion in the query box at semisimple category

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeOct 30th 2013
    • (edited Oct 30th 2013)

    I have touched the formatting at semisimple category slightly, and hereby I am moving that old query box from there to here:

    begin forwarded discussion

    What does it mean to ’have subobjects’? (I assume that the ’direct sums’ are biproducts.) —Toby

    have subobjects = idempotents split and yes, finite biproducts. Simple objects are ones in which End(X) = k.

    Urs: shouldn’t we say something like: a category is semisimple if each object is a direct sum of finitely many simple objects?

    Bruce: Urs, you’re right, and that’s indeed the way one morally thinks about it, but it’s a less canonical way of proceeding. We ask ourselves: given a linear category with direct sums and subobjects, and a chosen maximal collection {X i}\{X_i\} of nonisomorphic simple objects, how can we check if its semisimple? In the one way, we have to check whether a certain canonically defined map is an isomorphism. In the other way, we have to check if each object VV can be expressed as a direct sum of the X iX_i’s. Actually finding such a decomposition would be a noncanonical operation. So your shorter more snappy definition would force an auditor to perform an evil thing if he actually wanted to check it :-) It’s that old thing about “only that part of a representation which behaves like an irreducible ρ\rho is canonical, the actual break-down of that rep into direct sums of ρ\rho’s is noncanonical”. That is, what is canonical is Hom(V,X i)Hom(V, X_i) and not V= in iX iV = \bigoplus_i n_i X_i.

    Also, if we just had “a category is semisimple if each object is a direct sum of finitely many simple objects” without the conditions on direct sums and subobjects then we could have someone who nastily removes, say, all three-dimensional vector spaces from VectVect. It would still satisfy “each object is a direct sum of finitely many simple objects” but it shouldnt be regarded as a semisimple category since there are “holes”.

    Urs: okay, so “each object is finite sum of simples” is the right idea, while the right defintion is a bit different. I have accoridngly now created a section “Idea” with the former statement. (Every entry should start with a section “Idea”!) See if you like this. Otherwise, feel free to adjust.

    But in any case, it would be nice to have a discussion on how the “right definition” implies that every object is isomorphic to a finite direct sum of representables.

    Chris Schommer-Pries: Shouldn’t you require that for simples, End(X,X) is a simple algebra, not necessarily the ground field? For example the category of H-modules where H is the quaternion algebra over the reals. Shouldn’t this be semi-simple?

    Chris: It is not standard to assume the direct sum decomposition is finite in defining a semisimple category. For instance, the category of all vector spaces is semisimple. What’s written here at present looks like a definition of semisimple for categories where it has already been assumed that every object has finite length.

    end forwarded discussion

    • CommentRowNumber3.
    • CommentAuthorTobyBartels
    • CommentTimeOct 31st 2013

    I agree with Chris at the end here. We should not restrict ourselves to finite sums, although that might be an interesting variation. (κ\kappa-semisimple, anybody?)

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 31st 2013

    There are different traditions. In references like those listed at FFRS-formalism “semisimple” implies the finiteness clause.

    (Though just checking with their latest I see that they did switch to saying “finitely semismple” now.)

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeNov 1st 2013

    they did switch to saying “finitely semismple”

    That seems like a reasonable use of language. Of course, in some contexts, that could be the default, just like ‘ring’ means something commutative to some people.