---
title: Restricted Twisted Heisenberg Lie Superalgebras
url: https://www.emergentmind.com/topics/restricted-twisted-heisenberg-lie-superalgebras
type: topic
---

# Restricted Twisted Heisenberg Lie Superalgebras

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 [2509.03016]. 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 [1109.2452].

## 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
\[
\lambda=(\lambda_1,\dots,\lambda_m)\in(\mathbb F^\times)^m,\qquad
\kappa=(\kappa_1,\dots,\kappa_n)\in(\mathbb F^\times)^n,
\]
the twisted Heisenberg Lie superalgebra \(\mathfrak h^{\lambda,\kappa}_{m,n,t}\) is the superspace
\[
\mathfrak h^{\lambda,\kappa}_{m,n,t}
=
\operatorname{span}_{\mathbb F}\{e_1,\ldots,e_{2m},e_{2m+1},e_{2m+2}\}
\;\oplus\;
\operatorname{span}_{\mathbb F}\{\omega_1,\ldots,\omega_{2n},\eta_1,\ldots,\eta_t\},
\]
with parity decomposition
\[
(\mathfrak h^{\lambda,\kappa}_{m,n,t})_{\bar 0}
=\operatorname{span}\{e_1,\dots,e_{2m+2}\},\qquad
(\mathfrak h^{\lambda,\kappa}_{m,n,t})_{\bar 1}
=\operatorname{span}\{\omega_1,\dots,\omega_{2n},\eta_1,\dots,\eta_t\}.
\]
Its nonzero brackets are [2509.03016]
\[
[e_i,e_{m+i}]=e_{2m+1}\qquad (1\le i\le m),
\]
\[
[\omega_j,\omega_j]=e_{2m+1},\quad
[\omega_{n+j},\omega_{n+j}]=-e_{2m+1},\quad
[\eta_k,\eta_k]=e_{2m+1},
\]
\[
[e_{2m+2},e_i]=\lambda_i e_{m+i},\qquad
[e_{2m+2},e_{m+i}]=\lambda_i e_i,
\]
\[
[e_{2m+2},\omega_j]=\kappa_j\omega_{n+j},\qquad
[e_{2m+2},\omega_{n+j}]=\kappa_j\omega_j.
\]

The even center is one-dimensional, spanned by \(e_{2m+1}\). The ideal
\[
\mathfrak h_{m,2n+t}
=\operatorname{span}\{e_1,\ldots,e_{2m+1}\mid \omega_1,\ldots,\omega_{2n},\eta_1,\ldots,\eta_t\}
\]
is a Heisenberg Lie superalgebra with even center, and \(\mathfrak h^{\lambda,\kappa}_{m,n,t}\) is a \(1\)-dimensional extension of \(\mathfrak h_{m,2n+t}\) by \(e_{2m+2}\) [2509.03016].

In this terminology, the untwisted Heisenberg Lie superalgebra \(\mathfrak h_{m,2n+t}\) has only the central brackets \([e_i,e_{m+i}]\), \([\omega_j,\omega_j]\), and \([\eta_k,\eta_k]\), whereas the twisted version adds the outer derivation \(e_{2m+2}\) acting semisimply on the even pairs \((e_i,e_{m+i})\) with eigenvalues \(\pm\lambda_i\) and on the odd pairs \((\omega_j,\omega_{n+j})\) with eigenvalues \(\pm\kappa_j\) [2509.03016]. Over \(\mathbb C\), twisted Heisenberg superalgebras were introduced precisely by adding derivations to Heisenberg superalgebras, and their fine gradings and Weyl groups were classified [1809.02166].

## 2. Restricted structures and the \([p]\)-map

A restricted Lie superalgebra is a Lie superalgebra \(L=L_{\bar 0}\oplus L_{\bar 1}\) equipped with a map
\[
[p]:L_{\bar 0}\to L_{\bar 0}
\]
such that \((L_{\bar 0},[p])\) is a restricted Lie algebra and \(L_{\bar 1}\) is a restricted module over \(L_{\bar 0}\) with respect to the adjoint action [1109.2452]. In particular, the \([p]\)-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 [2509.03016].

