Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras
Here, in every simple finite-dimensional vectorial Lie superalgebra considered with the standard grading where every indeterminate is of degree 1, the maximal graded solvable subalgebras are classified over ℂ.
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
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Інститут математики НАН України
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| Цитувати: | Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras. Irina Shchepochkina. SIGMA 21 (2025), 047, 11 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860271731693846528 |
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| author | Shchepochkina, Irina |
| author_facet | Shchepochkina, Irina |
| citation_txt | Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras. Irina Shchepochkina. SIGMA 21 (2025), 047, 11 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Here, in every simple finite-dimensional vectorial Lie superalgebra considered with the standard grading where every indeterminate is of degree 1, the maximal graded solvable subalgebras are classified over ℂ.
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| first_indexed | 2026-03-21T11:42:52Z |
| format | Article |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 21 (2025), 047, 11 pages
Maximal Graded Solvable Subalgebras of Simple
Finite-Dimensional Vectorial Lie Superalgebras
Irina SHCHEPOCHKINA
Independent University of Moscow, 11 B. Vlasievsky per., 119 002 Moscow, Russia
E-mail: iparam53@gmail.com
Received March 06, 2025, in final form June 19, 2025; Published online June 24, 2025
https://doi.org/10.3842/SIGMA.2025.047
Abstract. Here, in every simple finite-dimensional vectorial Lie superalgebra considered
with the standard grading where every indeterminate is of degree 1, the maximal graded
solvable subalgebras are classified over C.
Key words: Lie superalgebra; Lie algebra; maximal subalgebra; solvable subalgebra
2020 Mathematics Subject Classification: 17B20; 17B30
1 Introduction
Hereafter, the ground field is C, all spaces are finite-dimensional. For the notation of simple Lie
superalgebras and related basics of linear superalgebra, see [8].
In [5], Dynkin solved the problem (now classical) “describe maximal subalgebras of simple
finite-dimensional Lie algebras”. In [14], the superanalog of the problem Dynkin solved for
subalgebras of any type, was solved for the maximal non-simple irreducible subalgebras in
linear (realised by means of linear operators or matrices) Lie superalgebras gl(p|q) and sl(p|q),
in their queer analogs q(n) and sq(n), in pe(n) that preserves the non-degenerate odd bilinear
form, in its supertraceless subalgebra spe(n), in a version peλ(n) of pe(n), and in the simple Lie
superalgebra osp(m|2n) that preserves the non-degenerate even bilinear form.
In this note, there are described and classified maximal graded solvable Lie subsuperalgebras
in the finite-dimensional simple vectorial (i.e., realised by means of vector fields) Lie superalge-
bras of the four analogs of the Cartan series of simple Lie algebras. These ambients are considered
with their standard grading where every indeterminate is of degree 1. The particular case of
vect(0|2) ≃ sl(1|2) was solved in [13] in terms of supermatrices. Here, the answer is the union of
five propositions which require different prerequisites in various cases and are proved differently.
2 Related results
S. Lie was the first to try to classify maximal subalgebras of simple vectorial Lie algebras
with polynomial coefficients. To classify all maximal subalgebras seems to be a wild (hopeless)
problem: there are too many such subalgebras. Several researchers distinguished various classes
of maximal subalgebras important in applications (e.g., [9, 16]) or dealt with cases feasible as, for
example, are the classifications of the maximal graded subalgebras in finite-dimensional simple
vectorial Lie superalgebras of Cartan type over C, see [2], and over algebraically closed fields of
characteristic p > 3, see [1].
In [13], the maximal, not necessarily graded, solvable Lie subsuperalgebras of gl(p|q) and
sl(p|q) are described. These maximal solvable Lie subsuperalgebras are sometimes larger and
sometimes smaller (see [13, Theorem 3]) than the super analog of the Borel subalgebra defined
mailto:iparam53@gmail.com
https://doi.org/10.3842/SIGMA.2025.047
2 I. Shchepochkina
by Penkov as any Lie subsuperalgebra spanned by the maximal torus and root vectors corre-
sponding to either all positive or all negative roots, see [10]; recall, for comparison, that the
Borel subalgebras of simple Lie algebras are maximal solvable. The case of h′(0|4) ≃ psl(2|2)
solved here is not considered in [13], where the closely related cases of gl(2|2) and sl(2|2) are
solved, yielding, however, totally distinct types of maximal non-graded solvable subalgebras, see
Proposition 6.1 and an elucidation in its proof.
Kuznetsov [7] classified maximal graded solvable subalgebras of finite codimension in simple
vectorial Lie algebras over algebraically closed fields of characteristic p > 3: a problem analogous
to the one solved here for the graded solvable Lie algebra s in the case where s−1 = 0, see
Section 4 (i).
3 The ambients
There are four series of simple finite-dimensional vectorial Lie superalgebras; let us recall their
description, see [8] containing also details of history.
