Restricted Heisenberg Lie Superalgebras

• Restricted Heisenberg Lie superalgebras are Lie superalgebras defined over fields of characteristic p>2, equipped with an even restricted [p]-operator and having a one-dimensional center.
• They combine the classical Heisenberg bracket with restricted cohomology methods to classify both splitting and non-splitting one-dimensional central extensions.
• The framework distinguishes two fundamental families (even-center and odd-center) and extends to twisted variants that require extra parameter conditions for their restricted structures.

Searching arXiv for papers on restricted Heisenberg Lie superalgebras, restricted cohomology, and central extensions. Search results:

• (Yang, 30 Aug 2025): Restricted Central Extensions of Restricted Heisenberg Lie Superalgebras
• (Yang, 3 Sep 2025): On the Restricted Cohomology of Twisted Heisenberg Lie Superalgebras
• (Bouarroudj et al., 2024): Central extensions of restricted Lie superalgebras and classification of p-nilpotent Lie superalgebras in dimension 4

Restricted Heisenberg Lie superalgebras are Heisenberg Lie superalgebras over an algebraically closed field $F$ of characteristic $p>2$ equipped with a restricted structure, i.e. an even $[p]$-operator on the even part making the odd part a restricted module over the even part. In the setting developed by Yong Yang, the basic objects are two families with one-dimensional center, one with an even center and one with an odd center; their ordinary and restricted low-degree cohomology determine the corresponding restricted one-dimensional central extensions (Yang, 30 Aug 2025).

1. Algebraic setting

A general Heisenberg Lie superalgebra may be presented as

$H_{m,n}=\Span\{p_i,q_i,Z\}\oplus\Span\{\psi_a\}, \quad [p_i,q_j]=\delta_{ij}Z, \quad [\psi_a,\psi_b]=B_{ab}Z, \quad Z\in\mathfrak z,$

with $|p_i|=|q_i|=|Z|=\bar0$ and $|\psi_a|=\bar1$ (Bouarroudj et al., 2024). This description isolates the characteristic feature of the Heisenberg type: the brackets close on a one-dimensional central subspace.

For restricted Lie superalgebras, the even part carries the $[p]$-map. In Yang’s framework, the relevant cohomological input is the ordinary Chevalley–Eilenberg complex with trivial coefficients,

$$C^q(\mathfrak g)=\wedge^q(\mathfrak g)^*, \qquad H^q(\mathfrak g)=\ker(d^q)/\mathrm{im}(d^{q-1}),$$

together with restricted cochain spaces $C^2_{\mathrm{res}}(\mathfrak g)$ or $C^2_*(\mathfrak g)$, depending on notation, where a $p>2$0-cochain is paired with a $p>2$1-compatible map on $p>2$2 (Yang, 30 Aug 2025).

The subject therefore combines three structures: the Heisenberg bracket, the restricted $p>2$3-map, and the cohomological passage from ordinary to restricted extension theory. This is the mechanism through which one-dimensional restricted central extensions are classified.

2. Two basic families with one-dimensional center

Yang’s treatment distinguishes an even-center family $p>2$4 and an odd-center family $p>2$5 (Yang, 30 Aug 2025).

Family	Underlying superspace and brackets	Restricted structure
$p>2$6	$p>2$7, with $p>2$8 and $p>2$9	For $[p]$0, $[p]$1
$[p]$2	$[p]$3, with $[p]$4	$[p]$5 for all $[p]$6, hence $[p]$7 on $[p]$8

For the even-center family, the one-dimensional even center is spanned by $[p]$9. Fixing

$H_{m,n}=\Span\{p_i,q_i,Z\}\oplus\Span\{\psi_a\}, \quad [p_i,q_j]=\delta_{ij}Z, \quad [\psi_a,\psi_b]=B_{ab}Z, \quad Z\in\mathfrak z,$0

one obtains the unique $H_{m,n}=\Span\{p_i,q_i,Z\}\oplus\Span\{\psi_a\}, \quad [p_i,q_j]=\delta_{ij}Z, \quad [\psi_a,\psi_b]=B_{ab}Z, \quad Z\in\mathfrak z,$1-map on $H_{m,n}=\Span\{p_i,q_i,Z\}\oplus\Span\{\psi_a\}, \quad [p_i,q_j]=\delta_{ij}Z, \quad [\psi_a,\psi_b]=B_{ab}Z, \quad Z\in\mathfrak z,$2 by