For the even twisted Heisenberg Lie algebra \(\mathfrak h_m^\lambda\), one has the criterion:
\[
\mathfrak h_m^\lambda \text{ is restricted iff } p>2 \text{ and }
\lambda_1^{p-1}=\cdots=\lambda_m^{p-1}.
\]
Then there is a parameter \(\mu=(\mu_1,\dots,\mu_{2m+2})\in\mathbb F^{2m+2}\) such that
\[
e_i^{[p]}=\mu_i e_{2m+1}\qquad (1\le i\le 2m+1),
\]
\[
e_{2m+2}^{[p]}=|\lambda|\,e_{2m+2}+\mu_{2m+2}e_{2m+1},
\]
where
\[
|\lambda|:=\lambda_1^{p-1}=\cdots=\lambda_m^{p-1}.
\]
For a general even element \(g=\sum_{i=1}^{2m+2}a_i e_i\in\mathfrak h_m^\lambda\),
\[
\begin{split}
g^{[p]}
&=
a_{2m+2}^{p-1}|\lambda|\sum_{i=1}^{2m}a_i e_i
+a_{2m+2}^{p}|\lambda| e_{2m+2} \\
&\quad+
\Bigg(
\sum_{i=1}^{2m+2} a_i^p\mu_i
+2^{-1} a_{2m+2}^{p-2}\sum_{i=1}^m \lambda_i^{p-2}(a_i^2-a_{m+i}^2)
\Bigg)e_{2m+1}.
\end{split}
\tag{\(\ast\)}
\]
This same formula gives the \([p]\)-map on the even part of the superalgebra \(\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}\) [2509.03016].

The full restriction criterion is:
\[
\mathfrak h^{\lambda,\kappa}_{m,n,t}\text{ is restricted iff }p>2
\]
and
\[
\lambda_1^{p-1}=\cdots=\lambda_m^{p-1}
=
\kappa_1^{p-1}=\cdots=\kappa_n^{p-1}.
\]
Writing this common value again as
\[
|\lambda|
=
\lambda_1^{p-1}=\cdots=\lambda_m^{p-1}
=
\kappa_1^{p-1}=\cdots=\kappa_n^{p-1},
\]
the resulting restricted Lie superalgebra is denoted \(\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t}\) [2509.03016].

The equality of all \((p-1)\)-st powers is forced by the compatibility condition
\[
[g^{[p]},h]=[\underbrace{g,[\cdots[g,[g}_{p},h]]\cdots]]
\]
for even \(g\) and odd \(h\), applied in particular to \(g=e_{2m+2}\) and odd generators \(\omega_j\). A recurrent misconception is that “restricted” should provide a \([p]\)-operator on odd generators; the standard super definition excludes this and instead imposes restricted-module compatibility on the odd part [1109.2452].

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

For a Lie superalgebra \(L\) with trivial coefficients, the ordinary cochains are
\[
C^0(L)=\mathbb F,\qquad C^q(L)=\wedge^q L^*\quad (q\ge 1),
\]
with differentials
\[
d^1(\psi)(g\wedge h)=\psi([g,h]),
\]
and, for \(\varphi\in C^2(L)\),
\[
\begin{split}
d^2(\varphi)(g\wedge h\wedge f)
&=\varphi([g,h]\wedge f)
-(-1)^{|f||h|}\varphi([g,f]\wedge h) \\
&\quad+(-1)^{|g|(|h|+|f|)}\varphi([h,f]\wedge g).
\end{split}
\]
Thus \(H^q(L)=\ker d^q/\operatorname{im}d^{q-1}\) for \(q=1,2\) [2509.03016].