Let Λ(n) or Λ(ξ) designate the Grassmann algebra generated by ξ := (ξ1, . . . , ξn) with the
standard grading (deg ξi = 1 for all i) and parity p equal to the degree modulo 2. Here, the
maximal graded solvable Lie subsuperalgebras of simple Lie superalgebras of vector fields on the
0|n-dimensional supermanifold C0|n are classified, the ambient superalgebras being considered
with their Z-grading corresponding to the standard grading of Λ(n), the algebra of functions on
C0|n. This grading, called Weisfeiler grading, is associated with the Weisfeiler filtration con-
structed by means of the maximal subalgebra consisting of the fields vanishing at the origin, see,
e.g., [3]. Note that the Lie superalgebra svect(0|n) of divergence-free vector fields has a filtered
deformation s̃vect(0|n) which can be endowed with Z-gradings, but these gradings are not Weis-
feiler gradings. In [2], a natural Z/nZ-grading of s̃vect(0|n), induced by the standard Z-grading
of svect(0|n), is considered and the maximal subalgebras of s̃vect(0|n) are described for n even.
Let vect(0|n) := derΛ(n), also denoted vect(ξ), be the Lie superalgebra of superderivations
of Λ(n), its elements can be expressed as
∑
fi∂i, where fi ∈ Λ(n) and ∂iξj := δij . The symbol
b(ξ) designates the subalgebra b ⊂ vect(ξ) realized in coordinates ξ.
Let svect(0|n) := {D ∈ vect(0|n) | divD = 0}, where the divergence is
div
(∑
fi∂i
)
:=
∑
(−1)p(fi)∂i(fi).
Let po(0|n) be the Poisson Lie superalgebra realized on the superspace Λ(n) generated for
n = 2k + l by
ξ := (ξ1, . . . , ξk), η := (η1, . . . , ηk), ζ := (ζ1, . . . , ζl), (3.1)
with the Poisson bracket
{f, g} := (−1)p(f)
∑
1≤i≤k
(
∂f
∂ξi
∂g
∂ηi
+
∂f
∂ηi
∂g
∂ξi
)
+
∑
1≤j≤l
∂f
∂ζj
∂g
∂ζj
.
Let h(0|n) := Span{Hf | f ∈ Λ(n)}, where
Hf := (−1)p(f)
∑
1≤i≤k
(
∂f
∂ξi
∂ηi +
∂f
∂ηi
∂ξi
)
+
∑
1≤j≤l
∂f
∂ζj
∂ζj
.
Clearly, h(0|n) is the quotient of po(0|n) modulo center (spanned by constants). Although in
this note we only need linear algebra and no geometry, recall, for the sake of completeness, that
h(0|n) preserves a non-degenerate super-anti-symmetric closed differential 2-form ω on a (0|n)-
dimensional supermanifold, see [8].
Maximal Graded Solvable Subalgebras 3
Hereafter, g designates one of the Lie superalgebras vect(0|n) (resp. svect(0|n) or h(0|n)) in
their standard gradings induced by the standard Z-grading of Λ(n) for n > 1 (resp. n > 2 or
n > 3) when these Lie superalgebras are simple, except h(0|n). Note that the description of
maximal graded solvable subalgebras in h(0|n) yields the description of same type of subalgebras
in the simple derived algebra h′(0|n); we describe the elementsHf of h(0|n) in terms of generating
functions f .
Let Ξ = ξ1 · · · ξn, let t be an extra indeterminate of parity n (mod 2) and deg(t) = 0. The
Lie superalgebras s̃vect(0|2n) := (1 + tΞ)svect(0|2n) are isomorphic for any t ̸= 0, so we assume
t = 1. The standard Z-grading of svect(0|n) induces a Z/n-grading of s̃vect(0|n), the degrees
being represented by integers from −1 to n−2. The Z- or Z/n-grading of the Lie superalgebra s
is said to be compatible (with parity) if s0̄ =
⊕
i≥0 s2i. Clearly,
s̃vecti(0|n) =
{
svecti(0|n) for n even,
svecti(0|n)⊗ C[t] for n odd,
for i ̸= −1,
s̃vect−1(0|n) = Span
(
∂̃1, . . . , ∂̃n
)
, where ∂̃i := (1 + tΞ)∂i.
Here, we will not consider deformations with odd parameter.
Lemma 3.1. Let s :=
⊕
i≥−1 si be a compatible with parity Z- or Z/nZ-grading of a Lie super-
algebra s. Then, s is solvable if and only if so is s0.
Proof. In [6, Proposition 1.3.3], based on the description of irreducible modules over solvable
Lie superalgebras proved later, see [11], it is proved that “the Lie superalgebra g is solvable if
and only if its even component g0̄ is solvable”. (Note that this statement, true over any field of
characteristic not 2, does not hold over fields of characteristic 2; for examples where g is simple
while g0̄ is solvable, see [3, 4].) Since the grading of s is compatible with parity, then s0̄ contains
a nilpotent ideal n :=
⊕
i>0 s2i and s0̄/n = s0. So, s is solvable if and only if so is s0. ■
4 The three possible cases
Let s be a maximal graded solvable subalgebra of g, i.e., si = s ∩ gi for all i. Let us consider
the possible cases: (i) s−1 = 0, (ii) s−1 = g−1, and (iii) 0 ̸= V = s−1 ⊊ g−1. Since each case is
determined by its (−1)st component, we accordingly designate the maximal solvable subalgebra
of g as (i) ms0g, (ii) mscg, (iii) msV g, where ms means “maximal solvable” (subalgebra), with 0,
or complete, or equal to V partial (−1)st component, respectively.
