Restricted Lie-Rinehart Superalgebra

• Restricted Lie-Rinehart superalgebras are structures that blend supercommutative algebras, Lie superalgebra theory, and restricted Lie algebra concepts in characteristic p > 2.
• They feature a p|2p–structure with explicitly defined p-maps and superized Hochschild conditions ensuring compatibility between algebra actions and module derivations.
• This framework underpins the construction of universal enveloping superalgebras and provides concrete examples such as differential operators and Witt-type superalgebras.

A restricted Lie-Rinehart superalgebra is a structure unifying super-commutative algebra, Lie superalgebra theory, and the notion of restriction in positive characteristic, extending classical restricted Lie algebra and Lie-Rinehart algebra concepts to the super context. It is defined over a field $\K$ of characteristic $p>2$ and is rooted in a “superized” version of Hochschild's lemma, which motivates the specific compatibility constraints on the $p$-maps and module actions. This framework supports the construction of universal enveloping superalgebras with a universal property analogous to the classical case and admits various natural and explicit examples, including differential operators and Witt-type superalgebras.

1. Structural Foundation: Lie-Rinehart Superalgebras

A Lie-Rinehart superalgebra over $\K$ consists of a triple $(A,L,\rho)$ where $A=A_{\bar0}\oplus A_{\bar1}$ is a unital, associative, supercommutative $\K$-algebra, $L=L_{\bar0}\oplus L_{\bar1}$ is a Lie superalgebra and a graded $A$-module, and $\rho:L\to\Der_\K(A)$ is an $A$-linear Lie superalgebra morphism called the anchor. The central compatibility is the Leibniz rule: $$[\,x,\,a\,y\,] =(-1)^{|x|\,|a|}\,a\,[\,x,\,y\,]+\rho(x)(a)\,y, \quad x,y\in L,\ a\in A.$$ Representations are $A$-modules $M$ admitting an $A$-linear Lie action $\phi:L\to\End_\K(M)$ obeying the analogous module Leibniz rule. Morphisms in this category consist of compatible pairs of algebra and Lie superalgebra morphisms that respect the anchor.

2. Restricted Lie Superalgebras: $p|2p$–Structures

A restricted Lie superalgebra $L=L_{\bar0}\oplus L_{\bar1}$ over characteristic $p>2$ is equipped with a $p|2p$–structure. The even component $L_{\bar0}$ carries the usual restricted structure with a $p$-map $[p]:L_{\bar0}\to L_{\bar0}$, satisfying

$(\lambda\,x)^{[p]}=\lambda^p\,x^{[p]},\quad \ad_{x^{[p]}}=(\ad_x)^p,\quad (x+y)^{[p]}=x^{[p]}+y^{[p]}+\sum_{i=1}^{p-1}s_i(x,y).$

For the odd component, the map $x^{[2p]}:=(x^2)^{[p]}$ for $x\in L_{\bar1}$ is defined via $x^2=\frac12[x,x]$; it encodes the super analog of the $p$-map. The structure is specified to ensure module and morphism compatibility: morphisms and modules require parity-specific compatibility conditions with $[p]$ and $[2p]$ respectively.

3. Superized Hochschild's Lemma and Its Role

The classical Hochschild lemma links associative algebra powers in characteristic $p$ with restricted Lie algebra actions. Its superization, as established in (Bouarroudj et al., 23 Nov 2025), delineates the interaction between module powers and the anchor map for all parity combinations, yielding the following cases:

• if $a,x$ are both even: $$\phi(a\,x)^{p} = a^p\,\phi(x)^p + \rho(a x)^{p-1}(a)\,\phi(x)$$,
• if $a$ even, $x$ odd: $\phi(a\,x)^{2p}$ receives nontrivial correction terms involving explicit combinatorial constants $\lambda_i$,
• if $a$ odd, $x$ even: $\phi(a\,x)^{2p}=0$,
• if both are odd: $\phi(a\,x)^{p} = a(\rho(x)(a))^{p-1}\,\phi(x)$.

This result is derived via analysis of smash product algebras and combinatorial recurrence for algebraic coefficients, ensuring all restriction maps and module interactions remain consistent in the super context.

