Proper Levi Decomposition

• Proper Levi decomposition is the structural partition of groups and Lie algebras into a semisimple (or reductive) Levi factor and a maximal solvable or unipotent radical.
• It guarantees that the multiplication map between the Levi factor and radical is an isomorphism while satisfying strict compatibility with additional algebraic structures.
• This decomposition underpins advances in representation theory, cohomological analysis, and classification by enabling clear separation and understanding of complex algebraic behaviors.

A proper Levi decomposition refers to the structural partition of an algebraic group, a Lie algebra, or closely related structures such as group schemes, into a semisimple or reductive "Levi" component and a maximal solvable or unipotent radical component, under conditions ensuring the decomposition respects additional algebraic, geometric, or module-theoretic constraints. This concept underpins classification results and ensures tractability of representation theory, module decomposition, and cohomological analysis, particularly in nonzero characteristic or where additional structure (grading, o-minimality, conformal structure) is present.

1. Formal Definition and Classical Context

Given a connected linear algebraic group $G$ over a field $k$, a Levi factor $M$ is a reductive subgroup that complements the unipotent radical $R = R_u(G)$ such that the multiplication map

$M \times R \to G, \quad (m, r) \mapsto m r$

is an isomorphism of algebraic groups. This realizes $G \cong M \ltimes R$, with $M \simeq G/R_u(G)$. In characteristic zero, Levi’s theorem asserts the existence and uniqueness (up to conjugacy) of Levi factors, ensuring every finite-dimensional Lie algebra splits as $L = S \oplus R$, where $S$ is a semisimple subalgebra and $R$ the radical.

In positive characteristic or more general settings, neither existence nor uniqueness is guaranteed. A decomposition is called "proper" if it meets the following conditions:

• The Levi factor $M$ intersects $R$ trivially: $M \cap R = \{1\}$ (or, on Lie algebras, $L \cap R = 0$).
• $M$ is a complement to $R$ (and $R$ is defined and split over the base field/structure).
• The multiplication map above is a bijective algebraic (or scheme-theoretic) morphism.

Furthermore, in algebraic group schemes or group-theoretic contexts with additional structure, a proper Levi decomposition is typically required to be compatible with that structure.

2. Existence and Uniqueness Criteria

When $k$ has positive characteristic, the existence of a Levi factor is obstructed by the possible nonvanishing of certain cohomology groups. Sufficient (and in certain regimes, necessary) conditions are given in terms of a filtration (the "linearizable splitting sequence") of the unipotent radical $R$: $$1 = R_n \subset R_{n-1} \subset \cdots \subset R_0 = R$$ such that $V_i = R_i / R_{i+1}$ are vector groups, and $G_{red} = G/R$ acts linearly on each $V_i$. The main statements are:

• If $H^2(G_{red}, V_i) = 0$ for all $i$, then a Levi factor exists.
• If, in addition, $H^1(G_{red}, V_i) = 0$ for all $i$, then any two Levi factors are conjugate under $R(k)$.

This cohomological approach captures both the splitting (existence) and conjugacy (uniqueness) of Levi factors in families and under field extensions, see (1007.2777).

In o-minimal and Lie-theoretic structures, one also achieves a canonical decomposition $G = R \cdot S$ with $R$ a definable solvable radical and $S$ a unique (up to conjugacy) ind-definable semisimple subgroup, with similar properties at the level of the associated Lie algebra, see (1111.2369). Here, existence and uniqueness are managed through definability properties and the realization of semisimple structure via countable unions of definable sets and discrete centers.

3. Proper Levi Extensions in Lie Algebras and Nilpotent Cases

Levi’s theorem in characteristic zero ensures that any finite-dimensional Lie algebra can be decomposed as $L = S \oplus R$. The properness of the decomposition is tied to compatibility:

• $S$ acts as derivations on $R$, i.e., $[S, R] \subset R$.
• The adjoint representation $p: S \to \mathrm{gl}(R)$, $p(s)(a) = [s, a]$, must satisfy $p([x, y]) = [p(x), p(y)]$.

Given a solvable (often nilpotent) Lie algebra $R$, a semisimple Lie algebra $S$ is a Levi extension if there exists a structure on $S \oplus R$ such that $\rho(S) \subseteq \mathrm{Der}(R)$. The critical criterion is the existence of such a $\rho$, often analyzed via S-module structure on the lower central series of $R$ and realized using free nilpotent Lie algebras for explicit classification (Benito et al., 2013).

