Skew-Symmetric Super-Biderivations

• Skew-symmetric super-biderivations are bilinear operators on Lie superalgebras that extend derivations while preserving the graded structure and obeying skew-symmetry via the Koszul sign rule.
• They are proven to be inner in key classes using weight space decomposition, centroid analysis, and abelian subalgebra techniques, demonstrating strong structural rigidity.
• Their analysis impacts deformation theory and representation classification, with non-inner cases arising only in exceptional parameterized scenarios.

A skew-symmetric super-biderivation is a bilinear operator on a Lie (super)algebra, designed to generalize the concept of derivation to higher order and to respect the superalgebra’s graded structure. These maps exhibit an interplay of algebraic, combinatorial, and geometric features, illuminating the structure and rigidity properties of broad classes of Lie superalgebras, especially in the modular context and over fields of nonzero characteristic.

1. Definitions and Formal Properties

Let $L$ be a Lie superalgebra over a field $F$ (usually of characteristic $p > 2$ for Cartan type superalgebras). A $\mathbb{Z}_2$-homogeneous bilinear map $\varphi: L \times L \to L$ of parity $d(\varphi)$ is a super-biderivation if it satisfies graded derivation properties in each argument. Specifically, for homogeneous $x, y, z\in L$,

$$\varphi(x, [y, z]) = [\varphi(x, y), z] + (-1)^{(d(\varphi)+d(x))d(y)} [y, \varphi(x, z)].$$

The skew-symmetry adapts the Koszul sign rule: $$\varphi(x, y) = -(-1)^{d(\varphi)d(x) + d(\varphi)d(y) + d(x)d(y)} \varphi(y, x).$$ Typical examples include inner biderivations of the form $\varphi_\lambda(x, y) = \lambda [x, y]$ for $\lambda \in F$.

For clarity, the structure can be compared side-by-side:

Notion	Formula	Context
Skew-symmetric super-biderivation	$$\varphi(x, y) = -(-1)^{d(\varphi)d(x) + d(\varphi)d(y) + d(x)d(y)} \varphi(y, x)$$	Modular Lie superalgebras
Derivation	$D([x, y]) = [D(x), y] + [x, D(y)]$	Classical Lie algebras
Inner biderivation	$\varphi(x, y) = \lambda [x, y]$	Universal in classification

These axioms ensure that for parity-homogeneous maps, the biderivation identity and skew-symmetry are preserved under all graded permutations.

2. Structural Classification and Weight Space Decomposition

For Cartan-type Lie superalgebras (including the special, Hamiltonian, and odd Hamiltonian superalgebras), the technical approach leverages the presence of distinguished abelian subalgebras (e.g., $T_S$, $T_H$, $T_{SHO}$), enabling a fine weight decomposition:

• For each $t \in T$, the adjoint action decomposes $L$ into weight spaces: $L = \bigoplus_\lambda L_\lambda$, where $[t, x] = \lambda(t)x$ for $x\in L_\lambda$.
• If $T$ is abelian and $x,y\in T$, then $[x,y]=0$, enforcing by the biderivation axioms that $\varphi(x, y)=0$ for abelian pairs.

Crucially, the weight decomposition allows for the explicit tracking of the action of $\varphi$ on basis elements. In the modular Cartan-type settings, the following paradigm emerges:

• Application of $\varphi$ to carefully selected weight homogeneous pairs (especially when one lies in the toral subalgebra) leads to rigid constraints, often immediately showing that non-zero $\varphi(x, y)$ must be proportional to $[x, y]$.
• Through the use of explicit commutator formulas and the Jacobi identity, all coefficients arising from possible biderivation images coalesce to a single scalar.

Hence, every skew-symmetric super-biderivation $\varphi$ is forced to be inner, that is,

$\varphi(x, y) = \lambda [x, y],$

for all $x, y \in L$ (Xu et al., 16 Aug 2025, Xu et al., 16 Aug 2025, Xu et al., 16 Aug 2025, Zhao et al., 2019, Xu et al., 2023).

3. Centroid, Innerness, and General Rigidity

The categorical source of this rigidity is the centroid $T(L)$ of the superalgebra, the set of linear maps $f: L \to L$ commuting with all inner derivations: $f([x, y]) = [f(x), y] = [x, f(y)].$ For finite-dimensional simple Lie superalgebras, the centroid reduces to scalar multiples of the identity ($T(L)_0 = F\cdot \mathrm{Id}$, $T(L)_1 = 0$), implying every such $f$ is proportional to the identity (Xu et al., 2023). Thus, every biderivation factoring through the centroid is inner.

