Abelian Subalgebra TSHO in SHO

• Abelian Subalgebra TSHO is an abelian, toral subalgebra in the odd Hamiltonian Lie superalgebra SHO that enables efficient weight space decomposition.
• Its diagonal, even, pairwise-commuting elements facilitate the classification of derivations, biderivations, and automorphisms in modular Lie superalgebra theory.
• The innerness of skew-symmetric super-biderivations, revealed through TSHO, simplifies cohomological analysis and extension problems in Cartan-type modular Lie superalgebras.

The Abelian Subalgebra TSHO arises naturally within the framework of the special odd Hamiltonian Lie superalgebra SHO($n$, $n; t$) over a field of characteristic $p > 2$. TSHO is defined as the abelian subalgebra spanned by certain even, pairwise-commuting elements, specifically $T_H(x_i x_{i'} - x_j x_{j'})$ for $i, j \in Y_0 = \{1, \ldots, n\}$, where $T_H$ denotes the embedding map from functions in the divided powers/Grassmann algebra into the odd Hamiltonian superalgebra HO($n$, $n; t$). This subalgebra serves as a toral subalgebra, facilitating a diagonalizable adjoint-action and enabling a weight space decomposition of HO and its subalgebra SHO. The structure and symmetry provided by TSHO are pivotal for the classification and analysis of derivations, biderivations, and automorphisms in this family of Cartan-type modular Lie superalgebras.

1. Definition and Algebraic Role of TSHO

The abelian subalgebra TSHO is explicitly given by: $$\mathrm{TSHO} = \mathrm{span}_F \left\{ T_H(x_i x_{i'} - x_j x_{j'}) : i, j \in Y_0 \right\}.$$ Every generator $T_H(x_i x_{i'} - x_j x_{j'})$ has parity zero and their mutual commutators vanish under the Lie bracket, establishing TSHO as an abelian subalgebra. TSHO is constructed to act diagonally by adjunction on HO, rendering the weight space decomposition tractable and robust for further algebraic manipulations. Its “toral” character mirrors the function of diagonal Cartan subalgebras in ordinary Lie theory, but adapted to the peculiarities of modular and superalgebraic contexts. This diagonal action forms the backbone for all subsequent weight space analysis in both HO and SHO.

2. Weight Space Decomposition Relative to TSHO

The adjoint action of TSHO on basis elements of HO can be described as follows. For any $T_H(x^{(\alpha)} x^u)$, with $x^{(\alpha)}$ a divided power and $x^u$ a Grassmann monomial, their commutator with $T_H(x_i x_{i'} - x_j x_{j'})$ yields: $$[T_H(x_i x_{i'} - x_j x_{j'}), T_H(x^{(\alpha)} x^u)] = (\delta_{i' \in u} - \alpha_i + \alpha_j - \delta_{j' \in u}) \cdot T_H(x^{(\alpha)} x^u),$$ where $\delta_{P}$ is 1 if $P$ holds and zero otherwise. Thus, $T_H(x^{(\alpha)} x^u)$ is an eigenvector for the adjoint action by every generator of TSHO, and the corresponding eigenvalue is expressed as $$(\alpha + \langle u \rangle)(T_H(x_i x_{i'} - x_j x_{j'}))$$.

This enables the following weight decomposition: $$HO = \bigoplus_{(\alpha + \langle u \rangle)} HO_{(\alpha + \langle u \rangle)},$$ where

$$HO_{(\alpha + \langle u \rangle)} = \left\{ T_H(x^{(\beta)} x^v) : \text{commutator as above yields the corresponding eigenvalue}\right\}.$$

Correspondingly, the special odd Hamiltonian superalgebra satisfies

$$SHO_{(\alpha + \langle u \rangle)} = HO_{(\alpha + \langle u \rangle)} \cap SHO.$$

Weight space decomposition simplifies the paper of derivations and biderivations, since actions may be expressed and compared directly in terms of their weights with respect to TSHO.