$H_{m,n}=\Span\{p_i,q_i,Z\}\oplus\Span\{\psi_a\}, \quad [p_i,q_j]=\delta_{ij}Z, \quad [\psi_a,\psi_b]=B_{ab}Z, \quad Z\in\mathfrak z,$3

Because $H_{m,n}=\Span\{p_i,q_i,Z\}\oplus\Span\{\psi_a\}, \quad [p_i,q_j]=\delta_{ij}Z, \quad [\psi_a,\psi_b]=B_{ab}Z, \quad Z\in\mathfrak z,$4, this defines the restricted superalgebra $H_{m,n}=\Span\{p_i,q_i,Z\}\oplus\Span\{\psi_a\}, \quad [p_i,q_j]=\delta_{ij}Z, \quad [\psi_a,\psi_b]=B_{ab}Z, \quad Z\in\mathfrak z,$5.

For the odd-center family, the one-dimensional odd center is spanned by $H_{m,n}=\Span\{p_i,q_i,Z\}\oplus\Span\{\psi_a\}, \quad [p_i,q_j]=\delta_{ij}Z, \quad [\psi_a,\psi_b]=B_{ab}Z, \quad Z\in\mathfrak z,$6. Since the odd part must be a restricted module over the even part and the center is odd, one checks that $H_{m,n}=\Span\{p_i,q_i,Z\}\oplus\Span\{\psi_a\}, \quad [p_i,q_j]=\delta_{ij}Z, \quad [\psi_a,\psi_b]=B_{ab}Z, \quad Z\in\mathfrak z,$7 for all $H_{m,n}=\Span\{p_i,q_i,Z\}\oplus\Span\{\psi_a\}, \quad [p_i,q_j]=\delta_{ij}Z, \quad [\psi_a,\psi_b]=B_{ab}Z, \quad Z\in\mathfrak z,$8 is forced. The resulting restricted superalgebra is denoted

$H_{m,n}=\Span\{p_i,q_i,Z\}\oplus\Span\{\psi_a\}, \quad [p_i,q_j]=\delta_{ij}Z, \quad [\psi_a,\psi_b]=B_{ab}Z, \quad Z\in\mathfrak z,$9

An important structural distinction already appears here: in the two basic families, the restricted structure is unique once the family is fixed, while later twisted variants admit restricted structures only under an additional parameter condition.

3. Ordinary cohomology in degrees $|p_i|=|q_i|=|Z|=\bar0$0 and $|p_i|=|q_i|=|Z|=\bar0$1

The ordinary cohomology used in the classification of restricted extensions is computed with trivial coefficients via the Chevalley–Eilenberg differential

$|p_i|=|q_i|=|Z|=\bar0$2

(Yang, 30 Aug 2025).

For the even-center family $|p_i|=|q_i|=|Z|=\bar0$3, one has

$|p_i|=|q_i|=|Z|=\bar0$4

Writing $|p_i|=|q_i|=|Z|=\bar0$5,

$|p_i|=|q_i|=|Z|=\bar0$6

A basis of cocycles may be taken to be the classes of

$|p_i|=|q_i|=|Z|=\bar0$7

$|p_i|=|q_i|=|Z|=\bar0$8

For the odd-center family $|p_i|=|q_i|=|Z|=\bar0$9,

$|\psi_a|=\bar1$0

If $|\psi_a|=\bar1$1, then

$|\psi_a|=\bar1$2

where

$|\psi_a|=\bar1$3

These formulas show that the ordinary $|\psi_a|=\bar1$4-cohomology is already sufficiently explicit to distinguish bracket deformations by parity and by the position of the center. A plausible implication is that the restricted extension problem is tractable because the ordinary cocycle representatives can be lifted concretely.

