New simple modular Lie superalgebras as generalized prolongs
NEW SIMPLE MODULAR LIE SUPERALGEBRAS AS GENERALIZED
PROLONGS
SOFIANE BOUARROUDJ1, PAVEL GROZMAN2, DIMITRY LEITES3
Abstract. Over algebraically closed fields of characteristic p > 2, prolongations of the
simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied
for certain simplest gradings of these algebras. Several new simple Lie superalgebras are
discovered, serial and exceptional, including superBrown and superMelikyan superalgebras.
Simple Lie superalgebras with Cartan matrix of rank 2 are classified.
1. Introduction
1.1. Setting. We use standard notations of [FH, S]; for the precise definition (algorithm) of
generalized Cartan-Tanaka–Shchepochkina (CTS) complete and partial prolongations, and
algorithms of their construction, see [Shch]. Hereafter K is an algebraically closed field
of characteristic p > 2, unless specified. Let g′ = [g, g], and c(g) = g ⊕ center, where
dim center = 1. Let n)g denote the incarnation of the Lie (super)algebra g with the n)th
Cartan matrix, cf. [GL4, BGL1]. On classification of simple vectorial Lie superalgebras
with polynomial coefficients (in what follows referred to as vectorial Lie superalgebras of
polynomial vector fields over C, see [LSh, K3]).
The works of S. Lie, Killing and È. Cartan, now classical, completed classification over C
(1) simple Lie algebras of finite dimension and of polynomial vector fields.
Lie algebras and Lie superalgebras over fields in characteristic p > 0, a.k.a. modular Lie
(super)algebras, were first recognized and defined in topology, in the 1930s. The simple
Lie algebras drew attention (over finite fields K) as a step towards classification of simple
finite groups, cf. [St]. Lie superalgebras, even simple ones and even over C or R, did not
draw much attention of mathematicians until their (outstanding) usefulness was observed by
physicists in the 1970s. Meanwhile mathematicians kept discovering new and new examples
of simple modular Lie algebras until Kostrikin and Shafarevich ([KS]) formulated a conjecture
embracing all previously found examples for p > 7. Its generalization reads: select a Z-form
gZ of every g of type
1) (1), take gK := gZ⊗Z K and its simple finite dimensional subquotient
si(gK) (there can be several such si(gK)). Together with deformations
2) of these examples
we get in this way all simple finite dimensional Lie algebras over algebraically closed fields
if p > 5. If p = 5, we should add to the above list Melikyan’s examples.
1991 Mathematics Subject Classification. 17B50, 70F25.
Key words and phrases. Cartan prolongation, nonholonomic manifold, Lie superalgebra.
We are thankful to I. Shchepochkina for help; DL is thankful to MPIMiS, Leipzig, for financial support
and most creative environment.
1)Observe that the algebra of divided powers (the analog of the polynomial algebra for p > 0) and hence
all prolongs (Lie algebras of vector fields) acquire one more — shearing — parameter: N , see [S].
2)It is not clear, actually, if the conventional notion of deformation can always be applied if p > 0 (for
the arguments, see [LL]; cf. [Vi]); to give the correct (better say, universal) notion is an open problem, but
in some cases it is applicable, see [BGL4].
http://arxiv.org/abs/0704.0130v1
2 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites
Having built upon ca 30 years of work of several teams of researchers, and having added
new ideas and lots of effort, Block, Wilson, Premet and Strade proved the generalized KSh
conjecture for p > 3, see [S]. For p ≤ 5, the above KSh-procedure does not produce all
simple finite dimensional Lie algebras; there are other examples. In [GL4], we returned to
É. Cartan’s description of Z-graded Lie algebras as CTS prolongs, i.e., as subalgebras of
vectorial Lie algebras preserving certain distributions; we thus interpreted the “mysterious”
at that moment exceptional examples of simple Lie algebras for p = 3 (the Brown, Frank,
Ermolaev and Skryabin algebras), further elucidated Kuznetsov’s interpretation [Ku1] of
Melikyan’s algebras (as prolongs of the nonpositive part of the Lie algebra g(2) in one of
its Z-gradings) and discovered three new series of simple Lie algebras. In [BjL], the same
approach yielded bj, a simple super versions of g(2), and Bj(1;N |7), a simple p = 3 super
Melikyan algebra. Both bj and Bj(1;N |7) are indigenous to p = 3, the case where g(2) is
not simple.
1.2. Classification: Conjectures and results. Recently, Elduque considered super analogs
of the exceptional simple Lie algebras; his method leads to a discovery of 10 new simple
(presumably, exceptional) Lie superalgebras for p = 3. For a description of the Elduque
superalgebras, see [CE, El1, CE2, El2]; for their description in terms of Cartan matrices and
analogs of Chevalley relations and notations we use in what follows, see [BGL1, BGL2].
In [L], a super analog of the KSh conjecture embracing all types of simple (finite dimen-
sional) Lie superalgebras is formulated based on an entirely different idea in which the CTS
prolongs play the main role:
F o r e v e r y s i m p l e f i n i t e d i m e n s i o n a l L i e ( s u p e r ) a l g e b r a o f t h e
f o r m g(A) , t a k e i t s n o n - p o s i t i v e p a r t w i t h r e s p e c t t o a c e r t a i n
s i m p l e s t Z - g r a d i n g , c o n s i d e r i t s c o m p l e t e a n d p a r t i a l p r o l o n g s
a n d t a k e t h e i r s i m p l e s u b q u o t i e n t s .
The new examples of simple modular Lie superalgebras (BRJ, Bj(3;N |3), Bj(3;N |5))
support this conjecture. (This is how Cartan got all simple Z-graded Lie algebras of poly-
nomial growth and finite depth — the Lie algebras of type (1) — at the time when the root
technique was not discovered yet.)
1.2.1. Yamaguchi’s theorem ([Y]). This theorem, reproduced in [GL4, BjL], states that
for almost all simple finite dimensional Lie algebras g over C and their Z-gradings g = ⊕
of finite depth d, the CTS prolong of g≤ = ⊕
−d≤i≤0
gi is isomorphic to g, the rare exceptions
being two of the four series of simple vectorial algebras (the other two series being partial
prolongs).
1.2.2. Conjecture. In the following theorems, we present the results of SuperLie-assisted
([Gr]) computations of the CTS-prolongs of the non-positive parts of the simple finite di-
mensional Lie algebras and Lie superalgebras g(A); we have only considered Z-grading cor-
responding to each (or, for larger ranks, even certain selected) of the simplest gradings
r = (r1, . . . , rrkg), where all but one coordinates of r are equal to 0 and only one — selected
— is equal to 1, and where we set degX±i = ±ri for the Chevalley generators X±i of g(A),
see [BGL1].
O t h e r g r a d i n g s ( a s w e l l a s a l g e b r a s g(A) o f h i g h e r r a n k s ) d o
n o t y i e l d n e w s i m p l e L i e ( s u p e r ) a l g e b r a s a s p r o l o n g s o f t h e n o n -
p o s i t i v e p a r t s.
New simple modular Lie superalgebras 3
1.3. Theorem. The CTS prolong of the nonpositive part of g returns g in the following
cases: p = 3 and g = f(4), e(6), e(7) and e(8) considered with the Z-grading with one selected
root corresponding to the endpoint of the Dynkin diagram.
1.3.1. Conjecture. [The computer got stuck here, after weeks of computations] To the
cases of Theorem 1.3, one can add the case for p = 5 and g = el(5) (see [BGL2]) in its
Z-grading with only one odd simple root and with one selected root corresponding to any
endpoint of the Dynkin diagram.
1.4. Theorem. Let p = 3. For the previously known (we found more, see Theorems 1.6,
1.7) simple finite dimensional Lie superalgebras g of rank ≤ 3 with Cartan matrix and for
their simplest gradings r, the CTS prolongs (of the non-positive part of g) different from g
are given in the following table elucidated below.
1.5. Melikyan superalgebras for p = 3. There are known the two constructions of the
Melikyan algebra Me(5;N) = ⊕
Me(5;N)i, defined for p = 5:
1) as the CTS prolong of the triple Me0 = cvect(1; 1), Me−1 = O(1; 1)/const and the
trivial module Me−2, see [S]; this construction would be a counterexample to our conjecture
were there no alternative:
2) as the complete CTS prolong of the non-negative part of g(2) in its grading r = (01),
with g(2) obtained now as a partial prolong, see [Ku1, GL4].
In [BjL], we have singled out Bj(1;N |7) as a p = 3 simple analog of Me(5;N) as a partial
CTS prolongs of the pair (the negative part of k(1;N |7), Bj(1;N |7)0 = pgl(3)), and bj as a
p = 3 simple analog of g(2) whose non-positive part is the same as that of Bj(1;N |7), i.e.,
bj and Bj(1;N |7) are analogs of the construction 2).
