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

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]

is finite, and the multiplicity[4]
is also finite.
such that
  1. for n > 1, for some μ = μ(n) > λ
  2. for each μ in Λ, μ(n) = μ for only finitely many n

Examples

Notes

  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. Cline & Scott 1988, §3
  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.

References

See also

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