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 factorsS(μ) of A(λ)/S(λ) satisfy μ < λ.[3]
A finite-dimensional -algebra 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.
^Here, a composition factor of an object A in C is, by definition, a composition factor of one of its finite length subobjects.
^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.
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 factorsS(μ) of A(λ)/S(λ) satisfy μ < λ.[3]
A finite-dimensional -algebra 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.
^Here, a composition factor of an object A in C is, by definition, a composition factor of one of its finite length subobjects.
^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.