The original Melikyan’s construction 1) also has its super analog for p = 3 (only in the
situation described in Theorem 1.6) and it yields a new series of simple Lie superalgebras as
the complete prolongs, with another simple analog of g(2) as a partial prolong.
Recall ([BGL1]) that we normalize the Cartan matrix A so that Aii = 1 or 0 if the ith
root is odd, whereas if the ith root is even, we set Aii = 2 or 0 in which case we write 0̄
instead of 0 in order not to confuse with the case of odd roots.
1.6. Theorem. A p = 3 analog of the construction 1) of the Melikyan algebra is given by
setting g0 = ck(1; 1|1), g−1 = O(1; 1|1)/const and g−2 being the trivial module. It yields a
simple super Melikyan algebra that we denote by Me(3;N |3), non-isomorphic to a superMe-
likyan algebra Bj(1;N |7).
The partial prolong of the non-positive part of Me(3;N |3) is a new (exceptional) simple
Lie superalgebra that we denote by brj(2; 3). This brj(2; 3) has the three Cartan matrices:
and 2)
−1 0̄
joined by an odd reflection, and
−1 0̄
. It is a super
analog of the Brown algebra br(2) = brj(2; 3)0̄, its even part.
The CTS prolongs for the simplest gradings r of 1)brj(2; 3) returns known simple Lie
superalgebras, whereas the CTS prolong for a simplest grading r of 2)brj(2; 3) returns, as a
partial prolong, a new simple Lie superalgebra we denote BRJ.
Unlike br(2), the Lie superalgebra brj(2; 3) has analogs for p 6= 3, e.g., for p = 5, we
get a new simple Lie superalgebra brj(2; 5) such that brj(2; 5)0̄ = sp(4) with the two Cartan
matrices 1)
and 2)
. The CTS prolongs of brj(2; 5) for all its Cartan
matrices and the simplest r return brj(2; 5).
4 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites
Having got this far, it was impossible not to try to get classification of simple g(A)’s. Here
is its beginning part, see [BGL5].
1.7. Theorem. If p > 5, every finite dimensional simple Lie superalgebra with a 2 × 2
Cartan matrix is isomorphic to osp(1|4), osp(3|2), or sl(1|2). If p = 5, we should add
brj(2; 5). If p = 3, we should add brj(2; 3).
Remark. For details of description of the new simple Lie superalgebras of types Bj and
Me and their subalgebras, in particular, presentations of brj(2; 3) and brj(2; 5), and proof of
Theorem 1.7 and its generalization for higher ranks, see [BGL4, BGL5].
The new simple Lie superalgebras obtained are described in the next subsections.
g Cartan matrix r prolong
osp(3|2)
k(1|3)
k(1|3; 1)
osp(3|2)
k(1|3; 1)
sl(1|2)
vect(0|2)
vect(1|1)
vect(1|1)
osp(1|4)
k(3|1)
osp(1|4)
brj(2; 3)
0̄ −1
Me(3;N |3)
Brj(4|3)
Brj(4;N |3)
Brj(3;N |4) ⊃ BRJ
brj(2; 3)
0̄ −1
Brj(3;N |3)
Brj(3;N |4) ⊃ BRJ
brj(2; 5)
brj(2; 5)
brj(2; 5)
brj(2; 5)
brj(2; 5)
New simple modular Lie superalgebras 5
g Cartan matrix r prolong
sl(1|3)
0 −1 0
−1 2 −1
0 −1 2
(100)
(010)
(001)
vect(0|3)
sl(1|3)
vect(2|1)
0 −1 0
−1 0 −2
0 −1 2
(100)
(010)
(001)
vect(2|1)
sl(1|3)
vect(2|1)
psl(2|2) any matrix
(100)
(010)
(001)
svect(1|2)
psl(2|2)
svect(1|2)
osp(1|6)
2 −1 0
−1 2 −1
0 −1 1
(100)
(010)
(001)
k(5|1)
osp(1|6)
osp(1|6)
osp(3|4)
2 −1 0
−1 0 −1
0 −2 2
0 −1 0
−1 0 1
0 −1 1
(100)
(010)
(001)
k(3|3)
osp(3|4)
osp(3|4)
0 −1 0
−1 2 −1
0 −1 1
(100)
(010)
(001)
osp(3|4)
osp(5|2)
2 −1 0
−1 0 1
0 −1 1
0 −1 0
−1 0 1
0 −2 2
(100)
(010)
(001)
osp(5|2)
0 −1 0
−1 2 −1
0 −2 2
(100)
(010)
(001)
osp(5|2)
k(1|5)
osp(4|2;α)
α generic
2 −1 0
α 0 −1− α
0 −1 2
0 1 −1− α
−1 0 −α
−1− α α 0
(100)
(010)
(001)
osp(4|2;α)
osp(4|2;α)
α = 0,−1
1) The simple part of 1)osp(4|2;α) is sl(2|2);
for the CTS of psl(2|2), see above
2) 2)osp(4|2;α) ≃ sl(2|2);
for the CTS of sl(2|2), see above
6 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites
g Cartan matrix r prolong
osp(2|4)
0 1 0
−1 2 −2
0 −1 2
0 1 0
−1 0 2
0 −1 2
0 −2 1
−2 0 1
−1 −1 2
(100)
(010)
(001)
(100)
(010)
(001)
(100)
(010)
(001)
osp(2|4)
osp(2|4) if p > 3
Bj(3;N |3) if p = 3
osp(2|4)
k(3|2)
osp(2|4) if p > 3
Bj(3;N |3) if p = 3
osp(2|4)
osp(2|4)
osp(2|4)
k(3|2)
g(2|3) 3)
0 0 −1
0 0 −2
−1 −2 2
(100)
(010)
(001)
Bj(2|4)
Bj(3|5)
1.8. A description of Bj(3;N |3). For g = 1)osp(2|4) and r = (0, 1, 0), we have the
following realization of the non-positive part:
gi the generators (even | odd)
g−2 Y6 = ∂1 | Y8 = ∂4
g−1 Y2 = ∂2, Y5 = x2∂1 + x5∂4 + ∂3, | Y4 = ∂5, Y7 = 2x2∂4 + ∂6,
g0 ≃ Y3 = x22∂1 + x2x5∂4 + x2∂3 + 2x5∂6, Z3 = x32∂1 + 2x3x6∂4 + x3∂2 + 2x6∂5
sl(1|1) ⊕ sl(2) ⊕K H2 = 2x1∂1 + 2x2∂2 + x4∂4 + x5∂5 + 2x6∂6, H1 = [Z1, Y1],H3 = [Z3, Y3] |
Y1 = x1∂4 + 2x2∂5 + x3∂6, Z1 = 2x4∂1 + 2x5∂2 + x6∂3
The g0-module g−1 is irreducible, having one highest weight vector Y2.
Let p = 3. The CTS prolong gives sdim(g1) = 4|4. The g0-module g1 has the following
two lowest weight vectors:
V ′1 x1x2∂4 + 2x1∂6 + 2x2
2∂5 + x2x3∂6
V ′′1 x1x2∂1 + x1x5∂4 + 2x2x4∂4 + x1∂3 + x2
2∂2 + 2x2x5∂5 + x2x6∂6 + x3x5∂6 + x4∂6
Since g1 generates the positive part of the CTS prolong, [g−1, g1] = g0, and [g−1, g−1] = g−2,
the standard criteria of simplicity ensures that the CTS prolong is simple. Since none of
the Z-graded Lie superalgebras over C of polynomial growth and finite depth has grading
of this form (with g0 ≃ sl(1|1)⊕ sl(2)⊕ K), we conclude that this Lie superalgebra is new.
We denote it by Bj(3;N |3), where N is the shearing parameter of the even indeterminates.
