Highest-weight category explained

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

B\cap\left( c up _{\alpha} A _{\alpha\right)=cup} _{\alpha\left(B\cap} A _{\alpha\right)}

for all subobjects B and each family of subobjects of each object Xand 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 in C.
• Λ also indexes a collection of objects 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.^[2]
• For all μ, λ in Λ,

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

is finite, and the multiplicity^[3]

[A( λ ):S(\mu)]

is also finite.

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

0= F _{0(λ)\subseteq} F _{1(λ)\subseteq...\subseteq} I( λ )

such that

F _{1(λ)=A(λ)}

1. for n > 1,

F _{n(λ)/F} _{n-1} ( λ )\cong A(\mu) for some μ = λ(n) > λ

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

c up _{iF} _{i(λ)=I(λ).}

Examples

• The module category of the

k -algebra of upper triangular n x 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.

References

• Cline . E. . Parshall . B. . Scott . L. . January 1988 . Finite-dimensional algebras and highest-weight categories . . 1988 . 391 . 85–99 . . . 0075-4102 . 1782270 . 10.1515/crll.1988.391.85 . 10.1.1.112.6181 . 118202731 . 2012-07-17 .

See also

• Category O

Notes and References

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. Here, a composition factor of an object A in C is, by definition, a composition factor of one of its finite length subobjects.
3. 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.