In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

, a highest-weight category is a ''k''-linear category C (here ''k'' is a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

) that *is ''locally artinian'' *has

enough injectives In mathematics, especially in the field of category theory, the concept of injective object is a generalization of the concept of injective module. This concept is important in cohomology, in homotopy theory and in the theory of model categories ...

*satisfies ::

B\cap\left(\bigcup_\alpha A_\alpha\right)=\bigcup_\alpha\left(B\cap A_\alpha\right)

:for all subobjects ''B'' and each family of subobjects of each object ''X'' and such that there is a

locally finite poset In mathematics, a locally finite poset is a partially ordered set ''P'' such that for all ''x'', ''y'' ∈ ''P'', the interval 'x'', ''y''consists of finitely many elements. Given a locally finite poset ''P'' we can defin ...

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 factor In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natu ...

s ''S''(''μ'') of ''A''(''λ'')/''S''(''λ'') satisfy ''μ'' < ''λ''. * For all ''μ'', ''λ'' in Λ, ::

\dim_k\operatorname_k(A(\lambda),A(\mu))

:is finite, and the

multiplicity Multiplicity may refer to: In science and the humanities * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (philosophy), a philosophical concept * Multiplicity (psychology), having or using multi ...

Here, if ''A'' is an object in C and ''S'' is a simple object in C, the multiplicity :Sis, by definition, the supremum of the multiplicity of ''S'' in all finite length subobjects of ''A''. ::

(\lambda):S(\mu) /math>
:is also finite.
*Each ''S''(''λ'') has an

injective envelope In mathematics, particularly in algebra, the injective hull (or injective envelope) of a module is both the smallest injective module containing it and the largest essential extension of it. Injective hulls were first described in . Definition ...

''I''(''λ'') in C equipped with an increasing

filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...

0=F_0(\lambda)\subseteq F_1(\lambda)\subseteq\dots\subseteq I(\lambda)

:such that :#

F_1(\lambda)=A(\lambda)

:# for ''n'' > 1,

F_n(\lambda)/F_(\lambda)\cong A(\mu)

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

\bigcup_iF_i(\lambda)=I(\lambda).

Examples

* The module category of the

k

-algebra of upper triangular

n\times n

matrices over

k

. * This concept is named after the category of

highest-weight module In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplica ...

s 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 In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...

and

hereditary Heredity, also called inheritance or biological inheritance, is the passing on of traits from parents to their offspring; either through asexual reproduction or sexual reproduction, the offspring cells or organisms acquire the genetic infor ...

algebras are highest-weight categories. * A

cellular algebra In abstract algebra, a cellular algebra is a finite-dimensional associative algebra ''A'' with a distinguished cellular basis which is particularly well-adapted to studying the representation theory of ''A''. History The cellular algebras dis ...

over a field is quasi-hereditary (and hence its module category a highest-weight category)

iff In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...

its Cartan-determinant is 1.

Notes

References

*{{cite journal , last1 = Cline , first1 = E. , last2 = Parshall , first2 = B. , last3 = Scott , first3 = L. , date=January 1988 , title = Finite-dimensional algebras and highest-weight categories , journal =

Journal für die reine und angewandte Mathematik ''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English language, English: ''Journal for Pure and Applied Mathematics''). History The journal wa ...

, volume = 1988 , issue = 391 , pages = 85–99 , location =

Berlin, Germany Berlin is the capital and largest city of Germany, both by area and by population. Its more than 3.85 million inhabitants make it the European Union's most populous city, as measured by population within city limits having gained this stat ...

, publisher =

Walter de Gruyter Walter de Gruyter GmbH, known as De Gruyter (), is a German scholarly publishing house specializing in academic literature. History The roots of the company go back to 1749 when Frederick the Great granted the Königliche Realschule in B ...

, issn = 0075-4102 , oclc = 1782270 , doi = 10.1515/crll.1988.391.85 , citeseerx = 10.1.1.112.6181 , s2cid = 118202731 , url = http://u.math.biu.ac.il/~margolis/Representation%20Theory%20Seminar/Highest%20Weight%20Categories.pdf , access-date=2012-07-17

Examples

Notes

References

See also