Our calculations show that N2 = N3 = 1 always. For N1 = 1, 2, the super dimensions of the
positive components of Bj(3;N |3) are given in the following tables:
N = (1, 1, 1) g1 g2 g3 g4 g5 g6 – – –
sdim 4|4 5|5 4|4 4|4 2|2 0|3
N = (2, 1, 1) g1 g2 g3 g4 g5 g6 · · · g11 g12
sdim 4|4 5|5 4|4 5|5 4|4 5|5 · · · 2|2 0|3
New simple modular Lie superalgebras 7
Let V ′i , V
i and V
i be the lowest height vectors of gi with respect to g0. For N = (1, 1, 1),
these vectors are as follows:
gi lowest weight vectors
V ′2 x1
2∂4 + 2 x1x2∂5 + x1x3∂6 + x2x3
V ′′2 x1x2
2∂1 + x1x2x5∂4 + x1x2∂3 + 2 x1x5∂6 + x2
2x3∂3 + 2 x2
2x5∂5 + x2x3x5∂6
V ′′′2 x1
2∂1 + x2
2∂1 + x2
2x3∂2 + 2 x2
2x6∂5 + 2 x1x2∂2 + 2 x1x3∂3 + 2 x1x4∂4
+x2x3
2∂3 + x2x4∂5 + 2 x3x4∂6 + 2 x2
2x3x6∂4 + 2 x2x3x6∂6
V ′3 x1
2x2∂4 + 2 x1
2∂6 + 2 x1x2
2∂5 + x1x2x3∂6 + x2
V ′′3 x1
2x2∂1 + x1
2x5∂4 + x1
2∂3 + x1x2x3∂3 + 2 x1x2x5∂5 + x1x3x5∂6
+x2x3
2x5∂6 + x1x2x4∂4 + 2 x1x2
2∂2 + x1x2x3∂3 + 2 x1x2x6∂6
+2 x1x4∂6 + 2 x2
2∂3 + 2 x2
2x3x6∂6 + 2 x2
2x4∂5 + x2x3x4∂6
V ′4 x1
2∂1 + x1
2x2x5∂4 + x1
2x2∂3 + 2 x1
2x5∂6 + x1x2
2x3∂3 + 2 x1x2
2x5∂5
+x1x2x3x5∂6 + x2
2x5∂6
V ′′4 x1
2x4∂4 + x1
2x5∂5 + x1
2x6∂6 + x2
2x6∂6 + x1x2
2∂1 + x1x2
2x3∂2 + 2 x1x2
2x6∂5
+2 x1x3
2x5∂6 + 2 x1x2x3
2∂3 + 2 x1x2x4∂5 + x1x3x4∂6 + x2x3
2x4∂6 + 2 x1x2
2x3x6∂4
V ′5 x1
2∂2 + x1
2x4∂6 + 2 x1
2x2x3∂3 + 2 x1
2x2x4∂4 + x1
2x2x6∂6 + 2 x2
2x4∂6
+x1x2
2∂3 + x1x2
2x4∂5 + x1x2
2x3x6∂6 + 2 x1x2x3x4∂6
V ′6 x1
2x4∂1 + x1
2x5∂2 + 2 x1
2x6∂3 + x1
2x2x4∂3 + 2 x1
2x4x5∂6
+2 x1
2x2x3x5∂3 + x1
2x2x4x5∂4 + x1
2x2x5x6∂6 + x2
2x4x5∂6
+x1x2
2x5∂3 + x1x2
2x3x4∂3 + 2 x1x2
2x4x5∂5 + x1x2
2x3x5x6∂6 + x1x2x3x4x5∂6
For N = (2, 1, 1), the lowest hight vectors are as in the table above together with the
following ones
gi lowest weight vectors
V ′′′4 x1
3∂4 + 2 x1
2x2∂5 + x1
2x3∂6 + x1x2x3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V ′11 x1
2∂2 + x1
5x4∂6 + x1
2∂3 + x1
2x4∂5 + 2 x1
5x2x3∂3 + 2 x1
5x2x4∂4
5x2x6∂6 + 2 x1
2x4∂6 + x1
2x3x6∂6 + 2 x1
4x2x3x4∂6
V ′12 x1
2x4∂1 + x1
2x5∂2 + 2 x1
2x6∂3 + x1
5x2x4∂3 + 2 x1
5x4x5∂6 + x1
2x5∂3
2x3x4∂3 + 2 x1
2x4x5∂5 + 2 x1
5x2x3x5∂3 + x1
5x2x4x5∂4 + x1
5x2x5x6∂6
2x4x5∂6 + x1
2x3x5x6∂6 + x1
4x2x3x4x5∂6
Let us investigate if Bj(3;N |3) has partial prolongs as subalgebras:
(i) Denote by g′1 the g0-module generated by V
1 . We have sdim(g
1) = 2|2. The CTS par-
tial prolong (g−, g0, g
1)∗ gives a graded Lie superalgebra with the property that [g−1, g1] ≃
{Y1, h1} := aff. From the description of irreducible modules over solvable Lie superalge-
bras [Ssol], we see that the irreducible aff-modules are 1-dimensional. For irreducible aff-
submodules g′−1 in g−1 we have two possibilities: to take g
−1 = {Y4} or g′−1 = {Y7}; for both
of them, g′−1 is purely odd and we can never get a simple Cartan prolong.
(ii) Denote by g′′1 the g0-module generated by V
1 . We have sdim(g
1) = 2|2. The CTS
partial prolong (g−, g0, g
1)∗ returns osp(2|4).
1.9. A description of Bj(2|4). We consider 3)g(2|3) with r = (1, 0, 0). In this case,
sdim(g(2, 3)−) = 2|4. Since the g(2, 3)0-module action is not faithful, we consider the quo-
tient algebra g0 = g(2, 3)0/ann(g−1) and embed (g(2, 3)−, g0) ⊂ vect(2|4). This realization
8 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites
is given by the following table:
gi the generators (even | odd)
g−1 Y6 = ∂2, Y8 = ∂1 | Y11 = ∂3, Y10 = ∂4, Y4 = ∂5, Y1 = ∂6
g0 ≃ Y3 = x2∂1 + 2x4∂3 + x6∂5, Y9 = [Y2, [Y3, Y5]], Z3 = x1∂2 + 2x3∂4 + x5∂6, Z9 =
[Z2, [Z3, Z5]], H2 = [Z2, Y2],H3 = [Z3, Y3] | Y2 = x1∂4 + x5∂2, Y5 = [Y2, Y3],
osp(3|2) Y7 = [Y3, [Y2, Y3]], Z2 = x2∂5 + 2x4∂1, Z5 = [Z2, Z3], Z7 = [Z3, [Z2, Z3]]
The g0-module g−1 is irreducible, having one lowest weight vector Y11 and one highest weight
vector Y1. The CTS prolong (g−, g0)∗ gives a Lie superalgebra of superdimension 13|14.
Indeed, sdim(g1) = 4|4 and sdim(g2) = 1|0. The g0-module g1 has one lowest vector:
V1 = 2x1x2∂3 + x1x6∂1 + 2x2
2∂4 + x2x5∂1 + 2x2x6∂2 + x4x5∂3 + 2x4x6∂4 + x5x6∂5
The g2 is one-dimensional spanned by the following vector
2x2∂1 + x1
2x4∂3 + 2x1
2x6∂5 + x1x2
2∂2 + 2x1x2x3∂3 + x1x2x4∂4 + 2x1x2x5∂5 + x1x2x6∂6
+x1x3x6∂1 + 2x1x4x5∂1 + x1x4x6∂2 + 2x2
2x3∂4 + x2
2x5∂6 + x2x3x5∂1 + 2x2x3x6∂2 + x2x4x5∂2
+x3x4x5∂3 + 2x3x4x6∂4 + x3x5x6∂5 + 2x4x5x6∂6
Besides, if i > 2, then gi = 0 for all values of the sharing parameter N = (N1, N2). A
direct computation gives [g1, g1] = g2 and [g−1, g1] = g0. SuperLie tells us that this Lie
superalgebra has three ideals I1 ⊂ I2 ⊂ I3 with the same non-positive part but different
positive parts: sdim(I1) = 10|14, sdim(I2) = 11|14, sdim(I3) = 12|14. The ideal I1 is just
our bj, see [BjL, CE]. The partial CTS prolong with I1 returns I1 plus an outer derivation
given by the vector above (of degree 2). It is clear now that Bj(2|4) is not simple.
