# What is meant by “direct summand in a tensor product”?

I am currently working on the topic of Lie - Algebras and I have stumbled a few times over the expression "direct summand in a tensor product".

The text says that $\ V(\lambda)$ as an indecomposable direct summand is expressible as a tensor product.

$V(\lambda)$ denotes a semisimple L-module of highest weight $\lambda$.

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No offense, but I have to ask: How do you define a semi-simple module without knowing what a direct summand is? – t.b. Jul 8 '11 at 21:25
I'd wager that there is a mis-statement/question: probably $V(\lambda)$ is an indecomposable direct summand in a tensor product. – paul garrett Jul 8 '11 at 21:42
Please excuse my lack of English language skills. – nikki Jul 8 '11 at 22:16
Please excuse my lack of English language skills, it is not my native language. The topic Lie - Algebras is new to me and I have just begun reading a book about the topic. I meant "the direct summand in a tensor product". – nikki Jul 8 '11 at 22:22
There are two distinct notions here. Do you understand what a tensor product is? Do you understand what a direct summand is? – Qiaochu Yuan Jul 8 '11 at 22:45

To give a simple example, rather than a definition: with Lie algebra $g=sl(2)$, the "standard" repn $V(1)$ is 2-dimensional, and has highest weight $\pmatrix{1 & 0 \cr 0 & -1}\rightarrow 1$, where a highest-weight vector is $\pmatrix{1 \cr 0}$ and is annihilated by the raising operator $\pmatrix{0 & 1 \cr 0 & 0}$. The other weight is $-1$.
The tensor product of this std repn with itself is a _direct_sum_: $$V(1) \otimes V(1) \approx V(2) \oplus V(0)$$ More generally, for $g=sl(2)$ and $n\ge 1$, $$V(1) \otimes V(n) \approx V(n+1) \oplus V(n-1) \hskip30pt\hbox{(this was wrong earlier!)}$$ [The analogue for $V(1)$ replaced by $V(m)$ previously addled my wits, leading to an erroneous version of the previous.] Also, $V(2)$ has weights $2,0-2$, and $$V(2)\otimes V(n) \approx V(n+2) \oplus V(n) \oplus V(n-2) \hskip30pt\hbox{(for n\ge 2)}$$ Each $V(\lambda)$ is irreducible, in the sense that it has no proper subrepns/submodules.