3. Skew-Symmetric Super-Biderivations and Their Action on TSHO

A super-biderivation $\varphi$ on a Lie superalgebra $L$ is a bilinear mapping $\varphi: L \times L \rightarrow L$ satisfying a generalized Leibniz rule in each variable, together with the graded skew-symmetry: $$\varphi(x, y) = -(-1)^{d(\varphi)d(x) + d(\varphi)d(y) + d(x)d(y)} \varphi(y, x),$$ for homogeneous $x$, $y$, where $d(\cdot)$ is the parity. The canonical example of a skew-symmetric super-biderivation is the mapping $\varphi_\lambda(x, y) = \lambda [x, y]$, for $\lambda \in F$.

In the context of SHO, if $T_H(x^{(\alpha)} x^u) \in SHO$, Lemma 4.2 ensures: $$\varphi(T_H(x_i x_{i'} - x_j x_{j'}), T_H(x^{(\alpha)} x^u)) \in SHO_{(\alpha+\langle u \rangle)}.$$ Thus, any biderivation preserves the weight spaces determined by TSHO. More generally, via technical lemmas exploring commutation and the vanishing of the center, biderivation actions on commuting pairs must yield zero, heavily restricting their possible form.

4. Classification: Innerness of Skew-Symmetric Super-Biderivations in SHO

The principal structural result of the paper establishes that every skew-symmetric super-biderivation on SHO is inner. Explicitly, for any such $\varphi$ there exists $\lambda \in F$ such that

$$\varphi(T_H(f), T_H(g)) = \lambda [T_H(f), T_H(g)],$$

for all $T_H(f), T_H(g) \in SHO$.

The proof leverages the weight space decomposition with respect to TSHO and a series of technical lemmas aligning $\varphi$’s action on “toral” generators and other elements. For instance, one has

$$\varphi(T_H(x_i x_{i'} - x_j x_{j'}), T_H(x_i)) = \lambda [T_H(x_i x_{i'} - x_j x_{j'}), T_H(x_i)],$$

with the scalar factor $\lambda$ shown to be independent of $i$ via further decompositional arguments. Ultimately, any deviation from an inner form is excluded by the simplicity and vanishing center of SHO, and by compatibility across all weight spaces.

5. Structural Impact and Applications in Lie Superalgebra Theory

The rigidity implied by the innerness of all skew-symmetric super-biderivations underscores the deep influence of TSHO on the algebraic structure of SHO. First, this annihilates the potential for nontrivial second cohomology classes realized by skew-symmetric biderivations, simplifying the deformation and extension theory for SHO. The result streamlines the task of classifying outer automorphisms and provides crucial insight for the representation theory and symmetry analysis of SHO and related Cartan-type modular Lie superalgebras.

Moreover, the methodology — employing the abelian toral subalgebra TSHO for weight decompositions — presents a transferable framework for other classes of Lie superalgebras. Such decompositions are instrumental for the analysis of derivational and cohomological questions elsewhere in modular superalgebra theory, possibly with adaptations to cases lacking explicit diagonal tori.

6. Broader Mathematical and Physical Contexts

The findings regarding Abelian Subalgebra TSHO have ramifications beyond the immediate structure theory of SHO. In quantum algebra and supergeometry, such toral decompositions facilitate the construction and analysis of module categories, clarify branching rules, and are pivotal in the paper of higher symmetries, particularly in positive characteristic settings. The simplicity and rigidity proven for super-biderivations ensure that SHO maintains a robust symmetry profile, important for both mathematical classification and applications to theoretical physics models based on modular superalgebras.

In summary, the Abelian subalgebra TSHO in SHO($n$, $n; t$) provides a central “toral” framework for weight decomposition and symmetry analysis, enabling a full classification of skew-symmetric super-biderivations and contributing fundamentally to the cohomological and automorphism theory of Cartan-type modular Lie superalgebras (Xu et al., 16 Aug 2025).