1.10. A description of Bj(3|5). We consider 3)g(2|3) and r = (0, 1, 0). In this case,
sdim(g(2, 3)−) = 3|5. Since the g(2, 3)0-module action is again not faithful, we consider
the quotient module g0 = g(2, 3)0/ann(g−1) and embed (g(2, 3)−, g0) ⊂ vect(3;N |5). This
realization is given by the following table:
gi the generators (even | odd)
g−2 Y9 = ∂1 | Y10 = ∂3, Y11 = ∂2
g−1 Y8 = ∂4, Y6 = ∂5 | Y5 = 2x4∂2 + 2x5∂3 + 2x7∂1 + ∂7, Y2 = x4∂3 − 2x6∂1 + ∂8
Y7 = x5∂2 + ∂6
g0 ≃ sl(1|2) H1 = [Z1, Y1],H3 = [Z3, Y3], Y3 = 2x3∂2 + 2x7x8∂1 + x5∂4 + 2x7∂6 + x8∂7,
Z3 = 2x2∂3 + 2x6x7∂1 + x4∂5 + x6∂7 + 2x7∂8 | Y4 = [Y1, Y3], Z4 = [Z1, Z3],
Y1 = 2 (2x1∂3 + 2x6x7∂2 + x6∂4 + x7∂5)
Z1 = 2
x3∂1 + 2x4x5∂2 + 2x5
2∂3 + 2x5x7∂1 + 2x4∂6 + x5∂7
The g0-module g−1 is irreducible, having one highest weight vector Y2. We have sdim(g1) =
6|4. The g0-module g1 has two lowest weight vectors given by
V ′1 x1x5∂2 + 2x5x6x8∂2 + x5x7x8∂3 + 2x1∂6 + 2x3∂4 + x5x7∂4 + x5x8∂5 + 2x7x8∂7
V ′′1 x6x7x8∂2 + 2x1∂4 + x7x8∂5
Now, the g0-module generated by the the vectors V
1 and V
1 is not the whole g1 but a g0-
module that we denote by g′′1, of sdim = 4|4. The CTS prolong (g−, g0, g1)∗ is not simple, so
New simple modular Lie superalgebras 9
consider the Lie subsuperalgebra (g−, g0, g
1)∗; the superdimensions of its positive part are
adig′′1
(g′′1) g
1 adg′′1 (g
1) ad
(g′′1) ad
(g′′1) ad
(g′′1)
sdim 4|4 4|4 4|4 3|2 2|1
The lowest weight vectors of the above components are precisely {V ′2 , V ′′2 , V3, V4, V5} described
bellow:
adig1(g1) lowest weight vectors
V ′2 x1
2∂2 + 2x1x7∂4 + 2x1x8∂5 + x1x6x8∂2 + 2x1x7x8∂3
V ′′2 2x1
2∂1 + x1x2∂2 + x1x3∂3 + x1x6∂6 + x1x7∂7 + x1x8∂8 + 2x2x7∂4 + 2x2x8∂5
+2x3x6∂4 + 2x3x7∂5 + x2x6x8∂2 + 2x2x7x8∂3 + x3x6x7∂2 + x6x7x8∂7
V3 x1
2∂4 + 2x1x7x8∂5 + 2x1x6x7x8∂2
V4 x1
2x3∂2 + 2x1
2x5∂4 + x1
2x7∂6 + 2x1
2x8∂7 + x1
2x7x8∂1 + 2x1x3x7∂4 + 2x1x3x8∂5
+x1x3x6x8∂2 + 2x1x3x7x8∂3 + x1x5x7x8∂5 + x1x5x6x7x8∂2
V5 x1
2x2∂4 + 2x1
2x3∂5 + 2x1
2x6x7∂6 + x1
2x6x8∂7 + 2x1
2x7x8∂8 + 2x1
2x6x7x8∂1
+2x1x2x7x8∂5 + 2x1x3x6x7∂4 + 2x1x3x6x8∂5 + 2x1x2x6x7x8∂2 + 2x1x3x6x7x8∂3
Since none of the known simple finite dimensional Lie superalgebra over (algebraically closed)
fields of characteristic 0 or > 3 has such a non-positive part in any Z-grading, it follows that
Bj(3;N |5) is new.
Let us investigate if Bj(3;N |5) has subalgebras — partial prolongs.
(i) Denote by g′1 the g0-module generated by V
1 . We have sdim(g
1) = 2|3. The CTS
partial prolong (g−1, g0, g
1)∗ gives a graded Lie superalgebra with sdim(g
2) = 2|2 and g′i = 0
for i > 3. An easy computation shows that [g−1, g
1] = g0 and [g
1] ( g
2. Since we are
investigating simple Lie superalgebra, we take the simple part of (g−1, g0, g
1)∗. This simple
Lie superalgebra is isomorphic to g(2, 3)/c = bj.
(ii) Denote by g′′1 the g0-module generated by V
1 . We just saw that sdim(g
1) = 4|4. The
CTS partial prolong (g−1, g0, g
1)∗ gives also Bj(3|5).
r = (0, 0, 1). In this case, sdim(g(2, 3)−) = 4|5. Since the g(2, 3)0-module action is not
faithful, we consider the quotient algebra g0 = g(2, 3)0/ann(g−1) and embed (g(2, 3)−, g0) ⊂
vect(4;N |5). The CTS prolong returns bj := g(2, 3)/c.
1.11. A description of Me(3;N |3). 1) Our first idea was to try to repeat the above
construction with a suitable super version of g(2). There is only one simple super analog of
g(2), namely ag(2), but our attempts [BjL] to construct a super analog of Melikyan algebra
in the above way as Kuznetsov suggested [Ku1] (reproduced in [GL4]) resulted in something
quite distinct from the Melikyan algebra: The Lie superalgebras we obtained, an exceptional
one bj (cf. [CE, BGL1]) and a series Bj, are indeed simple but do not resemble either g(2)
or Me.
2) Our other idea is based on the following observation. The anti-symmetric form
(3) (f, g) :=
fdg =
fg′dt,
on the quotient space F/const of functions (with compact support) modulo constants on the
1-dimensional manifolds, has its counterpart in 1|1-dimensional case in presence of a contact
structure a n d o n l y i n t h i s c a s e as follows from the description of invariant bilinear
differential operators, see [KLV]. Indeed, the Lie superalgebra k(1|1) does not distinguish
10 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites
between the space of volume forms (let its generator be denoted vol) and the quotient Ω1/Fα,
where α = dt+ θdθ is the contact form.
For any prime p therefore, on the space g−1 := O(1;N |1)/ const of “functions (with com-
pact support) in one even indeterminate u and one odd, θ modulo constants”, the superanti-
symmetric bilinear form
(4) (f, g) :=
(fdg mod Fα) =
0 − f1g1)dt,
where f = f0(t) + f1(t)θ and g = g0(t) + g1(t)θ and where
′ := d
, is nondegenerate.
Therefore, we may expect that, for p small and N = 1, the Melikyan effect will reappear.
Consider p = 5 as the most plausible.
We should be careful with parities. The parity of vol is a matter of agreement, let it be
even. Then the integral is an odd functional but the factorization modulo Fα makes the form
(4) even. (Setting p(vol) = 1̄ we make the integral an even functional and the factorization
modulo Fα makes the form (4) even again.)
Since the form (4) is even, we get the following realization of
k(1; 1|1) ⊂ osp(5|4) ≃ k(5; 1, ..., 1|5)
by generating functions of contact vector fields on the 5|5-dimensional superspace with the
contact form, where the coefficients are found from the explicit values of
(p̂idq̂i − q̂2dp̂i) +
ξ̂jdηj + η̂jdξ̂j
− θ̂dθ̂.
The coordinates on this 5|5-dimensional superspace are hatted in order not to confuse them
with generating functions of k(1; 1|1):
gi basis elements
g−2 1̂
g−1 p̂1 = t, p̂2 = t
2, q̂1 = t
3, q̂2 = t
ξ̂1 = θ, ξ̂2 = tθ, θ̂ = t
2θ, η̂2 = t
3θ, η̂1 = t
We explicitly have:
(t, t4) =
t · t3dtN =
4t4dtN = 4 = −(t4, t);
(t2, t3) =
t2 · t2dtN =
6t4dtN = 1 = −(t3, t2);
(t4θ, θ) = −
t4 · 1dtN = −1 = (θ, t4θ);
(t3θ, tθ) = −
t3 · tdtN = −4 = (tθ, t3θ);
(t2θ, t2θ) = −
t2 · t2dtN = −6 = −1.
New simple modular Lie superalgebras 11
Now, let us realize k(1; 1|1) by contact fields in hatted functions:
gi basis elements
g−2 1̂
g−1 p̂1 = t, p̂2 = t
2, q̂2 = 4t
3, q̂1 = t
ξ1 = θ, ξ2 = tθ, θ̂ = t
2θ, η2 = 4t
3θ, η1 = t
g0 1 = 2 p̂1·q̂2 + 2p̂22 + 3 ξ1η2 + 3 ξ2θ̂; t = 2 p̂1q̂1 + 4 p̂2q̂2 + 4 ξ1η1 + 2 ξ2η2;
t2 = 2 p̂2q̂1 + 4 q̂
2 + 4 ξ2η1 + θ̂η2; t
3 = 3 q̂1q̂2 + 4 θ̂η1; t
4 = q̂21 + η2η1;
θ = p̂1η2 + p̂2θ̂ + q̂1ξ1 + q̂2ξ2; tθ = p̂1η1 + 2 p̂2η2 + q̂1ξ2 + 2 q̂2θ̂;
t2θ = p̂2η1 + q̂1θ̂ + 2 q̂2η2; t
3θ = 4 q̂1η2 + 4 q̂2η1; t
4θ = q̂1η1
The CTS prolong gives that g1 = 0.
The case where p = 3 is more interesting because it will give us the series Me(3;N |3).