Restricted cochains refine ordinary cochains by recording compatibility with the \([p]\)-map. For a restricted Lie superalgebra \((L,[p])\),
\[
C_*^2(L)=\{(\varphi,\omega)\mid \varphi\in C^2(L),\ \omega:L_{\bar 0}\to\mathbb F\text{ is \(\varphi\)-compatible}\},
\]
and
\[
C_*^3(L)=\{(\zeta,\eta)\mid \zeta\in C^3(L),\ \eta:L\times L_{\bar 0}\to\mathbb F\text{ is \(\zeta\)-compatible}\}.
\]
The space of Frobenius homomorphisms is
\[
\operatorname{Fr}(L_{\bar 0},\mathbb F)
=
\{f:L_{\bar 0}\to\mathbb F\mid f(ax+by)=a^pf(x)+b^pf(y)\}.
\]
There is a short exact sequence
\[
0\to \operatorname{Fr}(L_{\bar 0},\mathbb F)\xrightarrow{\iota} C_*^2(L)\xrightarrow{\pi} C^2(L)\to 0,
\]
hence
\[
\dim C_*^2(L)=\dim C^2(L)+\dim L_{\bar 0}.
\]
The restricted differentials are
\[
d_*^1(\psi)=(d^1\psi,{}^1(\psi)),\qquad {}^1(\psi)(g)=\psi(g^{[p]}),
\]
and
\[
d_*^2(\varphi,\omega)=(d^2\varphi,{}^2(\varphi,\omega)),
\]
with
\[
{}^2(\varphi,\omega)(g,h)
=
\varphi(g\wedge h^{[p]})-\varphi([g,\underbrace{h,\dots,h}_{p-1}],h).
\]
Then
\[
H_*^q(L)=\ker d_*^q/\operatorname{im} d_*^{q-1}\qquad (q=1,2)
\]
[2509.03016].

The low-degree relation between ordinary and restricted cohomology is the six-term exact sequence
\[
0\to H_*^1(L)\xrightarrow{\iota} H^1(L)\xrightarrow{D}\operatorname{Fr}(L_{\bar0},\mathbb F)
\to H_*^2(L)\xrightarrow{\pi}H^2(L)\xrightarrow{H}\operatorname{Fr}(L_{\bar0},H^1(L)),
\tag{\(\dagger\)}
\]
where
\[
D_\psi(g)=\psi(g^{[p]}),
\]
and, for \(\varphi\in Z^2(L)\),
\[
H_\varphi(g)\cdot h
=
\varphi(g\wedge (\mathrm{ad}\,g)^{p-1}(h))-\varphi(g^{[p]}\wedge h).
\]
This is the form used in the computation of \(H_*^2(\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t})\) [2509.03016]. Its general super-theoretic origin is the six-term exact sequence for restricted Lie superalgebras established in [1109.2452].

## 4. Ordinary and restricted low-degree cohomology

Let
\[
L=\mathfrak h^{\lambda,\kappa,\mu}_{m,n,t},\qquad
I=\mathfrak h_{m,2n+t}\trianglelefteq L,\qquad
L/I\cong \mathbb F\cdot e_{2m+2}.
\]
The Hochschild–Serre spectral sequence for the pair \((L,I)\) with trivial coefficients has
\[
E_2^{r,s}=H^r(L/I,H^s(I))\Longrightarrow H^{r+s}(L),
\]
and since \(L/I\) is \(1\)-dimensional even, one obtains
\[
H^k(L)\cong H^k(I)^{e_{2m+2}}
\;\oplus\;
\left(
e^{2m+2}\wedge \frac{H^{k-1}(I)}{e_{2m+2}\cdot H^{k-1}(I)}
\right).
\]
In particular, \(H^1(I)\) is spanned by the classes of
\[
\{e^1,\ldots,e^{2m}\mid \omega^1,\ldots,\omega^{2n},\eta^1,\ldots,\eta^t\},
\]
while \(e_{2m+2}\cdot H^1(I)=\operatorname{span}\{e^1,\dots,e^{2m},\omega^1,\dots,\omega^{2n}\}\). Therefore
\[
(H^1(I))^{e_{2m+2}}
=
\operatorname{span}\{\eta^1,\dots,\eta^t\},
\]
and
\[
H^1(L)
=
\operatorname{span}\{e^{2m+2}\mid \eta^1,\dots,\eta^t\}.
\]
Hence \(\dim H^1(L)=t+1\) [2509.03016].

