Английская Википедия:Highest-weight category

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

<math>B\cap\left(\bigcup_\alpha A_\alpha\right)=\bigcup_\alpha\left(B\cap A_\alpha\right)</math>
for 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:[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,
<math>\dim_k\operatorname{Hom}_k(A(\lambda),A(\mu))</math>
is finite, and the multiplicity[4]
<math>[A(\lambda):S(\mu)]</math>
is also finite.
<math>0=F_0(\lambda)\subseteq F_1(\lambda)\subseteq\dots\subseteq I(\lambda)</math>
such that
  1. <math>F_1(\lambda)=A(\lambda)</math>
  2. for n > 1, <math>F_n(\lambda)/F_{n-1}(\lambda)\cong A(\mu)</math> for some μ = λ(n) > λ
  3. for each μ in Λ, λ(n) = μ for only finitely many n
  4. <math>\bigcup_iF_i(\lambda)=I(\lambda).</math>

Examples

  • The module category of the <math>k</math>-algebra of upper triangular <math>n\times n</math> matrices over <math>k</math>.
  • This concept is named after the category of highest-weight modules of Lie-algebras.
  • A finite-dimensional <math>k</math>-algebra <math>A</math> 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

Шаблон:Reflist

References

See also

Партнерские ресурсы
Криптовалюты
Магазины
Хостинг
Разное

  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. ↑ Шаблон:Harvnb
  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.
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