---
title: Restricted Lie-Rinehart Superalgebra
url: https://www.emergentmind.com/topics/restricted-lie-rinehart-superalgebra
type: topic
---

# Restricted Lie-Rinehart Superalgebra

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 [2511.18372], 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 [2511.18372]).

## 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 [2511.18372] 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.

Source: https://www.emergentmind.com/topics/restricted-lie-rinehart-superalgebra