TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Generalized central-character decomposition of O

Statement

The category is the categorical direct sum $O = ⨁_{χ} O_{χ}$ : objects have finite support in the index $χ$ , morphisms between distinct components vanish, and the canonical component projections are exact.

Facts & Assumptions

Given: The setting above and the hypotheses in the statement.

[F1]

Fix a finite-dimensional complex semisimple Lie algebra $g$ , a Cartan subalgebra $h$ , and a positive Borel $b = h \oplus n^{+}$ . Write $Q^{+} = \sum_{i} Z_{\geq 0} α_{i}$ , $μ \leq λ$ when $λ - μ \in Q^{+}$ , and $w \cdot λ = w (λ + ρ) - ρ$ . Every $M \in O$ decomposes canonically into finitely many nonzero generalized central-character submodules: $M = ⨁_{χ} M_{χ} .$ For each summand there is a single $N \geq 1$ such that $m_{χ}^{N} M_{χ} = 0$ . The decomposition of zero is empty. (Generalized central-character summands)

Proof

1.1

Use the finite intrinsic decomposition of each object. If $f : M \to N$ is a module map, $m_{χ}^{N} v = 0$ implies $m_{χ}^{N} f (v) = 0$ , so $f (M_{χ}) \subset N_{χ}$ . Consequently all off-diagonal components of a map vanish, and maps between objects decompose uniquely into their same-character components.

2.1

For an exact sequence $0 \to A \to B \to C \to 0$ , the image and kernel equalities restrict to each character. In particular, to lift $c \in C_{χ}$ , lift it to $b \in B$ and decompose $b = \sum_{ψ} b_{ψ}$ . Preservation of characters and the direct decomposition of $C$ imply that $b_{χ}$ maps to $c$ . This proves surjectivity and hence exactness of every projection.

F1algebrastep 1.1

3.1

The functor taking a finitely supported family to its direct sum and the functor $M \mapsto (M_{χ})_{χ}$ are inverse up to the canonical isomorphisms. An empty family gives zero, and a single nonzero component is fixed by its projection.

F1algebrastep 2.1∎

Depends on

Generalized central-character summands

Used by

Every object of O has finite length Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

§6 Theorem 6.2(1), pp.9–10 (standard reference, not scraped)