Plesken Lie Algebra

• Plesken Lie algebra is a class of Lie algebras generated from 'skew' elements in group or associative algebras with anti-involution.
• Its structure is determined by decompositions into classical Lie algebras using character theory and representation metrics in both characteristic zero and modular cases.
• Recent research extends the construction to cellular algebras, examining central extensions and modular analogues with practical applications in symmetry analysis.

The Plesken Lie algebra is a class of Lie algebras constructed from group algebras, general associative algebras with anti-involution, or, in a modular variant, from group algebras over fields of positive characteristic. Initially introduced by Plesken and developed in detail by Cohen and Taylor, these algebras are built as canonical Lie subalgebras spanned by certain “skew” elements and have explicit structure governed by representation theory, module theory, and the presence of anti-involutions. Recent research extends the construction to cellular algebras and examines the implications of modular representation theory and central extensions.

1. Foundational Definitions and Construction

Given a finite group $G$ and a field $\mathbb{F}$ (often $\mathbb{C}$ or an algebraically closed field of characteristic $p$), the group algebra $\mathbb{F}[G]$ naturally carries a Lie algebra structure via the commutator: $$[a,b] = ab - ba \qquad \text{for } a, b \in \mathbb{F}[G].$$ Plesken's construction isolates, as a Lie subalgebra, the linear span of “skew” elements: $$\mathcal{L}(G) = \operatorname{span}_{\mathbb{F}}\{\,g - g^{-1} : g \in G\,\}.$$ This subspace, denoted $\mathcal{L}(G)$ (also $L(G)$ or $\mathcal{L}(\mathbb{F}[G])$), is closed under the commutator bracket: $$[g - g^{-1},\; h - h^{-1}] = (g - g^{-1})(h - h^{-1}) - (h - h^{-1})(g - g^{-1}) = (gh) - (gh^{-1}) - (g^{-1}h) + (g^{-1}h^{-1}),$$ which itself is expressed in terms of differences of group elements and their inverses. Thus, $\mathcal{L}(G)$ forms a Lie subalgebra of $\mathbb{F}[G]$ (Romeo et al., 2021, Romeo et al., 2022).

In the more general setting of an associative algebra $A$ with anti-involution $i: A \to A$, satisfying $i(ab)=i(b)i(a)$ and $i^2 = \text{id}$, the Plesken Lie algebra is

$$\mathcal{L}(A) = \operatorname{span}_{\mathbb{F}}\{\,a - i(a): a \in A\,\} = \{a \in A: i(a) = -a\}.$$

This generalizes the group algebra case, with the standard anti-involution $g \mapsto g^{-1}$ (Holm et al., 28 Aug 2025).

2. Structure Theory and Decomposition

Ordinary (Characteristic Zero) Case

For a finite group $G$ and $\mathbb{F} = \mathbb{C}$, $\mathbb{C}[G]$ is semisimple by Maschke’s theorem. The structure of $\mathcal{L}(G)$ is explicitly determined via the character theory of $G$:

• The irreducible characters of $G$ partition into subsets according to their Frobenius–Schur indicators ($+1$, $-1$, $0$).
• The Lie algebra decomposes as a direct sum of classical Lie algebras, where the type (general linear, orthogonal, symplectic) and size of each summand is dictated by the corresponding character’s degree and indicator:

$$\mathcal{L}(G) \cong \bigoplus_{x \in X_1} \mathfrak{gl}_{d_x} \oplus \bigoplus_{x \in X_{-1}} \mathfrak{so}_{d_x} \oplus \bigoplus_{(x,x') \in X_0} \mathfrak{sl}_{d_x},$$

with $d_x$ the degree (Cullinan, 20 Jun 2024, Holm et al., 28 Aug 2025).

Modular Case (Characteristic $p$)

With $\mathbb{F}$ algebraically closed of characteristic $p$ dividing $|G|$, $\mathbb{F}[G]$ decomposes into blocks rather than simple factors. The modular analog $L_p[G]$ is defined as above and projected onto each block: $L_p[G] = \bigoplus_{j=1}^r \mathfrak{a}_j,$ where each $\mathfrak{a}_j$ is a Lie subalgebra within the corresponding block. The composition factors of $L_p[G]$ are conjectured to be either abelian or of classical Lie type (with possible modifications when $p$ divides certain dimensions), and are predicted using the Brauer character theory and a modular version of the Frobenius–Schur indicator (Cullinan, 20 Jun 2024).

Generalization to Cellular Algebras

For a semisimple cellular algebra $A$ with anti-involution $i$, Artin–Wedderburn theory yields

$$A \cong \bigoplus_{\lambda \in \Lambda'} M(d_\lambda, \mathbb{C}),$$

where $d_\lambda = \dim W(\lambda)$ for cell modules $W(\lambda)$. The anti-involution induces the transpose or a variant thereof on each block, and the (–1)-eigenspace is exactly the space of skew-symmetric matrices: $$\mathcal{L}(A) \cong \bigoplus_{\lambda \in \Lambda'} \mathfrak{o}(d_\lambda, \mathbb{C}).$$ This result is canonical for semisimple cellular algebras and encompasses diagram algebras and other examples (Holm et al., 28 Aug 2025).

3. Categorical Perspective and Functoriality

Categorial frameworks have been developed relating the category of Lie algebras from group algebras, denoted $L_{\mathbb{F}G}$, and the corresponding category of Plesken Lie algebras, denoted $\mathcal{C}_\mathrm{PLG}$.

