Mutual fund separation theorem

In the mathematical field of representation theory, a highest-weight category is a k-linear category C (here k is a field) that

• is locally artinian
• has enough injectives
• satisfies

$B\cap \left(\bigcup _{\alpha }A_{\alpha }\right)=\bigcup _{\alpha }\left(B\cap A_{\alpha }\right)$

for all subobjects B and each family of subobjects {A_α} of each object X

and such that there is a locally finite poset Λ (whose elements are called the weights of C) that satisfies the following conditions:

• The poset Λ indexes an exhaustive set of non-isomorphic simple objects {S(λ)} in C.
• Λ also indexes a collection of objects {A(λ)} of objects of C such that there exist embeddings S(λ) → A(λ) such that all composition factors S(μ) of A(λ)/S(λ) satisfy μ < λ.
• For all μ, λ in Λ,

$\dim _{k}\operatorname {Hom} _{k}(A(\lambda ),A(\mu ))$

is finite, and the multiplicity

$[A(\lambda ):S(\mu )]$

is also finite.

• Each S(λ) has an injective envelope I(λ) in C equipped with an increasing filtration

$0=F_{0}(\lambda )\subseteq F_{1}(\lambda )\subseteq \dots \subseteq I(\lambda )$

such that

1. $F_{1}(\lambda )=A(\lambda )$
2. for n > 1, $F_{n}(\lambda )/F_{n-1}(\lambda )\cong A(\mu )$ for some μ = μ(n) > λ
3. for each μ in Λ, μ(n) = μ for only finitely many n
4. $\bigcup _{i}F_{i}(\lambda )=I(\lambda ).$

Examples

• The module category of the $k$-algebra of upper triangular $n\times n$ matrices over $k$.
• This concept is named after the category of highest-weight modules of Lie-algebras.
• A finite-dimensional $k$-algebra $A$ is quasi-hereditary iff its module category is a highest-weight category. In particular all module-categories over semisimple and hereditary algebras are highest-weight categories.
• A cellular algebra over a field is quasi-hereditary (and hence its module category a highest-weight category) iff its Cartan-determinant is 1.

Notes

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References

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See also

• Category O