(i) s−1 = 0. Obviously, if s is maximal, then
s0 is a maximal solvable subalgebra in g0 and si = gi for i > 0.
Let (ms0g)0 be a maximal solvable subalgebra in g0; set
ms0g := (ms0g)0 ⊕
(⊕
i>0
gi
)
. (4.1)
Proposition 4.1. Lie superalgebras ms0vect(0|n) and ms0svect(0|n) for n > 2, as well as ms0h(0|n)
for n > 4, are maximal solvable in vect(0|n), svect(0|n) and h(0|n), respectively. Moreover, the
algebra isomorphic to ms0svect(0|2n) is maximal solvable in s̃vect(0|2n).
Proof. From the classical results on maximal solvable subalgebras of sl(n) and o(n), we see
that the bases of s0 and g0 are as follows (for a reason the s0 are chosen differently — sometimes
upper-triangular, sometimes lower-triangular — see clarifications in Section 5):
4 I. Shchepochkina
g g0 basis of g0 basis of s0
vect(0|n) = vect(ξ) gl(n) ξi∂j , where 1 ≤ i, j ≤ n ξi∂j for i ≥ j
svect(0|n) = svect(ξ) sl(n) ξi∂j for i ̸= j and hi := ξi∂i − ξi+1∂i+1 ξi∂j for i > j and
for i ∈ {1, . . . , n− 1} hi for i ∈ {1, . . . , n− 1}
h(0|2k) = h(ξ, η) o(2k) ξiξj and ηiηj for i > j,
ξiηj where 1 ≤ i, j ≤ k ξiξj for i > j, ξiηj for i ≤ j
h(0|2k + 1) = h(ξ, η, ζ) o(2k + 1) ξiξj and ηiηj for i > j, ξiηj where ξiξj for i > j, ξiηj for i ≤ j
1 ≤ i, j ≤ k and ξiζ, ηiζ where 1 ≤ i ≤ k and ξiζ for i ∈ {1, . . . , k}
Table 1.
Assume the contrary:
let t be a solvable graded subsuperalgebra in g containing s. (4.2)
Since s0 is a maximal solvable subalgebra in g0, then t+ :=
⊕
i≥0 ti = s by Lemma 3.1.
Hence,
[t−1, g1] ⊂ s0. (4.3)
If n > 2, then for g = vect(0|n) and svect(0|n), the component g1 contains the elements
v1 := ξ1ξ2∂3 and vi := ξ1ξi∂2 for any i > 2. Clearly, the condition
[t−1, vi] ⊂ s0 for all i (4.4)
implies that t−1 = 0, so t = s.
Let ⟨S⟩ := Span(S) designate the space spanned by the set S; let Λi(⟨S⟩) be the component
of degree i in Λ(⟨S⟩). For the subspaces ⟨ξ⟩ ⊗Λ2(⟨η⟩) and Λ3(⟨η⟩) in the case of g = h(0|2k) for
k ≥ 3 as well as for the subspaces ⟨ξ⟩ ⊗Λ2(⟨η⟩) and ⟨ζ⟩ ⊗Λ2(⟨η⟩) in the case of g = h(0|2k+ 1)
for k ≥ 2, the condition (4.3) implies t−1 = 0. ■
5 Definitions needed in cases (ii) and (iii) (see [15])
The graded Lie (super)algebra b =
⊕
k⩾−d bk is said to be transitive if for all k ≥ 0 we have
{x ∈ bk | [x, b−] = 0} = 0, where b− := ⊕k<0 bk.
The maximal transitive Z-graded Lie (super)algebra g =
⊕
k⩾−1 gk, whose non-positive part
is the given g−1⊕ g0, is called the Cartan prolong — the result of Cartan prolongation — of the
pair (g−1, g0) and is denoted by (g−1, g0)∗.
Let g := vect(0|n) or svect(0|n) or h(0|n). Let, besides, b−1 ⊂ g−1 be a non-zero subspace
in g−1, and b0 ⊂ g0 a subalgebra (not necessarily maximal) preserving this b−1. Clearly,
b≤0 := b−1 ⊕ b0 is a subalgebra in g. Define the prolong of b≤0 in g as Z-graded subalgebra
b =
⊕
k≥−1 bk, where bk := {D ∈ gk | [D, b−1] ⊂ bk−1 for all k ≥ 1}.
If b−1 = g−1, we get the Cartan prolong (g−1, b0)∗. We retain the analog of this notation
for any non-zero subspace b−1 ⊂ g−1. If b0 is a maximal subalgebra of g0 preserving b−1, let us
shrink the notation (b−1, b0)∗ to (b−1)∗.
Lemma 5.1.
1) The bracket in g = vect(0|n) for n > 2 (resp. svect(0|n) for n > 2 or h(0|n) for n > 4),
induces a Lie superalgebra structure on the space (b−1, b0)∗.