The non-positive part is as follows:
gi basis elements
g−2 1̂
g−1 p̂1 = t, q̂2 = t
2, ξ1 = θ, θ̂ = tθ, η1 = t
g0 1 = p̂
1 + 2ξ̂1η̂1; t = 2 p̂1q̂1 + 2 ξ̂1η̂1; t
2 = 2 q̂21 + 2 θ̂η̂1; θ = 2 p̂1θ̂ + q̂1ξ̂1;
tθ = p̂1η̂1 + q̂1θ̂; t
2θ = q̂1η̂1
The Lie superalgebra g0 is not simple because [g−1, g1] = g0\{t2θ = q̂1η̂1}. Denote g′0 :=
[g−1, g1] ≃ osp(1|2). The CTS partial prolong (g−, g′0)∗ seems to be very interesting. First,
our computation shows that the parameter M = (M1,M2,M3) depends only on the first
parameter (relative to t). Namely, M = (M1, 1, 1). For M1 = 1, 2, the super dimensions of
the positive components of Bj(3;M |3) are given in the following table:
M = (1, 1, 1) g′1 g
5 – – – – –
sdim 2|4 4|2 2|4 3|2 0|1
M = (2, 1, 1) g′1 g
5 · · · g′14 g′15 g′16 g′17
sdim 2|4 4|2 2|4 4|2 2|4 · · · 4|2 2|4 3|2 0|1
Here we have that [g−1, g1] = g
0 and the g
1 generates the positive part. The standard criteria
for simplicity ensures that Me(3;N |3) is simple. For N = (1, a, b), the lowest weight vectors
12 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites
are as follows:
gi lowest weight vectors
V ′1 2p1
(2)η1 + 2 p1q1θ + q1
(2)ξ1 + ξ1θη1
V ′′1 2 p1q1η1 + q1
(2)θ + η1t
V ′2 2 p1q1θη1 + q1
(2)ξ1η1 + tq1
(2) + tθη1
V ′′2 2 p1
(2)q1
(2) + p1
(2)θη1 + p1q1ξ1η1 + 2q1
(2)ξ1θ + t
V ′3 2 tp1
(2)η1 + tp1q1θ + 2 tq1
(2)ξ1 + tξ1θη1
V ′′3 2 p1
(2)q1
(2)η1 + 2 tp1q1η1 + 2 q1
(2)ξ1θη1 + tq1
(2)θ + t(2)η1
V ′4 2 p1
(2)q1
(2)θη1 + 2 tp1q1θη1 + tq1
(2)ξ1η1 + t
(2)q1
(2) + t(2)θη1
V ′5 p1
(2)q1
(2)ξ1θη1 + t
(2)p1
(2)η1 + t
(2)p1q1θ + 2 t
(2)q1
(2)ξ1 + 2 t
(2)ξ1θη1
For N = (2, a, b), the lowest weight vectors are as above together with:
gi lowest weight vectors
V ′′4 2 tp1
(2)q1
(2) + tp1
(2)θη1 + tp1q1ξ1η1 + 2 tq1
(2)ξ1θ + t
V ′′5 2 tp1
(2)q1
(2)η1 + 2 t
(2)p1q1η1 + 2 tq1
(2)ξ1θη1 + t
(2)q1
(2)θ + t(3)η1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V ′15 2 t
(5)p1
(2)q1
(2)ξ1θη1 + 2t
(7)p1
(2)η1 + 2 t
(7)p1q1θ + t
(7)q1
(2)ξ1 + t
(7)ξ1θη1
2 t(6)p1
(2)q1
(2)η1 + 2 t
(7)p1q1η1 + 2 t
(6)q1
(2)ξ1θη1 + t
(7)q1
(2)θ + t(8)η1
V ′′16 2 t
(6)p1
(2)q1
(2)θη1 + 2 t
(7)p1q1θη1 + t
(7)q1
(2)ξ1η1 + t
(8)q1
(2) + t(8)θη1
V ′′17 2t
(6)p1
(2)q1
(2)ξ1θη1 + 2 t
(8)p1
(2)η1 + 2 t
(8)p1q1θ + t
(8)q1
(2)ξ1 + t
(8)ξ1θη1
Let us investigate the subalgebras of Me(3;N |3) — partial prolongs:
(i) Denote by g′1 the g0-module generated by V
1 . We have sdim(g
1) = 0|1 and gi = 0 for
all i > 1. The CTS partial prolong (g−, g0, g
1)∗ gives a graded Lie superalgebra with the
property that [g−1, g1] ≃ osp(1|2). The partial CTS prolong (g−, osp(1|2))∗ is not simple
(ii) Denote by g′′1 the g0-module generated by V
1 . We have sdim(g
1) = 3|2. The CTS
partial prolong (g−, g0, g
1)∗ returns brj(2; 3).
1.12. A description of Brj(4|3). We have the following realization of the non-positive
part inside vect(4|3):
gi the generators (even | odd)
g−4 Y8 = ∂1 | Y7 = ∂5
g−3 Y6 = ∂2 |
g−2 Y4 = ∂3 | Y5 = x3∂5 + x6∂1 + ∂6
g−1 Y3 = 2x2∂1 + 2x3∂2 + ∂4 |
Y2 = x2∂5 + 2x4
(2)x7∂1 + x4x6∂1 + x6x7∂5 + x4x7∂2 + x6∂2 + 2x4∂6 + 2x7∂3 + 1∂7,
g0 ≃ hei(0|2) ⊕ K H1 = [Z1, Y1], H2 = 2x5∂5 + x2∂2 + 2x3∂3 + 2x4∂4 + x7∂7, |
Y1 = 2x3
(2)∂5 + 2x3x6∂1 + 2x5∂1 + 2x3∂6 + x7∂4,
Z1 = x4
(2)∂6 + 2x4
(2)x6∂1 + x4
(2)x7∂2 + x4x7∂3 + 2x4x6x7∂5 + x1∂5 + 2x4∂7 + 2 x6∂3
The Lie superalgebra g0 is solvable, and hence the CTS prolong (g−, g0)∗ is NOT simple
since g1 does not generate the positive part. Our calculation shows that the prolong does
not depend on N , i.e., N = (1, 1, 1, 1). The simple part of this prolong is brj(2; 3). The sdim
New simple modular Lie superalgebras 13
of the positive parts are described as follows:
g1 g2 g3 g4 g5 g6 g7 g8 g9 g10
sdim 1|1 2|2 1|2 2|2 1|1 2|2 1|1 1|1 0|1 1|1
and the lowest weight vectors are as follows:
gi lowest weight vectors
V ′1 2x2x3∂5 + 2x2x6∂1 + x3x4∂6 + 2 x3x6∂2 + x3x7∂3 + x4x7∂4 + x3x4
(2)x7∂1 + 2x3x4x6∂1
+2x3x4x7∂2 + 2x3x6x7∂5 + 2x2∂6 + 2x3∂7 + 2x5∂2 + 2 x6∂4
V ′2 x4
(2)∂4 + x1x3∂5 + x1x6∂1 + x3x4
(2)∂6 + 2x3x4∂7 + 2x3x6∂3 + 2x4x6∂4 + 2x6x7∂7
+2x3x4
(2)x6∂1 + x3x4
(2)x7∂2 + x3x4x7∂3 + 2x3x4x6x7∂5 + x1∂6 + 2x5∂3
V ′′2 2x2
(2)∂1 + x3
(2)∂3 + 2x2x3∂2 + x3x4∂4 + x3x5∂5 + 2x3x7∂7 + x5x6∂1 + 2x6x7∂4 + x2∂4 + x5∂6
V ′3 2x1x2∂1 + 2x1x3∂2 + x2x3∂3 + 2 x2x4∂4 + 2x2x5∂5 + x2x6∂6 + 2x2x7∂7 + 2x3x6∂7 + x4x5∂6
+2x5x6∂2 + x5x7∂3 + x4
(2)x5x7∂1 + x3x4x6∂6 + x3x4x7∂7 + x3x6x7∂3 + 2x4x5x6∂1 + 2x4x5x7∂2
+2x4x6x7∂4 + 2x5x6x7∂5 + x3x4
(2)x6x7∂1 + 2x3x4x6x7∂2 + x1∂4 + 2x5∂7
V ′′3 2x3
(2)∂7 + x3
(2)x4∂6 + 2x3
(2)∂x6∂2 + x3
