Does anyone know what it means for a $V(1)$ to generate the tensor category of (finite dimensional) representations of $sl_2(\mathbb{C})$. Here $V(1)$ is the 2 dimensional irreducible Lie algebra representation of $sl_2(\mathbb{C})$
I (think I) can compute (by induction) that $V(1)^{ \otimes n} $ contains a vector of weight n, specifically $v_1 \otimes \ldots \otimes v_1$, where $v_1$ is the heighest weight vector, and this is highest possible weight of a vector in $V(1)^{\otimes n}$ (also this is the only weight $n$ vector in the decomposition). It follows that $V(1)$ contains $V(n)$ as a direct summand.
Is this sufficient to conclude that $V(1)$ generates the tensor category of finite dimensional $sl_2(\mathbb{C})$ representations?
I'm not really sure what it means to generate a tensor category (the notes I am reading don't explain), and I couldn't find it on google. Hopefully someone here knows off of the top of their head, or can suggest somewhere to look.
(I am trying to prove that the coordinate ring of $SL_2(\mathbb{C})$ is isomorphic to the algebra $O$ of matrix coefficients for the Hopf-algebra that is the universal enveloping algebra $U(sl_2(\mathbb{C})$, and part of the hint was to show this categorical fact first. I see how to conclude from my computation above that the map $\mathbb{C}[a,b,c,d] \to O$ is surjective - the matrix coefficients for $V(1)^{\otimes n}$ are the degree n homogeneous polynomials in the matrix coefficients for $V(1)$, and if by induction one already has all the matrix coefficients for $V(m)$, $m \in [0, n-1]$, one can subtract them off to isolate the matrix coefficients coming from $V(n)$ in $V(1)^{ \otimes n}$ - but I partially want to make sure I know what the word generates means in this context, and I'd like to learn of a nice reference if possible.)
There are a couple of things "generate" could mean here. The weakest reasonable one, which suffices in this case, is "generates under taking tensor products, finite direct sums, and direct summands." In general, depending on the situation, you might want to take other kinds of colimits.
For references try looking up anything that deals with Tannaka reconstruction for affine group schemes.