Representation theory

Highest-weight category

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 * satisfiesfor all subobjects B and each family of subobjects {Aα} 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,is finite, and the multiplicityis also finite. * Each S(λ) has an injective envelope I(λ) in C equipped with an increasing filtrationsuch that 1. * 2. * for n > 1, for some μ = λ(n) > λ 3. * for each μ in Λ, λ(n) = μ for only finitely many n 4. * (Wikipedia).

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From playlist Weightlifting

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From playlist Weightlifting

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From playlist Weightlifting

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From playlist Weightlifting

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From playlist Quantum Groups Seminar

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From playlist Mathematics

Related pages

Filtration (mathematics) | Cellular algebra | Mathematics | Field (mathematics) | Category O | Hereditary ring | Locally finite poset | Representation theory