@Noah: I prefer the generality of the Comes-Ostrik viewpoint, as explained in my comment to Torsten. My question was motivated by the unsolved problem of determining blocks for the parabolic subcategories of the BGG category, if "block" is defined in a general way. It's tempting to solve such a problem just by giving a definition to fit the situation. Given the finiteness properties in the BGG case one wants a parabolic subcategory to be a direct sum of indecomposable subcategories fitting a general notion of block. Describing those subcategories may be tricky.

@Torsten: Your helpful added comment is in the spirit of Jantzen's discussion, using finiteness conditions. I'd like to avoid idempotents or centers, which can make it easier (and more interesting) in some special cases to find block decompositions and describe the blocks. Without indecomposables the whole notion of "block" does lose interest. Still I'm inclined to start with the most general language and then verify in special cases that the category is a direct sum of blocks even if they are hard to classify. I'd prefer not to solve that problem by ad hoc definition.

Short answer: no. Books by Borel, Springer, me have limited coverage of structure over arbitrary fields. Try AMS online surveys by Tits (1965) PSPUM/9, Springer (1979) PSPUM/33.1: e-math.ams.org/publications/online-books/online_subject (and note Springer's 3.4 for your original question). Extensive details are in papers by Borel-Tits (IHES papers online at www.numdam.org): Groupes reductifs (Publ. Math. IHES 27, 1965), etc. Lie groups are just an example there. But books by Margulis, Zimmer focus more on ergodic theory, less on background. Good luck!

Local MacPherson students Braden and Gunnells tell me they took notes of their own (incomplete), but suggest that Vybornov's thesis would be a better source anyway if that's findable.

The structure of arbitrary real semisimple Lie groups is complicated to develop (while "unipotent" elements are more visible for linear algebraic groups), so your first question takes work to sort out. For compact Lie groups the special feature is that every element is conjugate to one in a fixed maximal torus, hence semisimple. In the noncompact case, generation by unipotents appears concretely for classical groups like SO$(n,1)$; but in general Borel-Tits structure theory for algebraic groups is probably most helpful and covers all fields of definition: noncompact becomes "isotropic".

Sorry, I was distracted by the header and didn't look far enough into the question. Some of my colleagues have been fond of sine-Gordon, but I'm an outsider.

@Victor: Thanks for the clarification. It's good to have a notion of block in a general situation, but always with the understanding that (for instance) finiteness conditions on the category may lead to better results. For me it's been confusing to encounter inconsistent uses of the label "block", with some meanings looser than others.

It's hard to say what is a "standard reference", which partly motivates my question. The approach of Comes-Ostrik seems reasonable, leaving aside their underlying field of characteristic 0. They emphasize indecomposable objects and morphisms rather than simple objects and extensions; both versions amount to the same thing in familiar examples, but their version may be more flexible.

@Theo: I'm not sure about CW, so I just changed tags. This is not meant for open-ended discussion but probably has no single "correct" answer.

@Noah: The examples I'm thinking about arise in various branches of representation theory, but "abelian category" is a good setting in view of Freyd-Mitchell embedding and recent work in other areas.

@xiyu The short answer is that Harish-Chandra's classical description of the center of a universal enveloping algebra of a finite dimensional semisimple Lie algebra doesn't generalize well. And a good definition of "block" might avoid referring to special features like central characters.

Fiebig's paper is definitely helpful and has been published (I think in the arXiv version): MR2205072 (2006k:17040) 17B67 Fiebig, Peter (D-FRBG), The combinatorics of category O over symmetrizable Kac-Moody algebras. Transform. Groups 11 (2006), no. 1, 29–49. Then try Frenkel-Gaitsgory. For blocks, note that already in the classical case, nonintegral weights require special arguments using Jantzen's integral Weyl subgroups. It's important to be precise about what you mean here by "block", since the term is sometimes used more loosely.

I agree that examples unrelated to Witt vectors might occur. There may even be examples which look entirely artificial, but of course it would be much nicer if instead they all arose naturally in a scheme-theoretic framework. Your last comment is important to keep in mind, since work from about 1989 by Gerry Schwarz and others has made it clear that some reductive group actions on affine space can't be linearized. Anyway, low degree cohomology is a natural tool if one can learn enough about it.

@Shizhuo: Sources for self-study of Kac-Moody theory are somewhat limited, but besides the book by Kac there are useful books by Moody-Pianzola and more recently Roger Carter (Cambridge Press). Carter's book is quite readable, but is limited to the most applicable cases: finite dimensional and affine Lie algebras.