TheoremStatement: Literature-sourcedProof: AI-adaptedverified 2026-09-23 (gpt-6-sol)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Triangular decomposition from a chosen positive root system

Statement

Let $Φ^{+}$ be a positive system in the root set from Root-space decomposition relative to a Cartan subalgebra and put

$n^{+} : = ⨁_{α \in Φ^{+}} g_{α}, n^{-} : = ⨁_{α \in Φ^{+}} g_{- α} .$

Then

$g = n^{-} \oplus h \oplus n^{+}$

as a direct sum of vector spaces, $n^{\pm}$ are Lie subalgebras, and multiplication induces a vector-space isomorphism

$U (n^{-}) \otimes U (h) \otimes U (n^{+}) \tilde{\to} U (g) .$

Facts & Assumptions

Given: A Cartan subalgebra $h$ of a complex semisimple Lie algebra $g$ , a supplied root-space decomposition $g = h \oplus ⨁_{α \in Φ} g_{α}$ as in the Statement, and a choice of positive roots $Φ^{+} \subseteq Φ$ . The assertion is conditional on this decomposition; it does not construct one for an arbitrary Cartan subalgebra.

Proof

technique · direct

1.1given

Group the positive, zero, and negative root spaces in the supplied decomposition. This gives the direct sum $g = n^{-} \oplus h \oplus n^{+}$ .

1.2givenalgebra

If $x \in g_{α}$ and $y \in g_{β}$ , Jacobi gives $[h, [x, y]] = [[h, x], y] + [x, [h, y]] = (α + β) (h) [x, y]$ for every $h \in h$ . Thus $[x, y]$ belongs to the supplied weight space for $α + β$ , and is zero if that nonzero weight is absent. A sum of two positive roots cannot be zero or negative, so $n^{+}$ is closed under brackets. The same argument applies to $n^{-}$ .

2.1step 1.1step 1.2∎

Choose ordered bases of $n^{-}$ , $h$ , and $n^{+}$ and concatenate them in that order. The ordered PBW monomials from PBW gives an ordered monomial basis for the enveloping algebra are then exactly products of a monomial in $U (n^{-})$ , one in $U (h)$ , and one in $U (n^{+})$ , so multiplication gives the stated vector-space isomorphism.

Depends on

Used by

Chevalley-contravariant forms Definition
Highest-weight vectors and cyclic highest-weight modules Definition
The classical BGG category O Definition
The Harish-Chandra projection Definition
The one-dimensional Borel module of weight lambda Definition
The standard induced resolution of the trivial module Definition
The Weyl vector rho for a chosen positive system Definition
Surjectivity modulo n-minus for free weight-generated modules (BGG 10.5) Lemma
The PBW model of a Verma module Theorem

Dependency tree · two levels

8 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

Pavel Etingof, Lie Groups and Lie Algebras I (standard reference, not scraped)