For \(H^2(L)\), one decomposes \(H^2(I)\) into \(e_{2m+2}\)-stable pieces and extracts the invariants. The resulting spanning cocycles are organized into five sets:
\[
\begin{aligned}
A_1 &=
\{
\delta_{\lambda_i=\pm\lambda_j}(e^{i,j}-\lambda_i\lambda_j^{-1}e^{m+i,m+j}), \\
&\qquad
\delta_{\lambda_i=\pm\lambda_j}(e^{i,m+j}+\lambda_i\lambda_j^{-1}e^{j,m+i})
\mid 1\le i\le j\le m
\},\\[0.3em]
A_2 &=
\{
\delta_{\lambda_i=\pm\kappa_j}(e^{m+i}\wedge\omega^j-\lambda_i\kappa_j^{-1}e^{i}\wedge\omega^{n+j}), \\
&\qquad
\delta_{\lambda_i=\pm\kappa_j}(e^{i}\wedge\omega^j-\lambda_i\kappa_j^{-1}e^{m+i}\wedge\omega^{n+j})
\mid 1\le i\le m,\ 1\le j\le n
\},\\[0.3em]
A_3 &=
\{
\delta_{\kappa_i=\pm\kappa_j}(\omega^{i,j}-\kappa_i\kappa_j^{-1}\omega^{n+i,n+j}), \\
&\qquad
\delta_{\kappa_i=\pm\kappa_j}(\omega^{i,n+j}-\kappa_i\kappa_j^{-1}\omega^{j,n+i})
\mid 1\le i\le j\le n
\},\\[0.3em]
A_4 &=\{\eta^{i,i},\eta^{k,l}\mid 1\le i\le t-1,\ 1\le k<l\le t\},\\[0.3em]
A_5 &=\{e^{2m+2}\wedge\eta^k\mid 1\le k\le t\}.
\end{aligned}
\]
Then \(H^2(L)\) is spanned by the classes of cocycles
\[
\bigcup_{i=1}^5 A_i.
\]
Its super-dimension is
\[
\mathrm{sdim}\,H^2(L)
=
\Big(
2\#\{(i,j)\mid\lambda_i=\pm\lambda_j,\ 1\le i<j\le m\}
\]
\[
+2\#\{(i,j)\mid\kappa_i=\pm\kappa_j,\ 1\le i\le j\le n\}
+\tfrac{t(t+1)}{2}+m-1,\ 
2\#\{(i,j)\mid\lambda_i=\pm\kappa_j,\ 1\le i\le m,\ 1\le j\le n\}+t
\Big),
\]
where \(A_1,A_3,A_4\) are even and \(A_2,A_5\) are odd [2509.03016].

Restricted \(1\)-cohomology is smaller:
\[
H_*^1(L)
=
(L/([L,L]+\langle L_{\bar 0}^{[p]}\rangle))^*.
\]
Since
\[
[L,L]+\langle L_{\bar0}^{[p]}\rangle
=
\operatorname{span}\{e_1,\dots,e_{2m+2}\mid \omega_1,\dots,\omega_{2n}\},
\]
the quotient has basis represented by \(\eta_1,\dots,\eta_t\), and
\[
H_*^1(L)=\operatorname{span}\{\eta^1,\dots,\eta^t\}.
\]
Thus the ordinary class \(e^{2m+2}\) survives in \(H^1(L)\) but not in \(H_*^1(L)\) [2509.03016].

For \(H_*^2(L)\), the map \(D\) in \((\dagger)\) has image
\[
\operatorname{im}D=\mathbb F\cdot \overline e^{2m+2}\subset \operatorname{Fr}(L_{\bar0},\mathbb F),
\]
because \(D(\eta^k)=0\) while
\[
D(e^{2m+2})(e_{2m+2})
=
e^{2m+2}(e_{2m+2}^{[p]})
=
|\lambda|.
\]
The map \(H:H^2(L)\to \operatorname{Fr}(L_{\bar0},H^1(L))\) kills \(\langle A_1\cup A_2\cup A_3\cup A_4\rangle\) and is nonzero on any class with a nonzero \(A_5\)-component. Therefore
\[
\ker H=\langle A_1\cup A_2\cup A_3\cup A_4\rangle,
\]
and the six-term sequence yields
\[
0\to \operatorname{Fr}(L_{\bar0},\mathbb F)/\mathbb F\overline e^{2m+2}
\to H_*^2(L)\to \langle A_1\cup A_2\cup A_3\cup A_4\rangle\to 0.
\]
Equivalently,
\[
H_*^2(L)
=
\operatorname{span}\Big(
\{(\varphi,\widetilde\varphi)\mid \varphi\in\bigcup_{i=1}^4 A_i\}
\cup
\{(0,\overline e^i)\mid 1\le i\le 2m+1\}
\Big),
\]
with super-dimension
\[
\mathrm{sdim}\,H_*^2(L)
=
\Big(
2\#\{(i,j)\mid\lambda_i=\pm\lambda_j,\ 1\le i<j\le m\}
\]
\[
+2\#\{(i,j)\mid\kappa_i=\pm\kappa_j,\ 1\le i\le j\le n\}
+\tfrac{t(t+1)}{2}+3m,\ 
2\#\{(i,j)\mid\lambda_i=\pm\kappa_j,\ 1\le i\le m,\ 1\le j\le n\}
\Big)
\]
[2509.03016].

