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 $\lambda$.
 A: 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.
In very crude terms, somewhat parallel to the appearance of these operations in quantum mechanics a long time ago: "irreducibles/indecomposables" are elementary particles,  the "tensor product" amounts to "crashing together" two repns, as you'd collide two particles, and "decomposing as a direct sum of irreducibles/indecomposables" is telling what elementary particles come out of the collision. Of course, this dramatically ignores the mathematical content... but perhaps the general shape of the action is of interest, prior to concrete definitions?