2) The Lie superalgebra (b−1, b0)∗ is maximal among all graded Lie subsuperalgebras b of g
with the given components b−1 and b0; the Lie superalgebra (b−1)∗ is maximal among all
subalgebras of g with given negative component.
Maximal Graded Solvable Subalgebras 5
Proof. Cf. [2]. Claim 1) is a direct corollary of the Jacobi identity. Claim 2) is evident. ■
We would like to select the maximal solvable subalgebra s0 of g0 so it would look as in Table 1.
Therefore, in case (iii), if s−1 = V , we’d like to numerate the basis g−1 beginning with that
of V .
If g is of series vect or svect, we assume that V is spanned by ∂1, . . . , ∂k. But then the
subalgebra of g0 preserving V contains the operators of the form ξj∂i for i ≤ k and j > k, as
commuting with V . Therefore, in these cases, it is convenient for us to select the lower -triangular
subalgebra in g0 for s0.
If g is of series h, the element ξ acts as ∂η. Therefore, if ξi for i = 1, . . . , k lie in V ∩V ⊥ (hence,
ηi for i = 1, . . . , k do not lie in V ), and there are also the ξj , ηj with i, j > k (wherever they
lie), then ξiηj with i ≤ k and j > k must lie in the subalgebra of g0 preserving V . Therefore, in
this case, it is convenient for us to select the upper -triangular subalgebra in g0 for s0.
(ii) s−1 = g−1.
Proposition 5.2.
1) Let g = vect(0|n) for n > 2 (resp. svect(0|n) for n > 2 or h(0|n) for n > 4), and let s0 be
the maximal solvable Lie subalgebra in g0, and mscg := (g−1, s0)∗. The Lie superalgebra
mscg is maximal solvable in g.
2) There are no maximal solvable Lie subalgebras s of g = s̃vect(0|2n) with s−1 = g−1.
Proof. 1) is a direct corollary of Lemma 3.1. 2) Observe that the (−1)st component of
s̃vect(0|2n) generates the whole algebra, so s̃vect(0|2n) has no maximal solvable subalgebra
containing the whole (−1)st component, see [2, Lemma 2.4]. ■
Explicit descriptions of mscg. If s0 is chosen as in Table 1, then for a basis of the
degree-(m− 1) component s
vect(0|n)
m−1 of mscvect(0|n) one can take monomials
ξi1 · · · ξim∂j , where ia ≥ j for any a ∈ {1, . . . ,m}.
Clearly,
mscsvect(0|n) = mscvect(0|n) ∩ svect(0|n).
For a basis of the (m− 1)st component msc
h(0|2k)
m−1 , we can take monomials
ξi1 · · · ξimηj , where ia ≤ j for any a ∈ {1, . . . ,m}, and ξi1 · · · ξim+1 for any ia.
For a basis of the (m−1)st component msc
h(0|2k+1)
m−1 , we can take the union of a basis of s
h(0|2k)
m−1
with the monomials
ξi1 · · · ξimζ for any ia.
Let hei(ζ) := hei(0|1) be the Heisenberg Lie superalgebra spanned by an odd ζ and even z with
the multiplication table
[z, hei(ζ)] = 0, [ζ, ζ] = z.
Let Λ(ξ) be the quotient of Λ(ξ) modulo constants. Then,
msch(0|2k) ≃ Λ(ξ) A mscvect(0|k)(ξ),
msch(0|2k+1) ≃ (Λ(ξ)⊗ hei(ζ)) A mscvect(0|k)(ξ). (5.1)
6 I. Shchepochkina
The generating functions whose degree with respect to η is ≤ 1 span a Lie subsuperalgebra g in
h(0|2k) isomorphic to Λ̄(ξ) A vect(ξ). Note that g is precisely the degree-0 component of h(ξ, η)
in the non-standard grading degns(f) := deg(f)− 1, where deg ξi = 0 while deg ηi = 1 for all i.
Note that if we take s0 as in Table 1, then s0 becomes a subalgebra in the Lie superalgebra
t0 = Span(ξiξj , ξiηj)1≤i,j≤k preserving the subspace V = Span(ξi)1≤i≤k. Let us introduce the
structure of the cotangent bundle on the space Span(ξi, ηj)1≤i,j≤k having identified ηi with ξ
∗
i .
Then, the Cartan prolong t := (g−1, t0)∗ is the Lie superalgebra preserving this structure of the
cotangent bundle. This remark clarifies the geometrical meaning of formulas (5.1).