## 5. Restricted one-dimensional central extensions

Restricted one-dimensional central extensions by an even, strongly abelian module \(\mathbb Fc\) are classified by the even part \(H_*^2(L)_{\bar 0}\). If \((\varphi,\omega)\in C_*^2(L)_{\bar 0}\), the corresponding extension \(\mathfrak G=L\oplus \mathbb Fc\) has
\[
[g,h]_{\mathfrak G}=[g,h]+\varphi(g\wedge h)c,
\qquad
g_0^{[p]}{}_{\mathfrak G}=g_0^{[p]}+\omega(g_0)c,
\]
for \(g,h\in L\) and \(g_0\in L_{\bar 0}\). Two such extensions are equivalent iff the cocycles are cohomologous in \(H_*^2(L)\) [2509.03016].

A basis for even restricted \(2\)-cocycles consists of
\[
(\varphi,\widetilde\varphi)\ \text{with}\ \varphi\in A_1\cup A_3\cup A_4,
\qquad
(0,\overline e^i)\ \text{for}\ 1\le i\le 2m+1.
\]
The odd cocycles from \(A_2\) do not classify restricted central extensions by an even \(1\)-dimensional center [2509.03016].

The Frobenius part \((0,\overline e^i)\) yields extensions \(\mathfrak G_i\) with
\[
[g,h]_{\mathfrak G_i}=[g,h],
\qquad
g_0^{[p]}{}_{\mathfrak G_i}=g_0^{[p]}+d_i^p c.
\]
These extensions are invisible on the Lie bracket level and deform only the \(p\)-structure. The set \(\{\mathfrak G_i\mid 1\le i\le 2m+1\}\) spans the space of restricted one-dimensional central extensions that are split as Lie superalgebra extensions [2509.03016]. The same phenomenon occurs for untwisted restricted Heisenberg Lie superalgebras, where Frobenius cocycles correspond to bracket-split but restricted-nontrivial extensions [2509.00426].

The \(A_1\)-classes produce even-even bracket extensions. For instance, if \(1\le i<j\le m\) and \(\lambda_i=\pm\lambda_j\),
\[
\varphi_{i,j}=e^{i,j}-\lambda_i\lambda_j^{-1}e^{m+i,m+j},
\]
and the associated extension \(\mathfrak G_{i,j}=L\oplus \mathbb Fc\) satisfies
\[
[g,h]_{\mathfrak G_{i,j}}
=
[g,h]
+
\big(
a_i a'_j-a_j a'_i
-\lambda_i\lambda_j^{-1}a_{m+i}a'_{m+j}
+\lambda_i\lambda_j^{-1}a_{m+j}a'_{m+i}
\big)c,
\]
with
\[
\begin{split}
g_0^{[p]}{}_{\mathfrak G_{i,j}}
&=
g_0^{[p]}
-2^{-1}d_{2m+2}^{p-2}\big(
\lambda_i^{p-2}d_{m+i}d_j
-2\lambda_j^{p-2}d_{m+j}d_i \\
&\qquad\qquad\qquad\qquad
+\lambda_i\lambda_j^{p-3}d_jd_{m+i}
\big)c.
\end{split}
\]
A second \(A_1\)-type,
\[
\varphi_{i,m+j}=e^{i,m+j}+\lambda_i\lambda_j^{-1}e^{j,m+i},
\]
gives
\[
[g,h]_{\mathfrak G_{i,m+j}}
=
[g,h]
+
(a_i a'_{m+j}-a_{m+j}a'_i-\lambda_i\lambda_j^{-1}a_{m+i}a'_j+\lambda_i\lambda_j^{-1}a_j a'_{m+i})c,
\]
together with the explicit \(p\)-formula recorded in the classification theorem [2509.03016].

