# How to correctly name “irreducible subrepresentation of an indecomposable representation”

I am studying the representation theory of some infinite dimensional algebras, for example infinite dimensional Lie-algebras, Kac-Moody algebras, W-algebras. These algebras arise as symmetry algebras of tow-dimensional conformally invariant quantum field theories.

I encountered the case that I have to study representations which are indecomposable, i.e. which cannot be decomposed into a direct sum of irreducible representations. However, in my cases, they always contain a irreducible sub-representation.

How can I name this irreducible sub-representation with a concise, short term so I don't have to write "the irreducible sub-representation of the indecomposable representation" all the time?

• "direct sum of all irreducible subrepresentations" is called socle, but I guess "indecomposable direct summand of the socle" is not shorter. – Julian Kuelshammer Nov 14 '12 at 10:08
• Are you saying that this irreducible subrep is unique? In that case it is just the socle. – Noah Snyder Nov 14 '12 at 12:28
• Your definition of indecomposable isn't the one normally used in mathematics: instead, this word refers to a module which can't be written as the direct sum of two proper submodules (nothing being said about the irreducibility of submodules). Aside from this, the long quoted description in your header is probably the best you can do in general. Naturally you can always make up new words, but if you use established terms you need to accept their standard meanings. – Jim Humphreys Nov 14 '12 at 15:53
• P.S. In most settings the language of "representations" is equivalent to the language of "modules" (over some associative algebra), in which case it's shorter to write "simple submodule" than "irreducible subrepresentation". On the other hand, physicists and sometimes others often shorten things by writing "irrep" and the like. But the word 'indecomposable" has no obvious substitute except the shorthand "indec" rarely used except in lectures. – Jim Humphreys Nov 14 '12 at 16:05
• Thank you, @Julian and @Noah for bringing the notion `socle' to my attention. And many thanks to @Jim for explaining to me that my language is mathematically imprecise. In fact, the situation I mentioned in my question is only a special case of indecomposable representations. The misleading wording happened because the typical situation we encounter in my physics context is, that all (relevant) representations can be written as direct sums of irreducible representations. By the way, I decided to use the long phrase given in my question. – Nithilher Aug 25 '13 at 19:54