CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedaudited 2026-09-22

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Every finite-dimensional module is a direct sum of highest-weight modules

Statement

Assume the Axiom of Choice. Let $g$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $h$ and a fixed positive system. Then every finite-dimensional representation $V$ of $g$ is a finite direct sum $V = V_{1} \oplus \dots \oplus V_{N}$ of irreducible submodules, where each $V_{j}$ is isomorphic to a highest-weight module $L (λ_{j})$ for a dominant integral weight $λ_{j}$ (Integral, dominant, and strictly dominant weights).

Facts & Assumptions

Given: The Axiom of Choice, such $g, h$ , a fixed positive system, and a finite-dimensional representation $V$ .

[A1]

The Axiom of Choice is assumed; it enters through the cited suppliers (The Axiom of Choice).

[L1]

Every finite-dimensional representation of a finite-dimensional semisimple Lie algebra over a characteristic-zero field is completely reducible: it is a direct sum of irreducible subrepresentations (Weyl's complete reducibility theorem, Irreducible, completely reducible, and faithful representations, The direct sum of an indexed family of modules).

[L2]

The finite-dimensional irreducible representations of $g$ are exactly the modules $L (λ)$ with $λ$ dominant integral (Highest-weight classification).

Proof

technique · direct

1.1

By [L1] the module $V$ is a direct sum of irreducible subrepresentations $V = ⨁_{i \in I} V_{i}$ .

L1A1

2.1

The index set $I$ is finite: each $V_{i}$ is nonzero, and a direct sum of nonzero subspaces of the finite-dimensional space $V$ has at most $\dim V$ summands; reindexing the finite set gives $V = V_{1} \oplus \dots \oplus V_{N}$ .

L1step 1.1

3.1

By [L2] each summand $V_{j}$ is isomorphic to $L (λ_{j})$ for a dominant integral weight $λ_{j}$ . Thus $V = V_{1} \oplus \dots \oplus V_{N}$ with $V_{j} ≅ L (λ_{j})$ ; equivalently, choosing these isomorphisms gives an isomorphism $V ≅ ⨁_{j = 1}^{N} L (λ_{j})$ .

L2step 2.1

4.1

This is the asserted finite decomposition.

step 3.1∎

Depends on

Used by

Top summand in a tensor product Proposition

Dependency tree · two levels

33 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

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed. (standard reference, not scraped)
Alexander Kirillov Jr., An Introduction to Lie Groups and Lie Algebras (standard reference, not scraped)