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In the [[mathematics|mathematical]] field of [[representation theory]], a '''highest-weight category''' is a [[linear category|''k''-linear category]] '''C''' (here ''k'' is a [[field (mathematics)|field]]) that
*is ''locally artinian''<ref>In the sense that it admits arbitrary [[direct limit]]s of [[subobject]]s and every object is a union of its subobjects of [[finite length object|finite length]].</ref>
*has [[enough injectives]]
*satisfies
::<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 {''A''<sub>&alpha;</sub>} 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:<ref name="dimensional-algebras">{{harvnb|Cline|Scott|1988|loc=§3}}</ref>
 
* The poset Λ indexes an exhaustive set of non-isomorphic [[simple object]]s {''S''(''λ'')} in '''C'''.
* Λ also indexes a collection of objects {''A''(''λ'')} of objects of '''C''' such that there exist embeddings ''S''(''λ'')&nbsp;→&nbsp;''A''(''λ'') such that all [[composition factor]]s ''S''(''μ'') of ''A''(''λ'')/''S''(''λ'') satisfy ''μ''&nbsp;<&nbsp;''λ''.<ref>Here, a composition factor of an object ''A'' in '''C''' is, by definition, a composition factor of one of its finite length subobjects.</ref>
* For all ''μ'', ''λ'' in Λ,
::<math>\dim_k\operatorname{Hom}_k(A(\lambda),A(\mu))</math>
:is finite, and the [[composition factor|multiplicity]]<ref>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''.</ref>
::<math>[A(\lambda):S(\mu)]</math>
:is also finite.
*Each ''S''(''λ'') has an [[injective envelope]] ''I''(''λ'') in '''C''' equipped with an increasing [[filtration (mathematics)|filtration]]
::<math>0=F_0(\lambda)\subseteq F_1(\lambda)\subseteq\dots\subseteq I(\lambda)</math>
:such that
:# <math>F_1(\lambda)=A(\lambda)</math>
:# for ''n'' > 1, <math>F_n(\lambda)/F_{n-1}(\lambda)\cong A(\mu)</math> for some ''μ'' = ''μ''(''n'') > ''λ''
:# for each ''μ'' in Λ, ''μ''(''n'') = ''μ'' for only finitely many ''n''
:# <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 module]]s 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 ring|semisimple]] and [[hereditary ring|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&nbsp;1.
 
==Notes==
{{Reflist}}
 
==References==
*{{cite journal
| last1 = Cline
| first1 = E.
| last2 = Parshall
| first2 = B.
| last3 = Scott
| first3 = L.
| year = 1988
| month = January
| title = Finite-dimensional algebras and highest-weight categories
| journal = [[Journal für die reine und angewandte Mathematik]]
| volume = 1988
| issue = 391
| pages = 85–99
| location = [[Berlin, Germany]]
| publisher = [[Walter de Gruyter]]
| format = pdf
| issn = 0075-4102
| oclc = 1782270
| doi = 10.1515/crll.1988.391.85
| accessdate = 7 January 2011
| id = {{citeseerx|10.1.1.112.6181}}
| url = http://u.math.biu.ac.il/~margolis/Representation%20Theory%20Seminar/Highest%20Weight%20Categories.pdf
| accessdate=2012-07-17
}}
 
== See also ==
*[[Category O]]
 
[[Category:Representation theory]]

Revision as of 06:16, 2 September 2013

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

B(αAα)=α(BAα)
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,
dimkHomk(A(λ),A(μ))
is finite, and the multiplicity[4]
[A(λ):S(μ)]
is also finite.
0=F0(λ)F1(λ)I(λ)
such that
  1. F1(λ)=A(λ)
  2. for n > 1, Fn(λ)/Fn1(λ)A(μ) for some μ = μ(n) > λ
  3. for each μ in Λ, μ(n) = μ for only finitely many n
  4. iFi(λ)=I(λ).

Examples

  • The module category of the k-algebra of upper triangular n×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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See also

  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. Template: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.