4. Restricted cohomology and the six-term exact sequence

The passage from ordinary to restricted cohomology is controlled by the restricted cochain complex

$|\psi_a|=\bar1$5

together with the six-term exact sequence

$|\psi_a|=\bar1$6

(Yang, 30 Aug 2025).

For the even-center family $|\psi_a|=\bar1$7, one checks that $|\psi_a|=\bar1$8 and that the last map $|\psi_a|=\bar1$9 vanishes because all $[p]$0-fold brackets land in the center spanned by $[p]$1. Hence

$[p]$2

Since

$[p]$3

and

$[p]$4

one obtains

$[p]$5

A homogeneous basis consists of the splitting part

$[p]$6

and the lifted ordinary cocycles

$[p]$7

where in fact each $[p]$8.

For the odd-center family $[p]$9, again $$C^q(\mathfrak g)=\wedge^q(\mathfrak g)^*, \qquad H^q(\mathfrak g)=\ker(d^q)/\mathrm{im}(d^{q-1}),$$0 and $$C^q(\mathfrak g)=\wedge^q(\mathfrak g)^*, \qquad H^q(\mathfrak g)=\ker(d^q)/\mathrm{im}(d^{q-1}),$$1, so

$$C^q(\mathfrak g)=\wedge^q(\mathfrak g)^*, \qquad H^q(\mathfrak g)=\ker(d^q)/\mathrm{im}(d^{q-1}),$$2

Here

$$C^q(\mathfrak g)=\wedge^q(\mathfrak g)^*, \qquad H^q(\mathfrak g)=\ker(d^q)/\mathrm{im}(d^{q-1}),$$3

and

$$C^q(\mathfrak g)=\wedge^q(\mathfrak g)^*, \qquad H^q(\mathfrak g)=\ker(d^q)/\mathrm{im}(d^{q-1}),$$4

In these two families, restricted $$C^q(\mathfrak g)=\wedge^q(\mathfrak g)^*, \qquad H^q(\mathfrak g)=\ker(d^q)/\mathrm{im}(d^{q-1}),$$5-cohomology splits into a Frobenius part and a lifted ordinary part. This contrasts sharply with the twisted case, where the exact sequence does not collapse in the same way.

5. Restricted one-dimensional central extensions

By general theory, equivalence classes of restricted one-dimensional central extensions of a restricted superalgebra $$C^q(\mathfrak g)=\wedge^q(\mathfrak g)^*, \qquad H^q(\mathfrak g)=\ker(d^q)/\mathrm{im}(d^{q-1}),$$6 by the trivial module $$C^q(\mathfrak g)=\wedge^q(\mathfrak g)^*, \qquad H^q(\mathfrak g)=\ker(d^q)/\mathrm{im}(d^{q-1}),$$7 are in bijection with $$C^q(\mathfrak g)=\wedge^q(\mathfrak g)^*, \qquad H^q(\mathfrak g)=\ker(d^q)/\mathrm{im}(d^{q-1}),$$8 (Yang, 30 Aug 2025). For a cocycle $$C^q(\mathfrak g)=\wedge^q(\mathfrak g)^*, \qquad H^q(\mathfrak g)=\ker(d^q)/\mathrm{im}(d^{q-1}),$$9, the extension

$C^2_{\mathrm{res}}(\mathfrak g)$0

is defined by

$C^2_{\mathrm{res}}(\mathfrak g)$1

For $C^2_{\mathrm{res}}(\mathfrak g)$2, the splitting extensions are the $C^2_{\mathrm{res}}(\mathfrak g)$3 classes $C^2_{\mathrm{res}}(\mathfrak g)$4, yielding superalgebras $C^2_{\mathrm{res}}(\mathfrak g)$5 in which the bracket is unchanged and

$C^2_{\mathrm{res}}(\mathfrak g)$6

The non-splitting extensions have $C^2_{\mathrm{res}}(\mathfrak g)$7. For each

$C^2_{\mathrm{res}}(\mathfrak g)$8

one obtains an extension $C^2_{\mathrm{res}}(\mathfrak g)$9 or $C^2_*(\mathfrak g)$0 with

$C^2_*(\mathfrak g)$1

These exhaust the

$C^2_*(\mathfrak g)$2

non-splitting classes.