A pivotal role is played by the structure of $\mathrm{Der}(R)$. If $\mathrm{Der}(R)$ is solvable (e.g., for filiform or characteristically nilpotent Lie algebras), nontrivial Levi extensions do not exist.

4. Compatibility with Additional Structure

In settings where $G$ or $L$ admits extra structure, such as gradings or module decompositions:

• For a graded Lie algebra $g = \bigoplus_n g_n$ with radical $t$ and semisimple complement $I$, there exists a Levi subalgebra $I = \bigoplus_n (I \cap g_n)$ that is invariant under the grading derivations. This makes the decomposition compatible with any grading derived from a family of commuting semisimple derivations $\{\delta_H\}$, see (Ciatti et al., 2017).
• For Lie conformal algebras $L = L_0 \oplus R$ with a splitting radical, one constructs finite faithful conformal representations. The split ensures that the semisimple part controls representation structure and allows standard inductive proofs analogous to the Ado theorem (Kolesnikov, 2012).

In the context of representation theory of current algebras, restriction of Weyl modules to a proper Levi subalgebra yields compatibility if certain admissibility conditions on weights are satisfied. This functorial behavior generalizes classical branching rules to current algebras and allows direct identification of restricted Weyl modules, providing explicit criteria for when such restrictions respect the Weyl structure (Fourier, 2012).

5. Proper Levi Decompositions in Algebraic Groups and Scheme Theory

For linear algebraic groups over fields of positive characteristic, a proper Levi decomposition often requires extra subtleties:

• When studying parahoric group schemes associated to a reductive group $G$ over a local field $K$ (with smooth integral model $P$ over discrete valuation ring $A$, special fiber $P/k$), one finds that $P/k$ may not be reductive, but under suitable splitting assumptions (e.g., $G$ splits over an unramified extension), $P/k$ has a Levi factor, uniquely defined up to geometric conjugacy (1007.2777).
• In centralizers of nilpotent elements in classical groups (especially in "bad" characteristic, e.g., characteristic $2$ for orthogonal or symplectic groups), the paper verifies that both point-level and scheme-theoretic criteria for properness are met: the Levi factor intersects the unipotent radical trivially at both the group and Lie algebra level. Isomorphism classes of reductive quotients are computed via explicit module-theoretic analysis (Babinski et al., 2016).

6. Applications and Ramifications

The presence of a proper Levi decomposition is fundamental in the following contexts:

• Cohomological and Structural Classification: Deciding existence and conjugacy of Levi factors, classifying solvable/nilpotent Lie algebras via extensions, and understanding the obstruction space via $H^2$ and $H^1$.
• Representation Theory: Reduction of module theory and representation classification to that of the Levi factors and their action on the radical; explicit construction of minimal faithful representations via block-matrix techniques (Ghanam et al., 2017).
• Algebraic Geometry: Decomposition of coordinate rings as modules over Levi subgroups, yielding explicit branching rules, combinatorial stratifications, and sphericity consequences for Schubert or determinantal varieties (Hodges et al., 2016).
• Arithmetic and Parahoric Theory: Descent, uniqueness, and conjugacy of Levi factors in integral models of reductive groups, with implications for reduction mod $p$ and theory of buildings.
• Model Theory and O-minimality: Extension of Lie-theoretic decompositions to definable and ind-definable settings, ensuring structural compatibility with model-theoretic dimensions (1111.2369).

7. Mathematical and Algorithmic Formulations

The following table summarizes the core algebraic characterizations of a proper Levi decomposition in several frameworks:

Structure	Core Decomposition	Properness Criteria
Algebraic group $G$	$G \cong M \ltimes R$	$M \cap R = \{1\}$, $M$ reductive, $R$ unipotent
Lie algebra $L$	$L = S \oplus R$	$[S, R] \subset R$, $S$ semisimple, $R$ solvable
Graded Lie algebra $g$	$g = \bigoplus_n (I_n \oplus t_n)$	$I = \bigoplus_n (I \cap g_n)$, invariance under grading derivations
Lie conformal algebra $L$	$L = L_0 \oplus R$	$L_0$ semisimple, $R$ splits as $L_0$-module
O-minimal group $G$	$G = R \cdot S$, $L(G) = L(R) \oplus s$	$S$ ind-definable semisimple, $R$ solvable radical

Proper Levi decompositions are central to advancing the structure theory in settings where classical assumptions no longer apply, particularly in positive characteristic, o-minimal frameworks, graded algebras, or generalized module-theoretic environments. Their existence, uniqueness, and compatibility with auxiliary structures underpin a wide range of results in both pure and applied mathematics.