Let us work over $\mathbb C$. Suppose that $G$ is a semisimple algebraic group and let $H \subset G$ be a maximal torus. Consider a dominant weight $\omega$, then one can associate a unique irreducible $G$-representation $V(\omega)$.

Consider the following example. Let $G=\mathrm{SL}_3(\mathbb C)$ and let $W=\mathbb C^3$ be the standard $G$-representation. We define
$$
V=\mathrm{Sym}^2W,
$$
it is a $6$-dimensional irreducible representation for $G$. Moreover, we can also see $V$ as the space of symmetric $3 \times 3$ matrices. In particular, it inherits the structure of unital ~~associative~~ algebra using the matrix product. This is an example of Jordan algebra.
On the other hand, if we define
$$
\tilde V=W \otimes W^\vee,
$$
it contains an irreducible $G$-representation that corresponds to the Lie algebra $\mathfrak g$.

In fact, these are all examples of $G$-representations that have also an algebra structure. My question is: given a group $G$ and a dominant weight $\omega$, can we know when $V(\omega)$ has a unital algebra structure?

Anyalgebra $V$ can be viewed as a map (the multiplication) $\mu: V \otimes V \to V$. If we don't require associativity OR any relation to $G$, then of course we can define an algebra structure on every irrep $V$ of $G$. Of course you do want some relation to $G$, I expect you want $\mu$ to be a $G$-map? Then the first step is to find out (using Weyl character formula or such) for which $V$ we have that $V$ appears as a summand in the decomposition of $V \otimes V$ into irreps. This is already a restriction. $\endgroup$3more comments