Restricted Twisted Heisenberg Lie Superalgebras

• Restricted twisted Heisenberg Lie superalgebras are algebraic structures defined over fields of prime characteristic, featuring a [p]-map on the even part and restricted module conditions on the odd part.
• They extend classical Heisenberg superalgebras by introducing a semisimple derivation that governs the interaction between even and odd generators through specific eigenvalue conditions.
• Their cohomological framework employs a six-term exact sequence to compute low-degree cohomology and classify one-dimensional central extensions via Frobenius maps and explicit cocycle data.

Restricted twisted Heisenberg Lie superalgebras are twisted Heisenberg Lie superalgebras equipped with a $[p]$-map on the even part such that the even part is a restricted Lie algebra and the odd part is a restricted module under the adjoint action. In the family studied over an algebraically closed field $\mathbb F$ of characteristic $p>0$, the underlying Lie superalgebra is denoted $\mathfrak h^{\lambda,\kappa}_{m,n,t}$, while the restricted form is denoted $\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}$, where $\mu$ parametrizes the $[p]$-map on the even part (Yang, 3 Sep 2025). Their low-degree cohomology is organized by the six-term exact sequence for restricted Lie superalgebras, which generalizes Hochschild’s sequence from restricted Lie algebras to the super setting (Liu, 2011).

1. Twisted Heisenberg Lie superalgebras and their presentation

Over an algebraically closed field $\mathbb F$ of characteristic $p>0$, for non-negative integers $m,n,t$ and parameters

$\mathbb F$0

the twisted Heisenberg Lie superalgebra $\mathbb F$1 is the superspace

$\mathbb F$2

with parity decomposition

$\mathbb F$3

Its nonzero brackets are (Yang, 3 Sep 2025)

$\mathbb F$4

$\mathbb F$5

$\mathbb F$6

$\mathbb F$7

The even center is one-dimensional, spanned by $\mathbb F$8. The ideal

$\mathbb F$9

is a Heisenberg Lie superalgebra with even center, and $p>0$0 is a $p>0$1-dimensional extension of $p>0$2 by $p>0$3 (Yang, 3 Sep 2025).

In this terminology, the untwisted Heisenberg Lie superalgebra $p>0$4 has only the central brackets $p>0$5, $p>0$6, and $p>0$7, whereas the twisted version adds the outer derivation $p>0$8 acting semisimply on the even pairs $p>0$9 with eigenvalues $\mathfrak h^{\lambda,\kappa}_{m,n,t}$0 and on the odd pairs $\mathfrak h^{\lambda,\kappa}_{m,n,t}$1 with eigenvalues $\mathfrak h^{\lambda,\kappa}_{m,n,t}$2 (Yang, 3 Sep 2025). Over $\mathfrak h^{\lambda,\kappa}_{m,n,t}$3, twisted Heisenberg superalgebras were introduced precisely by adding derivations to Heisenberg superalgebras, and their fine gradings and Weyl groups were classified (Xie et al., 2018).

2. Restricted structures and the $\mathfrak h^{\lambda,\kappa}_{m,n,t}$4-map

A restricted Lie superalgebra is a Lie superalgebra $\mathfrak h^{\lambda,\kappa}_{m,n,t}$5 equipped with a map

$\mathfrak h^{\lambda,\kappa}_{m,n,t}$6

such that $\mathfrak h^{\lambda,\kappa}_{m,n,t}$7 is a restricted Lie algebra and $\mathfrak h^{\lambda,\kappa}_{m,n,t}$8 is a restricted module over $\mathfrak h^{\lambda,\kappa}_{m,n,t}$9 with respect to the adjoint action (Liu, 2011). In particular, the $\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}$0-map is defined on the even part only; the odd part is required to satisfy the restricted module condition. The same formulation is used in the cohomological study of restricted twisted Heisenberg Lie superalgebras (Yang, 3 Sep 2025).

For the even twisted Heisenberg Lie algebra $\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}$1, one has the criterion: $\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}$2 Then there is a parameter $\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}$3 such that

$\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}$4

$\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}$5

where

$\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}$6

For a general even element $\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}$7,

$\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}$8

This same formula gives the $\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}$9-map on the even part of the superalgebra $\mu$0 (Yang, 3 Sep 2025).