A key functor $T: L_{\mathbb{F}G} \to \mathcal{C}_\mathrm{PLG}$ is defined by:

• On objects: $T(L_{\mathbb{F}G}) = \mathcal{L}(G)$.
• On morphisms: For a group homomorphism $f: G \to H$, the induced Lie algebra homomorphism $\overline{f}$ restricts to $f^\wedge: \mathcal{L}(G) \to \mathcal{L}(H)$ via $$f^\wedge(\sum a_i \widehat{g}_i) = \sum a_i \widehat{f(g_i)}$$.

This functor is full but not necessarily faithful, as different morphisms in $L_{\mathbb{F}G}$ may induce the same morphism in $\mathcal{C}_\mathrm{PLG}$, an effect observable in examples such as the Klein 4-group (Romeo et al., 2021).

4. Representation Theory and Central Extensions

Representations of a Plesken Lie algebra $L(G)$ are Lie algebra homomorphisms $\rho: L(G) \to \mathfrak{gl}(V)$ and can be induced from group representations $p: G \to \mathrm{GL}(V)$ via $\rho(\widehat{g}) = p(g) - p(g^{-1})$. Given a finite group $G$, the irreducibility of $\rho$ is closely linked to properties of $p$, yet the correspondence is not always exact: irreducibility may not be preserved under induction (Romeo et al., 2022, Arjun et al., 2023).

The framework is enriched by considering projective representations: homomorphisms $\pi: L(G) \to \mathfrak{pgl}(V)$, which can be lifted to maps $\widetilde{\pi}: L(G) \to \mathfrak{gl}(V)$ subject to a defect controlled by a 2-cocycle $a$: $$[\widetilde{\pi}(x), \widetilde{\pi}(y)] = \widetilde{\pi}([x,y]) + a(x,y) I_V.$$ The set of equivalence classes of central extensions of $L(G)$ by $\mathbb{C}$ is in bijection with the second cohomology group $H^2(L(G), \mathbb{C})$, which also classifies projectively equivalent projective representations (Romeo et al., 2022).

The theory further establishes a precise correspondence between irreducible projective representations of a Plesken Lie algebra and irreducible linear representations of its maximal central extension (the “cover”). This bijection is formally stated as $$\operatorname{Irr}(\mathcal{E}) \cong \bigcup_{\alpha \in H^2(L(G), \mathbb{C})} \operatorname{Irr}_\alpha(L(G))$$, linking algebraic and cohomological classifications (Arjun et al., 2023).

5. Examples and Structural Computations

Explicit examples illustrate the general theory:

• For $G = S_3$, the space $\mathcal{L}(S_3)$ is one-dimensional and brackets are trivial.
• For subgroups of the Heisenberg group $H(\mathbb{R})$, the Plesken Lie algebra $\mathcal{L}(H_i)$ is constructed as the span of differences $A - A^{-1}$ for $A \in H_i$, yielding concrete matrix Lie algebras (Romeo et al., 2021).
• In cellular algebras such as the planar rook algebra $\mathrm{PR}(n)$, the Plesken Lie algebra decomposes as a direct sum of orthogonal Lie algebras $\mathfrak{o}(\dim\,W(\lambda),\mathbb{C})$ correlating with the dimensions of cell modules (Holm et al., 28 Aug 2025).

Table: Key Constructions of Plesken Lie Algebra

Algebraic Context	Anti-involution	Plesken Lie Algebra Elements/Structure
$\mathbb{C}[G]$ (group algebra)	$g \mapsto g^{-1}$	$g - g^{-1}$, Lie bracket inherited
$A$ (assoc. algebra, general)	$i$ ($i^2 = \text{id}$)	$a - i(a)$; $(-1)$-eigenspace of $i$
$A = M(n, \mathbb{C})$	transpose	skew-symmetric matrices: $\mathfrak{o}(n)$
Planar rook algebra PR(n)	diagram reversal	$\oplus_k \mathfrak{o}({n \choose k})$

6. Modular Analogues and Current Conjectures

Recent developments introduce modular analogs $L_p[G]$ for $p$ dividing $|G|$. In this non-semisimple setting:

• The group algebra breaks into blocks, and $L_p[G]$ projects accordingly.
• Composition factors are conjectured to be either abelian (a modular “junk” phenomenon) or classical Lie algebras, governed by the modular character theory (Brauer characters) and indicators.
• For blocks of defect $0$, the corresponding projection $\mathfrak{a}_j$ is predicted to be classical, and divisibility conditions (e.g., $p \mid n+1$) alter the type of the summands (Cullinan, 20 Jun 2024).

This conjectural structure, with support from computational evidence, extends the character-theoretic determination of $\mathcal{L}(G)$ from the complex to the modular setting, forecasting a robust correspondence between modular group theory and Lie algebra composition series.

7. Broader Applications and Significance

The construction and paper of Plesken Lie algebras facilitate a direct connection between group theory, representation theory, and the structure theory of Lie algebras. They provide a setting in which cohomological invariants (e.g., the Schur multiplier) control essential representation-theoretic data, including central extensions and projective module theory (Romeo et al., 2022, Arjun et al., 2023). The extensions to cellular algebras and modular settings open new approaches in the analysis of symmetries in both algebraic and combinatorial settings, as well as computational methods for understanding invariants and decomposing group actions.

In summary, the Plesken Lie algebra serves as a canonical Lie theoretic object associated to (skew-)symmetric structures in group algebras and related associative objects with anti-involutions, with a rich interplay between block theory, module structure, and the theory of central extensions. Current research continues to elucidate its modular analogs and generalizations to broader classes of algebras, with conjectural frameworks guided by character theory, cohomology, and categorical relationships.