(2)x7∂3 + 2x2x3
(2)∂5 + 2x2x3∂6 + 2x2x5∂1 + x2x7∂4
+2x3x5∂2 + 2x3x6∂4 + x3
(2)x4
(2)x7∂1 + 2x3
(2)x4x6∂1 + 2x3
(2)x4x7∂2 + 2x3
(2)x6x7∂5 + 2x2x3x6∂1
+x3x4x7∂4 + x5∂4
V4 2x2
(2)∂6 + 2x2
(2)x3∂5 + 2x2
(2)x6∂1 + 2x3
(2)x4
(2)∂6 + x3
(2)x4∂7 + x3
(2)x6∂3 + 2x1x3
(2)∂5
+2x1x3∂6 + 2x1x5∂1 + x1x7∂4 + 2x2x3∂7 + 2x2x5∂2 + 2x2x6∂4 + 2 x3x5∂3 + x5x6∂6 + x5x7∂7
(2)x4
(2)x6∂1 + 2x3
(2)x4
(2)x7∂2 + 2x3
(2)x4x7∂3 + 2x1x3x6∂1 + x2x3x4∂6 + 2x2x3x6∂2
+x2x3x7∂3 + x2x4x7∂4 + x3x4
(2)x7∂4 + 2x3x4x6∂4 + x3
(2)x4x6x7∂5 + x2x3x4
(2)x7∂1 + 2 x2x3x4x6∂1
+2x2x3x4x7∂2 + 2x2x3x6x7∂5
V ′′4 x1
(2)∂1 + x4
(2)x5∂6 + 2x1x3∂3 + x1x4∂4 + x1x5∂5 + 2x1x6∂6 + x1x7∂7 + 2x4x5∂7 + 2x5x6∂3
(2)x5x6∂1 + x4
(2)x5x7∂2 + 2x4
(2)x6x7∂4 + x3x4
(2)x6∂6 + x3x4
(2)x7∂7 + 2x3x4x6∂7
+x4x5x7∂3 + x3x4
(2)x6x7∂2 + x3x4x6x7∂3 + 2x4x5x6x7∂5
V ′5 x1x2∂6 + x1x3∂7 + x1x5∂2 + x1x6∂4 + 2x2x5∂3 + x5x6∂7 + x1x2x3∂5 + x1x2x6∂1 + 2x1x3x4∂6
+x1x3x6∂2 + 2x1x3x7∂3 + 2x1x4x7∂4 + x2x4
(2)x7∂4 + x2x3x4
(2)∂6 + 2x2x3x4∂7 + 2x2x3x6∂3
+2x2x4x6∂4 + 2x2x6x7∂7 + 2x4x5x6∂6 + 2x4x5x7∂7 + 2x5x6x7∂3 + 2x4
(2)x5x6x7∂1
+2x1x3x4
(2)x7∂1 + x1x3x4x6∂1 + x1x3x4x7∂2 + x1x3x6x7∂5 + 2x2x3x4
(2)x6∂1 + x2x3x4
(2)x7∂2
+x2x3x4x7∂3 + 2x3x4x6x7∂7 + x4x5x6x7∂2 + 2x2x3x4x6x7∂5
V ′6 x1
(2)∂6 + x1
(2)x3∂5 + x1
(2)x6∂1 + 2x1x5∂3 + x4
(2)x5x6∂6 + x4
(2)x5x7∂7 + x1x4
(2)x7∂4
+x1x3x4
(2)∂6 + 2x1x3x4∂7 + 2x1x3x6∂3 + 2x1x4x6∂4 + 2x1x6x7∂7 + 2x4x5x6∂7
(2)x5x6x7∂2 + 2x1x3x4
(2)x6∂1 + x1x3x4
(2)x7∂2 + x1x3x4x7∂3 + x3x4
(2)x6x7∂7
+x4x5x6x7∂3 + 2x1x3x4x6x7∂5
V ′′6 x2
(2)x3∂3 + 2x2
(2)x4∂4 + 2x2
(2)x5∂5 + x2
(2)x6∂6 + 2x2
(2)x7∂7 + 2x1x2
(2)∂1 + x1x3
(2)∂3
+x1x2∂4 + x1x5∂6 + 2x2x5∂7 + 2x3
(2)x4
(2)x6∂6 + 2x3
(2)x4
(2)x7∂7 + x3
(2)x4x6∂7 + 2x1x2x3∂2
+x1x3x4∂4 + x1x3x5∂5 + 2x1x3x7∂7 + x1x5x6∂1 + 2x1x6x7∂4 + 2x2x3x6∂7 + x2x4x5∂6
+2x2x5x6∂2 + x2x5x7∂3 + x3x4
(2)x5∂6 + 2x3x4x5∂7 + 2x3x5x6∂3 + x5x6x7∂7
(2)x4
(2)x6x7∂2 + 2x3
(2)x4x6x7∂3 + x2x4
(2)x5x7∂1 + x2x3x4x6∂6 + x2x3x4x7∂7 + x2x3x6x7∂3
+2x2x4x5x6∂1 + 2x2x4x5x7∂2 + 2x2x4x6x7∂4 + 2x2x5x6x7∂5 + 2x3x4
(2)x5x6∂1 + x3x4
(2)x5x7∂2
+2x3x4
(2)x6x7∂4 + x3x4x5x7∂3 + x2x3x4
(2)x6x7∂1 + 2x2x3x4x6x7∂2 + 2x3x4x5x6x7∂5
14 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites
V ′7 x1
(2)∂4 + 2x1
(2)x2∂1 + 2x1
(2)x3∂2 + 2x1x5∂7 + x1x2x3∂3 + 2x1x2x4∂4 + 2x1x2x5∂5 + x1x2x6∂6
+2x1x2x7∂7 + 2x1x3x6∂7 + x1x4x5∂6 + 2x1x5x6∂2 + x1x5x7∂3 + 2x2x4
(2)x5∂6 + x2x4x5∂7
+x2x5x6∂3 + x1x4
(2)x5x7∂1 + x1x3x4x6∂6 + x1x3x4x7∂7 + x1x3x6x7∂3 + 2x1x4x5x6∂1 + 2x1x4x5x7∂2
+2x1x4x6x7∂4 + 2x1x5x6x7∂5 + x2x4
(2)x5x6∂1 + 2x2x4
(2)x5x7∂2 + x2x4
(2)x6x7∂4 + 2x2x3x4
(2)x6∂6
+2x2x3x4
(2)x7∂7 + x2x3x4x6∂7 + 2x2x4x5x7∂3 + x4x5x6x7∂7 + x1x3x4
(2)x6x7∂1 + 2x1x3x4x6x7∂2
+2x2x3x4
(2)x6x7∂2 + 2x2x3x4x6x7∂3 + x2x4x5x6x7∂5
V ′8 x1
(2)x3
(2)∂5 + x1
(2)x3∂6 + x1
(2)x5∂1 + 2x1
(2)x7∂4 + 2x2
(2)x5∂3 + x1x2
(2)∂6 + x1
(2)x3x6∂1
(2)x4
(2)x7∂4 + x2
(2)x3x4
(2)∂6 + 2x2
(2)x3x4∂7 + 2x2
(2)x3x6∂3 + 2x2
(2)x4x6∂4 + 2x2
(2)x6x7∂7
+x1x2
(2)x3∂5 + x1x2
(2)x6∂1 + x1x3
(2)x4
(2)∂6 + 2x1x3
(2)x4∂7 + 2 x1x3
(2)x6∂3 + x1x2x3∂7 + x1x2x5∂2
+x1x2x6∂4 + x1x3x5∂3 + 2x1x5x6∂6 + 2x1x5x7∂7 + x2x5x6∂7 + 2x2
(2)x3x4
(2)x6∂1 + x2
(2)x3x4
(2)x7∂2
(2)x3x4x7∂3 + x3
(2)x4
(2)x6x7∂7 + 2x1x3
(2)x4
(2)x6∂1 + x1x3
(2)x4
(2)x7∂2 + x1x3
(2)x4x7∂3
+2x1x2x3x4∂6 + x1x2x3x6∂2 + 2x1x2x3x7∂3 + 2x1x2x4x7∂4 + 2x1x3x4
(2)x7∂4 + x1x3x4x6∂4
+2x2x4x5x6∂6 + 2x2x4x5x7∂7 + 2x2x5x6x7∂3 + 2x3x4
(2)x5x6∂6 + 2x3x4
(2)x5x7∂7 + x3x4x5x6∂7
(2)x3x4x6x7∂5 + 2x1x3
(2)x4x6x7∂5 + 2x1x2x3x4
(2)x7∂1 + x1x2x3x4x6∂1 + x1x2x3x4x7∂2
+x1x2x3x6x7∂5 + 2x2x4
(2)x5x6x7∂1 + 2x2x3x4x6x7∂7 + x2x4x5x6x7∂2 + 2x3x4
(2)x5x6x7∂2+
2x3x4x5x6x7∂3
V ′9 x1
(2)x2x3∂5 + x1
(2)x2x6∂1 + 2x1
(2)x3x4
(2)x7∂1 + x1
(2)x3x4x6∂1 + x1
(2)x3x6x7∂5 + 2x1x2x3x4
(2)x6∂1
+2x1x2x3x4x6x7∂5 + 2x1x4
(2)x5x6x7∂1 + x1
(2)x3x4x7∂2 + x1
(2)x3x6∂2 + x1
(2)x5∂2
+x1x2x3x4
(2)x7∂2 + x1x4x5x6x7∂2 + x2x4
(2)x5x6x7∂2 + x1
(2)x2∂6 + 2x1
(2)x3x4∂6 + 2 x1
(2)x3x7∂3
+x1x2x3x4
(2)∂6 + x1x2x3x4x7∂3 + 2x1x2x3x6∂3 + 2x1x2x5∂3 + 2x1x4x5x6∂6 + 2x1x5x6x7∂3
+x2x4
(2)x5x6∂6 + x2x4x5x6x7∂3 + x1
(2)x3∂7 + 2x1
(2)x4x7∂4 + x1
(2)x6∂4 + 2x1x2x3x4∂7
+x1x2x4
(2)x7∂4 + 2x1x2x4x6∂4 + 2x1x2x6x7∂7 + 2x1x3x4x6x7∂7 + 2x1x4x5x7∂7 + x1x5x6∂7
+x2x3x4
(2)x6x7∂7 + x2x4
(2)x5x7∂7 + 2x2x4x5x6∂7
V ′10 x1
(2)x2
(2)∂1 + 2x1
(2)x3
(2)∂3 + 2 x1
(2)x2∂4 + 2x1
(2)x5∂6 + x1
(2)x2x3∂2 + 2x1
(2)x3x4∂4
(2)x3x5∂5 + x1
(2)x3x7∂7 + 2x1
(2)x5x6∂1 + x1
(2)x6x7∂4 + x2
(2)x4
(2)x5∂6 + 2x2
(2)x4x5∂7
(2)x5x6∂3 + 2x1x2
(2)x3∂3 + x1x2
(2)x4∂4 + x1x2
(2)x5∂5 + 2x1x2
(2)x6∂6 + x1x2
(2)x7∂7
+x1x2x5∂7 + 2x2
(2)x4
(2)x5x6∂1 + x2
(2)x4
(2)x5x7∂2 + 2x2
(2)x4