(iii) 0 ̸= s−1 ⊊ g−1 and g = vect(0|n) or svect(0|n) or s̃vect(0|n). Then, for any
solvable subalgebra t0 ⊂ g0 preserving s−1, we can chose a maximal solvable subalgebra s0 ⊂ g0
so that t0 ⊂ s0 and s0 also preserves s−1. Therefore, up to a renumbering of indeterminates, we
can assume that for 1 < k < n we have
s−1 =
{
⟨∂1, . . . , ∂k⟩ for vect or svect,
⟨∂1, . . . , ∂k⟩ ⊗ (1 + tΞ) for s̃vect(0|n),
and s0 is as described in Table 1. Then, we have the following direct sum, each of the summands
is a subalgebra, moreover, the sum of the first and second summands is semi-direct (the second
is an ideal), the sum of the first and third summands is semi-direct (the third is an ideal),
but the sum of the second and the third summands is only the sum of subspaces (recall the
definition (4.1) of ms0g)
(s−1, s0)
vect(0|n)
∗ = mscvect(0|k)(ξ1, . . . , ξk)⊕
(⊕
i>0
Λi(ξk+1, . . . , ξn)
)
⊗ vect(ξ1, . . . , ξk)
⊕ Λ(ξ1, . . . , ξk)⊗ms0vect(ξk+1, ..., ξn),
(s−1, s0)
svect(0|n)
∗ = (s−1, s0)
vect(0|n)
∗ ∩ svect(0|n),
(s−1, s0)
s̃vect(0|2n)
∗ = (s−1, s0)
vect(0|2n)
∗ ⊗ (1 + Ξ). (5.2)
Proposition 5.3. Let g = vect(0|n) or g = svect(0|n) for n ≥ 3 or s̃vect(0|2n) for n ≥ 2. Let
s−1 ⊊ g−1 be any non-zero proper subspace and s0 the maximal solvable Lie subalgebra in g0
preserving s−1. Then, (s−1, s0)
g
∗ is maximal solvable in g.
Proof. By Lemmas 3.1 and 5.1, the Lie superalgebra s := (s−1, s0)
g
∗ is maximal among all solv-
able subalgebras in g with the given negative component. Hence, the only thing we should
check now is that s is not contained in any subalgebra t of the same type, i.e., maximal
among all solvable subalgebras with a fixed but larger negative component. Indeed, if t−1 =
⟨∂1, . . . , ∂k; ∂k+1, . . . , ∂l⟩, then v := ξ1ξk+1∂2 belongs to s, but not to t, unless s = svect(0|n) and
dim s−1 = 1. In the latter case, v := ξ1ξ2∂2 − ξ1ξ3∂3 belongs to s, but not to t. ■
Let us elucidate formulas (5.2). Let V ⊂ g−1. By Lemma 5.1, if s−1 = V , then s ⊂ V g
∗ .
Clearly, the subalgebra V g
∗ consists of all the vector fields in g that preserve V , whereas s is
a maximal solvable subalgebra in V g
∗ . In the case where g = vect(0|n), the form of elements
of V g
∗ is absolutely clear. Let V := Span(∂1, . . . , ∂k), and vect≥0 :=
⊕
i≥0 vecti. Then, we have
the direct sum of subspaces (compare with formulas (5.2))
V
vect(0|n)
∗ = Λ(ξk+1, . . . , ξn)⊗ vect(ξ1, . . . , ξk)⊕ Λ(ξ1, . . . , ξk)⊗ vect≥0(ξk+1, . . . , ξn).
(iii) 0 ̸= V = s−1 ⊊ g−1 where g = h(0|n) for n > 4. As we have observed in Lem-
ma 5.1, in this case, the subalgebra s is contained in the maximal subalgebra V∗ ⊂ g with the
given negative component V and s is a maximal solvable subalgebra in V∗. In particular, s0 is
a maximal solvable subalgebra in the stabiliser St(V ) ⊂ g0 of V .
Maximal Graded Solvable Subalgebras 7
Let B be the non-degenerate symmetric and g0 = o(n)-invariant bilinear form on g−1 corre-
sponding to the symplectic form ω. Since St(V ), and therefore its subalgebra s0, preserve V ,
they also preserve the subspaces V ⊥, V ∩ V ⊥ = KerB|V and V + V ⊥. This means that, in gen-
eral, the subalgebra s0, being a maximal solvable subalgebra in St(V ), might be not maximal
solvable subalgebra in the whole component g0.
Introduce the following basis in g−1:
ξ = (ξ1, . . . , ξk) a basis in V ∩ V ⊥ = KerB|V ,
η = (η1, . . . , ηk) the dual to ξ basis in the complement to V + V ⊥ in g−1,
α =
(
ξa1 , . . . , ξ
a
l , η
a
1 , . . . , η
a
l , ζ
a
)
a basis in the complement to V ∩ V ⊥ in V ,
β =
(
ξb1, . . . , ξ
b
m, η
b
1, . . . , η
b
m, ζ
b
)
a basis in the complement to V ∩ V ⊥ in V ⊥. (5.3)
We will order the basis elements so: ξ, η, followed by α, and then β. Observe that all solvable
subalgebras of g0 are contained in the maximal solvable subalgebra from Table 1.
In certain cases, not all these basis elements are present. For example, if V ⊂ V ⊥, i.e., the
restriction of the form B to the subspace V is zero, there are no elements α. The other way
round, if V ⊥ ⊂ V , then there are no elements β. Finally, if V ⊥ = V , then the subspace V
is Lagrangian and only elements ξ, η remain. If V ∩ V ⊥ = 0, i.e., the restriction of B to V is
non-degenerate, then there are no elements ξ, η. Besides, if the codimension of V ∩ V ⊥ in V
(resp. in V ⊥) is even, then the element ζa (resp. ζb) is absent. Nevertheless, in order not to
overburden the text by considering various cases we will use all the basis elements assuming,
when needed, that some of them are absent, i.e., are equal to 0.