For $C^2_*(\mathfrak g)$3, the splitting classes are $C^2_*(\mathfrak g)$4, $C^2_*(\mathfrak g)$5, producing $C^2_*(\mathfrak g)$6 with only the $C^2_*(\mathfrak g)$7-map perturbed by $C^2_*(\mathfrak g)$8. The non-splitting classes come from ordinary cocycles $C^2_*(\mathfrak g)$9 or $p>2$00, yielding $p>2$01 or $p>2$02, in which only the bracket gains a $p>2$03-term and the $p>2$04-map remains zero on $p>2$05. Altogether these form the remaining $p>2$06 classes.

The classification separates “pure $p>2$07-map” deformations from bracket deformations. In the first type, the extension splits as an ordinary central extension but not as a restricted one; in the second, the bracket itself changes while the restricted structure on the even part is unchanged.

6. Twisted Heisenberg variants

The restricted cohomology of twisted Heisenberg Lie superalgebras displays a different pattern (Yang, 3 Sep 2025). For

$p>2$08

the even part is spanned by

$p>2$09

the odd part by

$p>2$10

and the nonzero brackets include

$p>2$11

$p>2$12

$p>2$13

Here a restricted structure exists if and only if

$p>2$14

Fixing

$p>2$15

the $p>2$16-map on generators is

$p>2$17

The ordinary cohomology is more selective than in the two basic families: $p>2$18 and $p>2$19 is grouped into five families $p>2$20. Restrictedly, one shows

$p>2$21

and the map $p>2$22 vanishes on the $p>2$23-components of $p>2$24. Therefore

$p>2$25

This computation isolates a genuine obstruction absent from the two basic families: the class corresponding to $p>2$26 is not annihilated by $p>2$27. A plausible implication is that twisted Heisenberg superalgebras provide the first natural setting in which restricted extension theory no longer reduces to a direct sum of Frobenius maps and ordinary cocycles.

The example

$p>2$28

makes this explicit. Restrictedness forces $p>2$29, and one finds

$p>2$30

Moreover, there are five independent restricted central extensions, three coming from “pure $p>2$31-map” deformations of $p>2$32, and two from the even cocycles above.

7. Low-dimensional cases and conceptual distinctions

The low-dimensional classification of $p>2$33-nilpotent restricted Lie superalgebras up to dimension $p>2$34 gives a concrete set of Heisenberg examples (Bouarroudj et al., 2024). The possibilities $p>2$35 are: $p>2$36 They correspond respectively to $p>2$37, $p>2$38, $p>2$39, and $p>2$40.

For $p>2$41 and $p>2$42, the possible $p>2$43-maps on the even generators fall into two classes: the trivial $p>2$44-map and the “step-one” $p>2$45-map

$p>2$46

up to automorphic variant. For $p>2$47 and $p>2$48, the even part is $p>2$49, so only $p>2$50 occurs.

The restricted cohomology in these low-dimensional cases is correspondingly small. In every Heisenberg case,

$p>2$51

where $p>2$52 is the cocycle defining the central bracket: for example,

$p>2$53

The classification states that, up to isomorphism and up to automorphisms of the underlying restricted structure, exactly four families occur, and that no further central or restricted deformations arise in dimension $p>2$54.

These low-dimensional results clarify two points. First, restricted Heisenberg Lie superalgebras are not determined solely by the underlying bracket; even in dimension $p>2$55, distinct $p>2$56-maps can yield non-isomorphic restricted structures. Second, the one-dimensionality of $p>2$57 in the $p>2$58-nilpotent small-dimensional cases should not be conflated with the higher-dimensional families, where Yang’s calculations produce large Frobenius summands and many non-splitting extension classes.