In modular Cartan-type superalgebras, similar use of the weight decomposition, simplicity, and centerlessness leads to the same outcome (Zhao et al., 2019, Xu et al., 16 Aug 2025, Xu et al., 16 Aug 2025, Xu et al., 16 Aug 2025), confirming that

$$\operatorname{BDer}_{\text{skew}}(L) = \operatorname{IBDer}(L),$$

where $\operatorname{IBDer}(L)$ is the space of inner biderivations.

4. Classification Results and Explicit Formulas

Across a diversity of settings—classical simple Lie superalgebras, modular Cartan-type Lie superalgebras, and their close relatives—the uniform conclusion is that all skew-symmetric super-biderivations are inner except in pathological or specially constructed infinite-dimensional or parameter-dependent algebras (as in specific variants of Schrödinger–Virasoro and $W(a, b)$ algebras over $\mathbb{C}$, where, for particular parameter values, non-inner biderivations arise (Fan et al., 2016, Tang, 2017)).

The formulas encapsulating the general result are:

• Simple Lie superalgebras (characteristic $\ne 2$):

$\varphi(x, y) = \lambda [x, y]$

for all $x,y$ (Xu et al., 2023).

• Cartan-type modular superalgebras ($K(m, n; t)$, $S(m, n; t)$, $H(m, n; t)$, $SHO(n, n; t)$, $p>2$):

$\varphi(x, y) = \lambda [x, y]$

for all $x, y$ (proofs via weight space and toral analysis) (Zhao et al., 2019, Xu et al., 16 Aug 2025, Xu et al., 16 Aug 2025, Xu et al., 16 Aug 2025).

• Deformative Schrödinger–Virasoro and $W(a, b)$ algebras: under generic parameters, only inner skew-symmetric biderivations occur, with explicit non-inner possibilities for exceptional parameters (Fan et al., 2016, Tang, 2017).

5. Methodological and Theoretical Impact

The proof techniques (using abelian subalgebras, weight space decomposition, explicit computation on basis elements, centroid analysis) form a toolkit adaptable to other classes of graded Lie (super)algebras. The implications are severalfold:

• There is structural rigidity: higher order (skew-)biderivation analogues of derivations admit no nontrivial algebraic "deformations" beyond those induced by the bracket, facilitating classification and automorphism studies.
• The interplay with module theory and the use of centroids highlight that obstructions to innerness are erased by simplicity or the triviality of the center/centroid.
• In cohomological and deformation-theoretic contexts, the vanishing of “obstructions” (i.e., the triviality of biderivation cohomology in the skew-symmetric case) further underlines the absence of higher order infinitesimal symmetries, simplifying calculations in modular representation theory and invariant theory.

6. Broader Consequences and Open Directions

While the result signals a closure under Lie superalgebra structure—innerness as a haLLMark of algebraic rigidity—several lines for future investigation remain active:

• Non-skew-symmetric (or symmetric) biderivations, for which the innerness property may fail in intriguing ways, especially in non-simple, non-perfect, or infinite-dimensional cases (Brešar et al., 2018).
• Extensions to Hom-type and quantum analogues, where the centroids and module structures differ, suggest possible new classes of "exotic" biderivations or broader centroids (Sun et al., 2020).
• Deformation and higher derived bracket frameworks: The Poisson superalgebra, derived bracket, and invariant form formalism provide a principled method to construct families of higher order derivations, embedding the investigation of super-biderivations into a larger algebraic supergeometry context (Vishnyakova, 2014).

7. Summary Table: Rigidity of Skew-Symmetric Super-Biderivations

Type/Algebra Class	All Skew-Sym. Super-Biderivations Inner?	Mechanism	Key Papers
Simple modular Cartan-type superalgebras	Yes	Weight space + abelian torus analysis	(Xu et al., 16 Aug 2025, Xu et al., 16 Aug 2025, Xu et al., 16 Aug 2025, Zhao et al., 2019)
Finite-dim. simple Lie superalgebras	Yes	Centroid reduces to scalars	(Xu et al., 2023)
Deformative Schrödinger–Virasoro	Generic: Yes; Exception: No	Explicit construction in exceptional cases	(Fan et al., 2016)
$W(a,b)$ algebras	Generic: Yes; Exception: No	Explicit construction with non-inner terms	(Tang, 2017)
General Hom-Lie superalgebras	Yes under centroid and module conditions	Centroid and module-theoretic identification	(Sun et al., 2020)

In summary, skew-symmetric super-biderivations are uniformly inner in the principal classes of simple and Cartan-type modular Lie superalgebras, demonstrating a strong rigidity in their higher-order derivation structure. This finding aligns with the centricity and graded decomposition approaches widespread in the literature, with only rare exceptions in specially parameterized algebras. The weight space and abelian subalgebra techniques used are foundational tools for continuing research in Lie superalgebra cohomology, deformation theory, and representation classification.