We say that the subspace V ⊂ g−1 is singular if dimKerB|V ≤ 1 and codimKerB|V = 1
in V ⊥. In coordinates, this means that of all basis elements of type β there is only one of ζb
and either no elements of type ξ, η at all, or there is just one of type ξ and one of type η.
Proposition 5.4. Let g = h(0|n) for n > 4. If the subspace V ⊂ g−1 is non-singular, then the
subalgebra msV h(0|n) is maximal solvable in h(0|n).
Proof. First, let us describe the Lie superalgebra V∗. First of all, note that among the elements
of the subspace V acting on functions generating h(0|n), there are derivations with respect to
all indeterminates of types η and α, but there are no derivations with respect to indeterminates
of types ξ and β. Therefore, the Lie superalgebra V∗ contains a subspace W :
W = Span(f ∈ h(0|n) | f is a monomial and degξ,β f ≥ 2).
Moreover, the subspaceW is an ideal in V∗ and is contained in (V∗)≥0. The idealW is the direct
sum of the three subspaces W =W 2,0 ⊕W 1,1 ⊕W 0,2, where
W 2,0 = Span(f ∈W | f is a monomial and degξ f ≥ 2, degβ f = 0),
W 1,1 = Span(f ∈W | f is a monomial and degξ f ≥ 1, degβ f ≥ 1),
W 0,2 = Span(f ∈W | f is a monomial and degξ f = 0, degβ f ≥ 2).
Observe that W 2,0, W 0,2 and W 2,0 +W 1,1 are subalgebras, but not ideals, in W .
The subalgebra W 0,2 is isomorphic to the tensor product h(β)≥0 ⊗ po(α)⊗Λ(η), understood
as the product of Poisson superalgebras, see [17], i.e., on each of the factors there are two
operations: a supercommutative and associative multiplication and the Poisson bracket (the
zero one on the third factor) and these two multiplications are related by means of the Leibniz
rule, so the Poisson bracket on the product is given by the formula
[f1(β)⊗ g1(α)⊗ h1(η), f2(β)⊗ g2(α)⊗ h2(η)]
= ±[f1(β), f2(β)]⊗ g1(α)g2(α)⊗ h1(η)h2(η)± f1(β)f2(β)⊗ [g1(α), g2(α)]⊗ h1(η)h2(η),
where the signs ± are governed by the sign rule.
8 I. Shchepochkina
Therefore, the solvable subalgebra msV h(0|n) contains the product ms0h(β) ⊗ po(α) ⊗ Λ(η)
coming from the subalgebra W 0,2.
The complement to the ideal W in V∗ is the direct sum of two subspaces W 1,0⊕W 0,0, where
W 1,0 = Span(f ∈W | f is a monomial and degξ f ≥ 1, degβ f = 0)
= Span(ψ(α)
∑
i
ξiφi(η)),
W 0,0 = Span(f ∈W | f is a monomial and degξ f = 0 = degβ f) = Span(f(α)) ≃ h(α).
Observe that Span(
∑
i ξiφi(η)) ≃ vect(η). The subspaceW 1,0 is not a subalgebra. A subalge-
bra containingW 1,0 is a semi-direct sumW 1,0 B W 2,0, and (W 1,0 B W 2,0)/W 2,0 ≃ vect(η)⊗Λ(α).
Therefore, we can represent W 1,0 B W 2,0 as vect(η)⊗ po(α) B W 2,0, so its solvable part is
mscvect(η) ⊗ po(α) B W 2,0.
Finally, each of the subspacesW 2,0,W 1,1,W 0,2 andW 1,0 B W 2,0 is invariant underW 0,0 ≃ h(α).
Thus,
V∗ = h(α) A (vect(η)⊗ po(α) +W 2,0 +W 1,1 + h(β)≥0 ⊗ po(α)⊗ Λ(η)),
and the maximal solvable subalgebra s = msV h(0|n) with the negative component s−1 = V is of
the form
s = msV h(0|n) = msch(α) A (mscvect(η) ⊗ po(α) +W 2,0 +W 1,1
+ms0h(β) ⊗ po(α)⊗ Λ(η)). (5.4)
This s is the maximal solvable subalgebra among all solvable subalgebras with a given negative
part. To see that s is maximal among all solvable subalgebras, it suffices to verify that if
t = (t−1, t0)∗ and t−1 ⊃ s−1 whereas t0 ⊃ s0 is solvable, then s is not contained in t.
First, note that the subspace t−1 is necessarily invariant with respect to t0, and hence with
respect to s0. Therefore, we are left with the three cases:
(i) if m ̸= 0, then the space t−1 must contain ξb1;
(ii) if m = 0, but V ⊥ is not contained in V , then the space t−1 must contain ζb;
(iii) if V ⊥ ⊂ V , then the space t−1 must contain ηk.
In each of these cases, we indicate an element u ∈ s1 which is contained in the subalgebra s,
but can not be contained in t. In the majority of these cases, u ∈ W . Each case contains also
subcases occasioned by a low dimension of the subspaces involved; some of these subcases are
exceptional because the subspace V is singular.