The full restriction criterion is: $\mu$1 and

$\mu$2

Writing this common value again as

$\mu$3

the resulting restricted Lie superalgebra is denoted $\mu$4 (Yang, 3 Sep 2025).

The equality of all $\mu$5-st powers is forced by the compatibility condition

$\mu$6

for even $\mu$7 and odd $\mu$8, applied in particular to $\mu$9 and odd generators $[p]$0. A recurrent misconception is that “restricted” should provide a $[p]$1-operator on odd generators; the standard super definition excludes this and instead imposes restricted-module compatibility on the odd part (Liu, 2011).

3. Cohomological framework: ordinary, restricted, and the six-term sequence

For a Lie superalgebra $[p]$2 with trivial coefficients, the ordinary cochains are

$[p]$3

with differentials

$[p]$4

and, for $[p]$5,

$[p]$6

Thus $[p]$7 for $[p]$8 (Yang, 3 Sep 2025).

Restricted cochains refine ordinary cochains by recording compatibility with the $[p]$9-map. For a restricted Lie superalgebra $\mathbb F$0,

$\mathbb F$1

and

$\mathbb F$2

The space of Frobenius homomorphisms is

$\mathbb F$3

There is a short exact sequence

$\mathbb F$4

hence

$\mathbb F$5

The restricted differentials are

$\mathbb F$6

and

$\mathbb F$7

with

$\mathbb F$8

Then

$\mathbb F$9

(Yang, 3 Sep 2025).

The low-degree relation between ordinary and restricted cohomology is the six-term exact sequence

$p>0$0

where

$p>0$1

and, for $p>0$2,

$p>0$3

This is the form used in the computation of $p>0$4 (Yang, 3 Sep 2025). Its general super-theoretic origin is the six-term exact sequence for restricted Lie superalgebras established in (Liu, 2011).

4. Ordinary and restricted low-degree cohomology

Let

$p>0$5

The Hochschild–Serre spectral sequence for the pair $p>0$6 with trivial coefficients has

$p>0$7

and since $p>0$8 is $p>0$9-dimensional even, one obtains

$m,n,t$0

In particular, $m,n,t$1 is spanned by the classes of

$m,n,t$2

while $m,n,t$3. Therefore

$m,n,t$4

and

$m,n,t$5

Hence $m,n,t$6 (Yang, 3 Sep 2025).

For $m,n,t$7, one decomposes $m,n,t$8 into $m,n,t$9-stable pieces and extracts the invariants. The resulting spanning cocycles are organized into five sets: $\mathbb F$00 Then $\mathbb F$01 is spanned by the classes of cocycles

$\mathbb F$02

Its super-dimension is

$\mathbb F$03

$\mathbb F$04

where $\mathbb F$05 are even and $\mathbb F$06 are odd (Yang, 3 Sep 2025).

Restricted $\mathbb F$07-cohomology is smaller: $\mathbb F$08 Since

$\mathbb F$09

the quotient has basis represented by $\mathbb F$10, and

$\mathbb F$11

Thus the ordinary class $\mathbb F$12 survives in $\mathbb F$13 but not in $\mathbb F$14 (Yang, 3 Sep 2025).

For $\mathbb F$15, the map $\mathbb F$16 in $\mathbb F$17 has image

$\mathbb F$18

because $\mathbb F$19 while

$\mathbb F$20

The map $\mathbb F$21 kills $\mathbb F$22 and is nonzero on any class with a nonzero $\mathbb F$23-component. Therefore

$\mathbb F$24

and the six-term sequence yields

$\mathbb F$25

Equivalently,

$\mathbb F$26

with super-dimension

$\mathbb F$27

$\mathbb F$28

(Yang, 3 Sep 2025).

5. Restricted one-dimensional central extensions

Restricted one-dimensional central extensions by an even, strongly abelian module $\mathbb F$29 are classified by the even part $\mathbb F$30. If $\mathbb F$31, the corresponding extension $\mathbb F$32 has

$\mathbb F$33

for $\mathbb F$34 and $\mathbb F$35. Two such extensions are equivalent iff the cocycles are cohomologous in $\mathbb F$36 (Yang, 3 Sep 2025).

A basis for even restricted $\mathbb F$37-cocycles consists of

$\mathbb F$38

The odd cocycles from $\mathbb F$39 do not classify restricted central extensions by an even $\mathbb F$40-dimensional center (Yang, 3 Sep 2025).

