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highest weight

Highest weight vectors are called vacuum vectors in physics/orbifold literature: attachments/Pasted%20image%2020220213161432.png

Highest weight categories

Abelian categories over fields $k$: categories $\mathsf{C}$ where $\mathsf{C}(x,y) \in {}_{k}{\mathsf{Mod}} $.
Locally artinian categories: admits arbitrary directed unions of subobjects, and every object is a union of its subobjects of finite length.
Highest weight categories: there exists an interval-finite poset $\Lambda$ of weights satisfying
- There is a complete collection of non-isomorphic simples $\left\{{S(\lambda)}\right\}_{\lambda \in \Lambda}$.
- There are objects with embeddings $S(\lambda) \hookrightarrow A(\lambda)$ such that all composition factors $S(\mu)$ of the quotient $A(\lambda)/ S(\lambda)$ satisfy $\mu < \lambda$. Consequently $\mathsf{C}(A(\lambda), A(\mu))$ has finite $k{\hbox{-}}$dimension, and $[A(\lambda): S(\mu)] < \infty$.
- Each simple has an injective envelope $I(\lambda)$ which admits a good filtration $0 = F_0(\lambda) \hookrightarrow F_1(\lambda) \cong A(\lambda) \hookrightarrow\cdots \hookrightarrow\bigcup F_i(\lambda) \cong I(\lambda)$ with associated graded pieces of the form ${\mathsf{gr}\,}_n = A(\mu(n))$ where $\mu(n) > \lambda$ and $\mu(n)$ occurs only finitely many times.
Not all objects are finite length.
$S(\lambda)$ is the socle of $A(\lambda)$.
The Exts between various $A$ and $S$ detect weight poset relations.
Example: take $A\in {}_{k} \mathsf{Alg} $ to be the $n\times n$ upper-triangular matrices and $A(i) = S(i)$ the irreducible 1-dimensional right $A{\hbox{-}}$module whose injective envelope has dimension $n-i+1$.