Mal'cev's criterion

In differential geometry, Mal'cev's criterion, proved by Anatoly Mal'cev, states that a simply connected nilpotent Lie group admits a lattice, i.e., a discrete co-compact subgroup, if and only if the associated Lie algebra admits a basis such that the structure constants are rational.

Version

v1 · 2026-09-28 · History

Domain-specific #

10537

Domain group

Formal Sciences

Origin domain

Mathematics

Subdomains

Lie Theory, Lattices in Lie Groups → Mathematics

Core Idea

Mal'cev's criterion is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In differential geometry, Mal'cev's criterion, proved by Anatoly Mal'cev, states that a simply connected nilpotent Lie group admits a lattice, i.e., a discrete co-compact subgroup, if and only if the associated Lie algebra admits a basis such that the structure constants are rational. In differential geometry, Mal'cev's criterion, proved by Anatoly Mal'cev, states that a simply connected nilpotent Lie group admits a lattice, i.e., a discrete co-compact subgroup, if and only if the associated Lie algebra admits a.

Scope of Application

• Relevance in Kähler geometry. Mal'cev's criterion is relevant in Kähler geometry because compact nilmanifolds with a Kähler structure must be diffeomorphic to a torus.

• Relevance in Kähler geometry. Therefore, when looking for manifolds that do not admit a Kähler structure, one may use Mal'cev's criterion to generate a compact nilmanifold from any rational Lie algebra.

• Relevance in Kähler geometry. By constructing other structures on the manifold, such as a complex or symplectic structure, one finds a non-Kähler manifold of said structure.

• Relevance in Kähler geometry. It therefore helps finding solutions to the Thurston–Weinstein problem, which concerns itself with the existence of non-Kähler symplectic manifolds.

• Documented setting. The relevance of Mal'cev's criterion comes from the fact that it gives us a one-to-one correspondence between isomorphism classes of Lie algebras with rational structure constants and compact nilmanifolds.

Clarity

A clear use of Mal'cev's criterion names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In differential geometry, Mal'cev's criterion, proved by Anatoly Mal'cev, states that a simply connected nilpotent Lie group admits a lattice, i.e., a discrete co-compact subgroup, if and only if the associated Lie algebra admits a basis such that the structure constants.

Manages Complexity

Mal'cev's criterion compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—in differential geometry, Mal'cev's criterion, proved by Anatoly Mal'cev, states that a simply connected nilpotent Lie group admits a lattice, i.e., a discrete co-compact subgroup, if and only if the associated Lie algebra admits a basis such that the structure constants are rational.—and the practical consequence—it therefore helps finding.

Abstract Reasoning

1. Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
2. State the relation. Use the source-grounded identity: In differential geometry, Mal'cev's criterion, proved by Anatoly Mal'cev, states that a simply connected nilpotent Lie group admits a lattice, i.e., a discrete co-compact subgroup, if and only if the associated Lie algebra admits a basis such that the structure constants are rational.
3. Check operation and conditions.

Knowledge Transfer

Within the home domain. Knowledge about Mal'cev's criterion transfers literally when a new case preserves the same carrier type, relation, and recognition test. Mal'cev's criterion is relevant in Kähler geometry because compact nilmanifolds with a Kähler structure must be diffeomorphic to a torus. Therefore, when looking for manifolds that do not admit a Kähler structure, one may use Mal'cev's criterion to generate a compact nilmanifold from any rational Lie algebra. Beyond the home domain. No canonical parent is asserted for Mal'cev's criterion.

Neighborhood in Abstraction Space

Mal'cev's criterion sits in a sparse region of the domain-specific corpus (67^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Sheaves & Birational Geometry (22 abstractions)

Nearest neighbors

• Nilmanifold — 0.86
• Complex Lie group — 0.85
• Essential manifold — 0.84
• Steenrod problem — 0.84
• Lie group — 0.83

Computed from structural-signature embeddings · 2026-10-08