4. Definition and Properties of Restricted Lie–Rinehart Superalgebras

A Lie-Rinehart superalgebra $(A,L,\rho)$ is restricted if:

1. $L$ carries a $p|2p$–structure, i.e., is a restricted Lie superalgebra.
2. The following Hochschild-type compatibility conditions hold for all homogeneous $a\in A$, $x\in L$:
   • $a$ even, $x$ even: $(a\,x)^{[p]}=a^p\,x^{[p]} + \rho(a x)^{p-1}(a)\,x$,
   • $a$ even, $x$ odd: $$(a\,x)^{[2p]}=a^{2p}\,x^{[2p]}+\rho(a x)^{2p-1}(a)\,x+\sum \lambda_i\,\rho(a x)^i(a)\,\rho(a x)^{2p-2-i}(a)\,x^2$$,
   • $a$ odd, $x$ even: $(a\,x)^{[2p]}=0$,
   • $a$ odd, $x$ odd: $(a\,x)^{[p]}=a(\rho(x)(a))^{p-1}\,x$.

For $x^2=\tfrac12[x,x]$ and $\lambda_i$ as combinatorial factors, these conditions ensure alignment between the superized restriction and underlying algebraic structure. This definition generalizes the classical restricted Lie-Rinehart algebra when restricted to the purely even case.

5. Modules and Semi-Direct Product Construction

Restricted representations of $(A,L,\rho)$ are $A$-modules $V$ with a map $\phi:L\to\End(V)$ that is both a restricted Lie superalgebra morphism and satisfies the LR-module compatibility rule. For each parity case, analogues of the superized Hochschild conditions must be satisfied. The semi-direct product $L\rtimes V$ acquires a natural Lie superalgebra structure with bracket: $$[(x+v), (y+w)]_{\rtimes} = [x,y] + \phi(x)(w) - (-1)^{|y||v|} \phi(y)(v),$$ and admits a restricted structure: for $e_i\in L_{\bar0}$, $v_j\in V_{\bar0}$,

$$(e_i+v_j)^{[p]} = e_i^{[p]} + \phi(e_i)^{p-1}(v_j).$$

A plausible implication is that, under reasonable center-freeness hypotheses, $(A, L\rtimes V, \widetilde\rho)$ is again a restricted Lie-Rinehart superalgebra.

6. Example Constructions

Representative examples include:

• $(A, \Der(A), \id)$ for any supercommutative algebra $A$, explicitly satisfying the superized Hochschild conditions.
• The restricted Witt superalgebra $W(n)$ over the Grassmann algebra $A=\Lambda(n)$, where $W(n)\subset \Der(\Lambda(n))$ has an induced restricted structure.
• Finite-dimensional “toy models” such as $A$ of dimension $1|1$, $L$ of dimension $2|1$ or $2|2$ with explicit presentations (cf. Examples 4.6–4.7 in (Bouarroudj et al., 23 Nov 2025)).

7. Universal Enveloping Algebra and Universal Property

For $(A,L,\rho)$, the ordinary universal enveloping superalgebra $U(A,L)$ is constructed via the semidirect sum $A\rtimes L$ and relations ensuring compatibility of algebra and Lie actions. The pairs of injections $\iota_A:A\hookrightarrow U(A,L)$, $\iota_L:L\hookrightarrow U(A,L)$ satisfy $\iota_A(a)\,\iota_L(x) = \iota_L(a\,x)$ and

$$\iota_L(x)\,\iota_A(a) - (-1)^{|x||a|}\iota_A(a)\,\iota_L(x) = \iota_A(\rho(x)(a)).$$

The restricted enveloping algebra $U_p(A,L)$ is obtained by further quotienting by the ideal generated by $\iota_L(x)^p-\iota_L(x^{[p]})$ for $x\in L_{\bar0}$. Proposition 5.1 in (Bouarroudj et al., 23 Nov 2025) establishes the universal property: any associative superalgebra $B$ with compatible maps $j_A, j_L$ factors uniquely through $U_p(A,L)$, provided $j_A$ is an algebra morphism and $j_L$ a restricted Lie morphism satisfying the prescribed relations.

This construction completes the parallel with classical theory and serves as a foundation for further study of representations, cohomology, and deformation of restricted Lie-Rinehart structures in the super setting.