(2)x6x7∂4 + x2
(2)x3x4
(2)x6∂6
(2)x3x4
(2)x7∂7 + 2x2
(2)x3x4x6∂7 + x2
(2)x4x5x7∂3 + x1x3
(2)x4
(2)x6∂6 + x1x3
(2)x4
(2)x7∂7
+2x1x3
(2)x4x6∂7 + x1x2x3x6∂7 + 2x1x2x4x5∂6 + x1x2x5x6∂2 + 2x1x2x5x7∂3 + 2x1x3x4
(2)x5∂6
+x1x3x4x5∂7 + x1x3x5x6∂3 + 2x1x5x6x7∂7 + x2
(2)x3x4
(2)x6x7∂2 + x2
(2)x3x4x6x7∂3
(2)x4x5x6x7∂5 + x1x3
(2)x4
(2)x6x7∂2 + x1x3
(2)x4x6x7∂3 + 2x1x2x4
(2)x5x7∂1 + 2x1x2x3x4x6∂6
+2x1x2x3x4x7∂7 + 2x1x2x3x6x7∂3 + x1x2x4x5x6∂1 + x1x2x4x5x7∂2 + x1x2x4x6x7∂4 + x1x2x5x6x7∂5
+x1x3x4
(2)x5x6∂1 + 2x1x3x4
(2)x5x7∂2 + x1x3x4
(2)x6x7∂4 + 2x1x3x4x5x7∂3 + 2x2x4x5x6x7∂7
+2x3x4
(2)x5x6x7∂7 + 2x1x2x3x4
(2)x6x7∂1 + x1x2x3x4x6x7∂2 + x1x3x4x5x6x7∂5
New simple modular Lie superalgebras 15
1.13. A description ofBrj(3;N |4). We have the following realization of the non-positive
part inside vect(3;N |4):
gi the generators (even | odd)
g−3 | Y6 = ∂4
g−2 Y5 = ∂1, Y6 = ∂2, Y7 = ∂3 |
g−1 | Y2 = 2x3∂4 + ∂5, Y3 = x2∂4 + x6∂1 + ∂6
Y4 = 2x1∂4 + 2x5x7∂4 + x5∂1 + x6∂2 + 2x7∂3 + ∂7
g0 ≃ hei(2|0)⊂+ KH2 H1 = [Z1, Y1], H2 = 2x1∂1 + x3∂3 + x4∂4 + x6∂6 + 2x7∂7
Y1 = 2x5x6x7∂4 + 2x1∂2 + 2x2∂3 + 2x5x6∂1 + x6x7∂3 + 2x5∂6 + 2x6∂7,
Z1 = 2x2∂1 + 2x3∂2 + x6x7∂1 + 2x6∂5 + x7∂6 |
The Lie superalgebra g0 is solvable with the property that [g0, g0] = hei(2|0). The CTS
prolong (g−, g0)∗ is NOT simple since g1 does not generate the positive part. Our calculation
shows that the prolong does not depend on N , i.e., N = (1, 1, 1, 1). The simple part of this
prolong is brj. The sdim of the positive parts are described as follows:
g1 g2 g3
sdim 0|3 3|0 0|2
and the lowest weight vectors are
V ′1 2x1
(2)∂4 + 2x1x5x7∂4 + x1x5∂1 + x1x6∂2 + 2x1x7∂3 + x4∂3 + x1∂7 + 2x5x6∂6 + x5x7∂7
V ′2 2x1x4∂4 + x2x5x6x7∂4 + 2x4x5x7∂4 + 2x1
(2)∂1 + x1x2∂2 + x2
(2)∂3 + x2x5x6∂1 + 2x2x6x7∂3
+x4x5∂1 + x4x6∂2 + 2x4x7∂3 + 2x1x5∂5 + x1x6∂6 + x2x5∂6 + x2x6∂7 + x4∂7 + x5x6x7∂6
V ′3 x1
(2)x2∂4 + x1x2x5x7∂4 + 2x4x5x6x7∂4 + x1
(2)x6∂1 + 2x1x2x5∂1 + 2x1x2x6∂2 + x1x2x7∂3
+2x1x4∂2 + 2 x1x5x6x7∂1 + 2x2x4∂3 + 2 x4x5x6∂1 + x4x6x7∂3 + x1
(2)∂6 + 2x1x2∂7
+x1x5x6∂5 + 2x1x5x7∂6 + x2x5x6∂6 + 2x2x5x7∂7 + 2x4x5∂6 + 2x4x6∂7
V ′′3 x1
(2)x3∂4 + x1x2
(2)∂4 + x1x3x5x7∂4 + x2
(2)x5x7∂4 + x1x2x6∂1 + 2x1x3x5∂1 + 2x1x3x6∂2
+x1x3x7∂3 + 2x1x4∂1 + 2x2
(2)x5∂1 + 2x2
(2)x6∂2 + x2
(2)x7∂3 + 2x2x4∂2 + 2x2x5x6x7∂1
+2x3x4∂3 + 2 x1
(2)∂5 + x1x2∂6 + 2x1x3∂7 + x1x6x7∂6 + 2x2
(2)∂7 + x2x5x6∂5 + 2x2x5x7∂6
+x3x5x6∂6 + 2x3x5x7∂7 + x4x5∂5 + x4x6∂6 + x4x7∂7
Let us study now the case where g′0 = der0(g−). Our calculation shows that g
0 is generated
by the vectors Y1, Z1, H1, H2 above together with V = 2x3∂1 + x7∂5. The Lie algebra g
solvable of sdim = 5|0. The CTS prolong (g−, g′0)∗ gives a Lie superalgebra that is not simple
because g′1 does not generate the positive part. Its simple part is a new Lie superalgebra
that we denote by BRJ, described as follows (here also N = (1, 1, 1):
g′1 adg′1(g
1) ad
(g′1) ad
(g′1) ad
(g′1) ad
(g′1)
sdim 0|6 6|0 0|5 3|0 0|3 1|0
16 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites
1.14. A description ofBrj(3;N |3). We have the following realization of the non-positive
part inside vect(3;N |3):
gi the generators (even | odd)
g−2 Y7 = ∂1 | Y5 = ∂4, Y8 = ∂5
g−1 Y1 = ∂2, Y6 = 2x2∂1 + ∂3 | Y3 = x2∂4 + x3∂5 + 2x6∂1 + ∂6
g0 H2 = [X2, Y2], H1 = x1∂1 + x3∂3 + 2x4∂4 + 2x6∂6, X4 = [X2,X2], Y4 = [Y2, Y2] |
Y2 = x
2 ∂4 + x2 x3∂5 + 2x2 x6∂1 + x1∂5 + x2∂6 + x4∂1 + x6∂3
X2 = x
3 ∂5 + 2x1∂4 + x3∂6 + x5∂1 + 2x6∂2
The Lie superalgebra g0 is isomorphic to osp(1|2)⊕ K. The CTS prolong (g−, g0)∗ is NOT
simple since it gives back brj(2; 3) + an outer derivation. The sdim of the positive parts are
described as follows:
g1 g2 g3
sdim 2|1 1|2 0|1
and the lowest weight vectors are
V ′1 2 x1 x2∂1 + x2 x4∂4 + x3 x4∂5 + 2x4 x6∂1 + x1∂3 + 2x2 x3∂3 + x2 x6∂6 + x4∂6
V ′2 x
1 ∂5 + x1 x
2 ∂4 + x1 x2 x3∂5 + 2x1 x2 x6∂1 + x1 x4∂1 + 2x
3 ∂5 + 2x
2 x5∂1 + 2x4 x5∂5 + x1 x2∂6
+x1 x6∂3 + 2x
2 x3∂6 + x
2 x6∂2 + x2 x3 x6∂3 + x2 x4∂2 + x2 x5∂3 + 2x4 x6∂6
V ′3 x
1 x2∂4 + x
1 x3∂5 + 2x
1 x6∂1 + 2x1 x2 x
3 ∂5 + 2x1 x2 x5∂1 + x2 x
3 x4∂1 + 2x2 x4 x5∂4 + 2x3 x4 x5∂5
+x4 x5 x6∂1 + x
1 ∂6 + 2x1 x2 x3∂6 + x1 x2 x6∂2 + x1 x3 x6∂3 + x1 x4∂2 + x1 x5∂3 + 2x2 x
3 x6∂3 + 2x2 x3 x4∂2
+2x2 x3 x5∂3 + x2 x5 x6∂6 + 2x3 x4 x6∂6 + 2x4 x5∂6
1.15. A description ofBrj(3;N |4). We have the following realization of the non-positive
part inside vect(3;N |4):
gi the generators (even | odd)
g−3 | Y8 = ∂4
g−2 Y4 = ∂1, Y6 = ∂2, Y7 = ∂3 |
g−1 | Y2 = x3∂4 + 2x5∂1 + ∂5, Y3 = ∂6 + x5 x6∂4 + x2∂4 + x5∂2 + 2x6∂3
Y5 = x1∂4 + x5∂3 + ∂7