In case (i) for k ≥ 1, we can take u = ξb1η
b
1η1 ∈ W 0,2. Then, [ξb1, u] = −ξb1η1 /∈ t0, because
the basis elements of type β go after ξ, η. If k = 0, i.e., the restriction of the form B to V is
non-degenerate, we can take (note that for l = 0 we have m ≥ 2, see (5.3), since n > 4)
u =
{
ξb1η
b
1η
a
1 ∈W 0,2 if l ≥ 1,
ηb1η
b
2ζ
a
1 ∈W 0,2 if l = 0,
Z=⇒ [ξb1, u] =
{
−ξb1ηa1 /∈ t0,
−ηb2ζa1 /∈ t0,
because the elements of type β follow the elements of type α.
In case (ii), of all elements of type β we have just one ζb. For k ≥ 2, we take u = ξ2η1ζ
b ∈W 1,1.
Then, [ζb, u] = ξ2η1 /∈ t0.
The subcases k = 1 and k = 0 are exceptional because the subspace V is singular.
Maximal Graded Solvable Subalgebras 9
For k = 1, the dimensions of the spaces spanned by each of the elements of type ξ and β are
equal to 1. Hence,W 2,0 =W 0,2 = 0 andW =W 1,1. The subspaceW 1,0 is a subalgebra, the sum
n =W 1,0+W 1,1 is a solvable ideal in the subalgebra V∗; this ideal consists of functions “divisible
by ξ1”. Accordingly, the maximal solvable subalgebra contained in V∗ is msV h(0|n) = msch(α) B n.
Note that this subalgebra is invariant under bracketing with ζb, so msV h(0|n) ⊂ msṼ h(0|n), where
Ṽ = V ⊕ ⟨ζb⟩.
For k = 0, there are no elements ξ, η, the codimension of V in g−1 is equal to 1, and
the restriction of the form B to V is non-degenerate. Therefore, the ideal in formula (5.4)
vanishes, and hence msV h(0|n) = msch(α). Since h(α) commutes with ζb, we see that in this case,
msV h(0|n) ⊂ msch(0|n).
In case (iii), there are no elements of type β and, if the elements of type α are present, i.e.,
when V ⊥ ̸= V , we can take
u =
{
ξkηkη
a
1 for l ≥ 1,
ξkηkζ
a for l = 0,
Z=⇒ [ηk, u] =
{
ηkη
a
1 /∈ t0,
ηkζ
a /∈ t0.
If V ⊥ = V , there remain only elements ξ, η; moreover, k ≥ 3 because n > 4. In this case, we
can take u = ξkξ2η1. Then, [ηk, u] = ξ2η1 /∈ t0. ■
6 The two remaining cases of small dimension:
vect(0|2) and h′(0|4)
For completeness, let us consider also the cases of vect(0|2) ≃ sl(1|2) and h′(0|4) ≃ psl(2|2)
whose maximal solvable subalgebras were described in their supermatrix realisation in [13].
Proposition 6.1. The maximal graded solvable subsuperalgebras in g = vect(0|2) are listed in
Table 2.
name of s basis of s−1 basis of s0 basis of s1
msV ∂1 ξ1∂1, ξ2∂1, ξ2∂2 ξ1ξ2∂1, ξ1ξ2∂2
msc ∂1, ∂2 ξ1∂1, ξ2∂1, ξ2∂2 ξ1ξ2∂1
Table 2.
The maximal graded solvable subsuperalgebras in g = h′(0|4) are listed in Table 3.
name of s basis of s−1 basis of s0 basis of s1
msV ξ1 ξ1η1, ξ1ξ2, ξ1η2, ξ2η2 ξ1η1η2, ξ1ξ2η1, ξ1ξ2η2, ξ2η1η2
msc ξ1, ξ2, η1, η2 ξ1η1, ξ1ξ2, ξ1η2, ξ2η2 ξ1ξ2η2
msṼ ξ1, ξ2, η2 ξ1η1, ξ1ξ2, ξ1η2, ξ2η2 ξ1η1η2, ξ1ξ2η1, ξ1ξ2η2
Table 3.
Proof. If g = vect(0|2), then the subalgebra ms0vect(0|2) is invariant under the action of ∂1, and
therefore is contained in msV , where V = ⟨∂1⟩, and is not maximal. Thus, we get precisely
two Z-graded maximal subalgebras, see Table 2. Obviously, although these subalgebras are
not isomorphic as Z-graded subalgebra, they are isomorphic as abstract subalgebras. This
corresponds with the description of maximal graded solvable subsuperalgebras of sl(1|2) obtained
in [13].
10 I. Shchepochkina
Now, consider g = h′(0|4). As stated in [13], both the maximal solvable subalgebra s of sl(2|2),
and of its quotient psl(2|2) modulo center, are described by the collection of superdimensions
of irreducible quotients under the tautological action of s in the 2|2-dimensional superspace.
For the 2|2-dimensional superspace, there are only three such collections: {(1|0), (1|1), (0|1)},
{(1|1), (1|1)} and {(2|2)}. Note that the last collection corresponds to a 1-parameter fam-
ily of subalgebras conjugate in gl(2|2) (resp. in pgl(2|2)), but not in sl(2|2) (resp. not in
psl(2|2)).
