“Highest-weight category”的意思、由来-开放百科全书

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Highest-weight category

释义

Examples
Notes
References
See also

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^[1]
has enough injectives
satisfies

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:^[2]

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.^[3]
For all μ, λ in Λ,

is finite, and the multiplicity^[4]

is also finite.

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

such that

for n > 1, for some μ = μ(n) > λ
for each μ in Λ, μ(n) = μ for only finitely many n

Examples

The module category of the -algebra of upper triangular matrices over .
This concept is named after the category of highest-weight modules of Lie-algebras.
A finite-dimensional -algebra 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

1. ^In the sense that it admits arbitrary direct limits of subobjects and every object is a union of its subobjects of finite length.
2. ^{{harvnb|Cline|Scott|1988|loc=§3}}
3. ^Here, a composition factor of an object A in C is, by definition, a composition factor of one of its finite length subobjects.
4. ^Here, if A is an object in C and S is a simple object in C, the multiplicity [A:S] is, by definition, the supremum of the multiplicity of S in all finite length subobjects of A.

References

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Examples

Notes

References

See also