The \(A_3\)-classes modify odd-odd \(\omega\)-brackets. For
\[
\varphi_{i,j}^\omega=\omega^{i,j}-\kappa_i\kappa_j^{-1}\omega^{n+i,n+j},
\]
the extension \(\mathfrak H_{i,j}\) has
\[
[g,h]_{\mathfrak H_{i,j}}
=
[g,h]
-
(b_i b'_j+b_j b'_i-\kappa_i\kappa_j^{-1}b_{n+i}b'_{n+j}-\kappa_i\kappa_j^{-1}b_{n+j}b'_{n+i})c,
\]
and
\[
g_0^{[p]}{}_{\mathfrak H_{i,j}}=g_0^{[p]}.
\]
Similarly, for
\[
\varphi_{i,n+j}^\omega=\omega^{i,n+j}-\kappa_i\kappa_j^{-1}\omega^{j,n+i},
\]
one gets another odd-odd family \(\mathfrak H_{i,n+j}\) with unchanged \(p\)-map [2509.03016].

The \(A_4\)-classes modify the \(\eta\)-\(\eta\) brackets. For
\[
\varphi_{k,l}^\eta=\eta^{k,l}\quad\text{or}\quad \eta^{k,k},
\]
the extension \(\mathfrak J_{k,l}\) satisfies
\[
[g,h]_{\mathfrak J_{k,l}}
=
[g,h]-(c'_k c_l+c'_l c_k)c,
\qquad
g_0^{[p]}{}_{\mathfrak J_{k,l}}=g_0^{[p]}.
\]
These formulas provide a complete description of restricted one-dimensional central extensions inside the chosen family [2509.03016].

## 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, \(\mathfrak h_{2m,n}\) and \(\mathfrak{ba}_n\), and their ordinary cohomology was computed by Hochschild–Serre methods relative to the center [1308.6681]. Second, restricted Heisenberg Lie superalgebras and their restricted one-dimensional central extensions were analyzed over an algebraically closed field of characteristic \(p>2\); in that untwisted setting, the restricted \(2\)-cohomology likewise splits into an ordinary part and a Frobenius part [2509.00426]. Third, twisted Heisenberg superalgebras over \(\mathbb C\) were defined by adjoining derivations to Heisenberg superalgebras and were studied through fine gradings and Weyl groups [1809.02166].

Within this landscape, the “twisted” and “restricted” modifiers refer to different structures. “Twisted” refers to the semisimple derivation \(e_{2m+2}\) acting on even and odd generators; “restricted” refers to the presence of a \([p]\)-map on the even part together with restricted-module compatibility on the odd part [2509.03016]. The recent cohomological classification shows that the restriction criterion is controlled by the equalities
\[
\lambda_i^{p-1}=\kappa_j^{p-1},
\]
while the low-degree cohomology is controlled by the eigenspace coincidences
\[
\lambda_i=\pm\lambda_j,\qquad \kappa_i=\pm\kappa_j,\qquad \lambda_i=\pm\kappa_j
\]
that appear in the sets \(A_1,A_2,A_3\) and in the dimension formulas for \(H^2\) and \(H_*^2\) [2509.03016]. This suggests that the parameter sets \(\lambda\), \(\kappa\), and the common scalar \(|\lambda|\) 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
\[
0\to H_*^1\to H^1\to \operatorname{Fr}(L_{\bar0},\mathbb F)\to H_*^2\to H^2\to \operatorname{Fr}(L_{\bar0},H^1)
\]
measures exactly how ordinary cocycles interact with the \(p\)-structure [1109.2452]. In the twisted Heisenberg case, this interaction becomes completely explicit: the \(A_5\)-part of \(H^2\) is killed when passing to \(H_*^2\), while the Frobenius classes \((0,\overline e^i)\) appear only in restricted cohomology [2509.03016]. 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 \(p\)-operators.

Source: https://www.emergentmind.com/topics/restricted-twisted-heisenberg-lie-superalgebras