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date: 2022-02-13 16:13
modification date: Sunday 13th February 2022 16:13:48
title: highest weight
aliases: [highest weight vector, vacuum, highest weight representation, highest weight category]
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# highest weight

Highest weight categories are [Morita equivalent](Unsorted/Morita%20equivalence.md) to[[quasi-hereditary]] algebras.

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 $\cat{C}$ where $\cat{C}(x,y) \in \kmod$.
- 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 $\ts{S(\lambda)}_{\lambda \in \Lambda}$.
	- There are objects with embeddings $S(\lambda) \injects A(\lambda)$ such that all composition factors $S(\mu)$ of the quotient $A(\lambda)/ S(\lambda)$ satisfy $\mu < \lambda$. Consequently $\cat{C}(A(\lambda), A(\mu))$ has finite $k\dash$dimension, and $[A(\lambda): S(\mu)] < \infty$.
	- Each simple has an injective envelope $I(\lambda)$ which admits a good filtration $0 = F_0(\lambda) \injects F_1(\lambda) \cong A(\lambda) \injects \cdots \injects \Union F_i(\lambda) \cong I(\lambda)$ with associated graded pieces of the form $\gr_n = A(\mu(n))$ 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 $A\in \kalg$ to be the $n\times n$ upper-triangular matrices and $A(i) = S(i)$ the irreducible 1-dimensional right $A\dash$module whose injective envelope has dimension $n-i+1$.