g0 ≃ hei(2|0)⊂+ KH2 H1 = [Z1, Y1], H2 = 2x1∂1 + x2∂2 + x4∂4 + x5∂5 + 2x7∂7
Y1 = x1∂2 + x2∂3 + x5 x6∂3 + 2x5∂6 + 2x6∂7
Z1 = 2x5 x6 x7∂4 + 2x2∂1 + x3∂2 + x5 x6∂1 + 2x6 x7∂3 + 2x6∂5 + x7∂6 |
The Lie superalgebra g0 is solvable with the property that [g0, g0] = hei(2|0). The CTS
prolong (g−, g0)∗ is NOT simple since g1 does not generate the positive part. Our calculation
shows that the prolong does not depend on N , i.e., N = (1, 1, 1, 1). The simple part of this
prolong is 3)brj(2; 3). The sdim of the positive parts are described as follows:
g1 g2 g3
sdim 0|3 3|0 0|2
New simple modular Lie superalgebras 17
and the lowest weight vectors are
V ′1 2x1 x3∂4 + x
2 ∂4 + x2 x5 x6∂4 + x1 x5∂1 + x2 x5∂2 + 2 x2 x6∂3 + 2x3 x5∂3 + x4∂3 + 2x1∂5 + x2∂6 + 2 x3∂7 + 2x5 x7∂7
V ′2 2x1 x4∂4 + x
1 ∂1 + 2x1 x2∂2 + 2x
2 ∂3 + 2x2 x5 x6∂3 + 2x4 x5∂3 + 2x1 x5∂5 + x1 x7∂7 + x2 x5∂6 + x2 x6∂7 + 2x4∂7
V ′3 x1 x
3 ∂4 + 2x
2 x3∂4 + 2x2 x3 x5 x6∂4 + 2x1 x3 x5∂1 + 2x1 x4∂1 + x
2 x5∂1 + 2x2 x3 x5∂2 + x2 x3 x6∂3 + 2x2 x4∂2
+2x2 x5 x6 x7∂3 + x
3 x5∂3 + 2x3 x4∂3 + x1 x3∂5 + x1 x6 x7∂6 + 2x
2 ∂5 + 2x2 x3∂6 + 2x2 x5 x6∂5 + x2 x5 x7∂6
+2x2 x6 x7∂7 + x
3 ∂7 + x3 x5 x7∂7 + x4 x5∂5 + x4 x6∂6 + x4 x7∂7
Let us study now the case where g′0 = der0(g−). The Lie algebra g
0 is solvable of sdim = 5|0.
The CTS prolong (g−, g
0)∗ gives a Lie superalgebra that is not simple because g
1 does not
generate the positive part. Its simple part is a new Lie superalgebra that we had denoted
by BRJ, described as follows (here also N = (1, 1, 1):
g′1 adg′1(g
1) ad
(g′1) ad
(g′1) ad
(g′1) ad
(g′1)
sdim 0|6 6|0 0|5 3|0 0|3 1|0
1.16. Constructing Melikyan superalgebras. Denote by F1/2 := O(1; 1)
dx the space
of semi-densities (weighted densities of weight 1
). For p = 3, the CTS prolong of the triple
(K,Π(F1/2), cvect(1; 1))∗ gives the whole k(1;N |3). For p = 5, let us realize the non-positive
part in k(1;N |5):
gi the generators
g−2 1
g−1 Π(F1/2)
g0 ∂1 ←→ 4 ξ1η2 + ξ2θ, x1∂1 ←→ 2 ξ1η1 + ξ2η2, x
1 ∂1 ←→ 2 ξ2η1 + 3 θη2, x
1 ∂1 ←→ 2 θη1
1 ∂1 ←→ 2 η2η1, t
The CTS prolong gives that gi=0 for all i > 0.
Consider now the case of (K,Π(F1/2), cvect(2; 1))∗, where p = 3. The non-positive part is
realized in k(1;N |9) as follows:
gi the generators
g−2 1
g−1 Π(F1/2)
g0 ∂1 ←→ 2 ξ1η3 + x2θ + 2 ξ3η4, x1∂1 ←→ ξ1η1 + ξ2η2 + 2 ξ4η4, x21∂1 ←→ ξ3η1 + ξ4η3 + θη2,
∂2 ←→ 2 ξ1η2 + ξ2ξ4 + ξ3θ, x2∂2 ←→ ξ1η1 + ξ3η3 + ξ4η4, x22∂2 ←→ ξ2η1 + θη3 + 2 η4η2,
x1x2∂1 ←→ ξ2η1 + η4η2, x1x2∂2 ←→ ξ3η1 + 2 ξ4η3, x21x2∂1 ←→ θη1 + 2 η3η2,
x21x2∂2 ←→ ξ4η1, x1x22∂1 ←→ 2 η4η1, x1x22∂2 ←→ θη1 + η3η2, x21x22∂1 ←→ η3η1,
2∂2 ←→ η2η1, t
The CTS prolong (g−, g0)∗ gives a Lie superalgebra that is not simple with the property that
sdim(g1) = 0|4 and gi = 0 for all i > 1. The generating functions of g1 are
ξ2η2η1 + 2 ξ3η3η1 + ξ4η4η1 + θη3η2, 2 ξ4η3η1 + θη2η1, θη3η1 + η4η2η1, η3η2η1.
1.17. Defining relations of the positive parts of brj(2; 3) and brj(2; 5). For the
presentations of the Lie superalgebras with Cartan matrix, see [GL1, BGL1]. The only non-
trivial part of these relations are analogs of the Serre relations (both the straightforward
18 Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites
ones and the ones different in shape). Here they are:
brj(2; 3); sdim brj(2; 3) = 10|8.
1) [[x1, x2] , [x2, [x1, x2]]] = 0,
[[x2, x2] , [[x1, x2] , [x2, x2]]] = 0.
2) ad3x2(x1) = 0,
[[x1, x2] , [[x1, x2] , [x1, x2]]] = 0,
[[x2, [x1, x2]] , [[x1, x2] , [x2, [x1, x2]]]] = 0.
3) ad3x1(x2) = 0,
[x2, [x1, [x1, x2]]]− [[x1, x2], [x1, x2]] = 0,
[[x1, x2], [x2, x2]] = 0.
brj(2; 5); sdim brj(2; 5) = 10|12.
1) [[x2, [x1, x2]] , [x2, [x1, x2]]] = 2 [[x1, x2] , [[x1, x2] , [x2, x2]]],
[[x2, x2] , [[x1, x2] , [x2, x2]]] = 0,
[[x2, [x1, x2]] , [[x1, x2] , [x2, [x1, x2]]]] = 0.
2 )ad4x2(x1) = 0,
[[x2, [x1, x2]] , [x2, [x2, [x1, x2]]]] = 0,
[[[x1, x2] , [x1, x2]] , [[x1, x2] , [x2, [x1, x2]]]] = 0.
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1Department of Mathematics, United Arab Emirates University, Al Ain, PO. Box: 17551;
Bouarroudj.sofiane@uaeu.ac.ae, 2Equa Simulation AB, Stockholm, Sweden; pavel@rixtele.com,
3MPIMiS, Inselstr. 22, DE-04103 Leipzig, Germany, on leave from Department of Mathe-
matics, University of Stockholm, Roslagsv. 101, Kräftriket hus 6, SE-106 91 Stockholm,
Sweden; mleites@math.su.se, leites@mis.mpg.de
http://arxiv.org/abs/math/0612861
http://arxiv.org/abs/math/0702499
http://arxiv.org/abs/math/0702594
http://arxiv.org/abs/math/0702755
	1. Introduction
	References