Now, let us look how the constructions, used in this note in other cases, work in the descrip-
tion of maximal graded solvable Lie superalgebras in h′(0|4). As in the case of vect(0|2), the
subalgebra ms0 is invariant under the action of ξ1, and therefore is contained in msV , where
V = ⟨ξ1⟩, and hence is not maximal. This subalgebra, as well as the isomorphic to it as ab-
stract, but not as Z-graded, subalgebra msc corresponds to the collection of superdimensions
{(1|0), (1|1), (0|1)}. To the subalgebra msṼ , where Ṽ = Span(ξ1, ξ
a, ηa), there corresponds the
collection {(1|1), (1|1)}. As a result, we get three maximal Z-graded solvable subalgebras, two
of which are isomorphic as abstract ones, see Table 3.
name of s basis of s−1 basis of s0 basis of s1
msV ξ1 ξ1η1, ξ1ξ2, ξ1η2, ξ2η2 ξ1η1η2, ξ1ξ2η1, ξ1ξ2η2, ξ2η1η2
msc ξ1, ξ2, η1, η2 ξ1η1, ξ1ξ2, ξ1η2, ξ2η2 ξ1ξ2η2
msṼ ξ1, ξ
a, ηa ξ1η1, ξ1ξ
a, ξ1η
a, ξaηa ξ1η1η
a, ξ1η1ξ
a, ξ1ξ
aηa
Table 4.
Note that the 1-parameter family of maximal solvable subalgebras in sl(2|2) (and in psl(2|2))
corresponding to the collection {(2|2)} consists of non-graded subalgebras, and therefore this
family is absent in this note devoted to the classification of graded subalgebras.
Observe also that five subalgebras of the form msV are not maximal. If dimV = 3, and
the restriction of the form B to the subspace V is non-degenerate, and also if dimV = 2 and
dimKerB|V = 1, then V is singular. The other three cases are occasioned by very small dimen-
sion of all subspaces involved.
name of s basis of s−1 basis of s0 basis of s1
msV ζ ξbηb, ξbζb ξbηbζb, ξbζbζ, ξbηbζ
msṼ ζ, ξb, ζb ξbηb, ξbζb, ξbζ, ζζb ξbηbζb, ξbζbζ, ξbηbζ
msV ξ1, ξ2 ξ1η1, ξ1ξ2, ξ1η2, ξ2η2, ξ1η1η2, ξ1ξ2η1, ξ1ξ2η2
msṼ ξ1, ξ2, η2 ξ1η1, ξ1ξ2, ξ1η2, ξ2η2 ξ1η1η2, ξ1ξ2η1, ξ1ξ2η2
msV ξa, ηa ξaηa, ξbηb ξbηbξa, ξbηbηa
msṼ ξa, ηa, ξb ξaηa, ξbηb, ξaξb, ξbηa ξbηbξa, ξbηbηa, ξbξaηa
Table 5.
Table 5 describes these three subalgebras msV and their ambients msṼ . ■
Acknowledgements
I am thankful to A.L. Onishchik who raised the problem, and D. Leites for help. A part of the
results of this note was announced in [12]; their proofs were sketched in Report no. 32/1988-15
of Department of Mathematics of Stockholm University to which I am thankful for hospitality.
Maximal Graded Solvable Subalgebras 11
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1 Introduction
2 Related results
3 The ambients
4 The three possible cases
5 Definitions needed in cases (ii) and (iii) (see Shch)
6 The two remaining cases of small dimension: vect(0|2) and h'(0|4)
References
|
| id | nasplib_isofts_kiev_ua-123456789-213529 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-21T11:42:52Z |
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| spelling | Shchepochkina, Irina 2026-02-18T11:26:00Z 2025 Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras. Irina Shchepochkina. SIGMA 21 (2025), 047, 11 pages 1815-0659 2020 Mathematics Subject Classification: 17B20; 17B30 arXiv:2503.03339 https://nasplib.isofts.kiev.ua/handle/123456789/213529 https://doi.org/10.3842/SIGMA.2025.047 Here, in every simple finite-dimensional vectorial Lie superalgebra considered with the standard grading where every indeterminate is of degree 1, the maximal graded solvable subalgebras are classified over ℂ. I am thankful to A.L. Onishchik, who raised the problem, and D. Leites for their help. A part of the results of this note was announced in [12]; their proofs were sketched in Report no. 32/1988-15 of the Department of Mathematics of Stockholm University, to which I am thankful for hospitality. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras Article published earlier |
| spellingShingle | Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras Shchepochkina, Irina |
| title | Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras |
| title_full | Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras |
| title_fullStr | Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras |
| title_full_unstemmed | Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras |
| title_short | Maximal Graded Solvable Subalgebras of Simple Finite-Dimensional Vectorial Lie Superalgebras |
| title_sort | maximal graded solvable subalgebras of simple finite-dimensional vectorial lie superalgebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213529 |
| work_keys_str_mv | AT shchepochkinairina maximalgradedsolvablesubalgebrasofsimplefinitedimensionalvectorialliesuperalgebras |