The Frobenius part $\mathbb F$41 yields extensions $\mathbb F$42 with

$\mathbb F$43

These extensions are invisible on the Lie bracket level and deform only the $\mathbb F$44-structure. The set $\mathbb F$45 spans the space of restricted one-dimensional central extensions that are split as Lie superalgebra extensions (Yang, 3 Sep 2025). The same phenomenon occurs for untwisted restricted Heisenberg Lie superalgebras, where Frobenius cocycles correspond to bracket-split but restricted-nontrivial extensions (Yang, 30 Aug 2025).

The $\mathbb F$46-classes produce even-even bracket extensions. For instance, if $\mathbb F$47 and $\mathbb F$48,

$\mathbb F$49

and the associated extension $\mathbb F$50 satisfies

$\mathbb F$51

with

$\mathbb F$52

A second $\mathbb F$53-type,

$\mathbb F$54

gives

$\mathbb F$55

together with the explicit $\mathbb F$56-formula recorded in the classification theorem (Yang, 3 Sep 2025).

The $\mathbb F$57-classes modify odd-odd $\mathbb F$58-brackets. For

$\mathbb F$59

the extension $\mathbb F$60 has

$\mathbb F$61

and

$\mathbb F$62

Similarly, for

$\mathbb F$63

one gets another odd-odd family $\mathbb F$64 with unchanged $\mathbb F$65-map (Yang, 3 Sep 2025).

The $\mathbb F$66-classes modify the $\mathbb F$67-$\mathbb F$68 brackets. For

$\mathbb F$69

the extension $\mathbb F$70 satisfies

$\mathbb F$71

These formulas provide a complete description of restricted one-dimensional central extensions inside the chosen family (Yang, 3 Sep 2025).

6. Relation to untwisted Heisenberg superalgebras and broader context

Restricted twisted Heisenberg Lie superalgebras sit at the intersection of three established strands. First, untwisted Heisenberg Lie superalgebras with one-dimensional center occur in two types, $\mathbb F$72 and $\mathbb F$73, and their ordinary cohomology was computed by Hochschild–Serre methods relative to the center (Bai et al., 2013). Second, restricted Heisenberg Lie superalgebras and their restricted one-dimensional central extensions were analyzed over an algebraically closed field of characteristic $\mathbb F$74; in that untwisted setting, the restricted $\mathbb F$75-cohomology likewise splits into an ordinary part and a Frobenius part (Yang, 30 Aug 2025). Third, twisted Heisenberg superalgebras over $\mathbb F$76 were defined by adjoining derivations to Heisenberg superalgebras and were studied through fine gradings and Weyl groups (Xie et al., 2018).

Within this landscape, the “twisted” and “restricted” modifiers refer to different structures. “Twisted” refers to the semisimple derivation $\mathbb F$77 acting on even and odd generators; “restricted” refers to the presence of a $\mathbb F$78-map on the even part together with restricted-module compatibility on the odd part (Yang, 3 Sep 2025). The recent cohomological classification shows that the restriction criterion is controlled by the equalities

$\mathbb F$79

while the low-degree cohomology is controlled by the eigenspace coincidences

$\mathbb F$80

that appear in the sets $\mathbb F$81 and in the dimension formulas for $\mathbb F$82 and $\mathbb F$83 (Yang, 3 Sep 2025). This suggests that the parameter sets $\mathbb F$84, $\mathbb F$85, and the common scalar $\mathbb F$86 serve as primary cohomological invariants within the family.

A further conceptual point is supplied by the general six-term exact sequence. For restricted Lie superalgebras, the sequence

$\mathbb F$87

measures exactly how ordinary cocycles interact with the $\mathbb F$88-structure (Liu, 2011). In the twisted Heisenberg case, this interaction becomes completely explicit: the $\mathbb F$89-part of $\mathbb F$90 is killed when passing to $\mathbb F$91, while the Frobenius classes $\mathbb F$92 appear only in restricted cohomology (Yang, 3 Sep 2025). A plausible implication is that restricted twisted Heisenberg Lie superalgebras provide a particularly transparent family in which the obstruction-theoretic meaning of the six-term sequence can be seen directly at the level of generators, brackets, and $\mathbb F$93-operators.