title: Duflo-Serganova homology for exceptional modular Lie superalgebras with Cartan matrix
authors: Andrey Krutov Dimitry Leites Jin Shang
text: 
  
  Duflo–Serganova homology]Duflo–Serganova homology for exceptional modular
   Lie superalgebras with Cartan matrix
  
  
  
  
  
   ^aDepartment of Mathematics, University of Zagreb, Bijenička 30, 10000 Zagreb, Croatia 
  
  ^b Independent University of Moscow, Bolshoj Vlasievskij per, dom
   11, RU-119 002 Moscow, Russia;
   andrey.krutov@math.hr
  
   
  ^cNew York University Abu Dhabi,
  Division of Science and Mathematics, P.O. Box 129188, United Arab
  Emirates; {dl146, js8544}@nyu.edu
  
  ^dDepartment of mathematics, Stockholm University, Roslagsv. 101, Stockholm, Sweden; mleites@math.su.se
  
  ^*The corresponding author
  
  
   [2020]Primary 17B50, 17B55, 17B56; Secondary 17B20
  
   
  For the exceptional finite-dimensional modular Lie superalgebras 𝔤(A) with indecomposable Cartan matrix A, and their simple subquotients, we computed non-isomorphic Lie superalgebras constituting the homologies of the odd elements with zero square.  These homologies are key ingredients in the Duflo–Serganova approach to the representation theory.
  
  There were two definitions of defect of Lie superalgebras in the literature with different ranges of application. We suggest a third definition and an easy-to-use way to find its value. 
  
  In positive characteristic, we found out one more reason to consider the space of roots  over reals, unlike the space of weights, which should be considered over the ground field.
  
  We proved that the  rank of the homological element (decisive in calculating the defect of a given Lie superalgebra)  should be considered in the adjoint module, not the irreducible module of least dimension (although the latter is sometimes possible to consider, e.g., for p=0). 
  
  We also computed the above homology for the only case of simple Lie superalgebras with symmetric root system not considered so far over the field of complex numbers, and its  modular versions: 𝔭𝔰𝔩(a|a+pk) for a and k small, and p=2, 3, 5.
  
  
  
  
  
  
  [
      Andrey Krutov^a,b,*, Dimitry Leites^c, d, Jin Shang^c
      Received December 30, 2023; accepted December 30, 2023
  ==========================================================
  
  
  
  A. Krutov D. Leites J. ShangDuflo–Serganova homology
  
  
  empty
  
  
  
  
  
  
  § INTRODUCTION
  
  
  For notation, see <cit.>. All vector spaces, in particular, Lie (super)algebras  we consider here, are finite-dimensional 
  over the ground field  of characteristic p. Let p denote also the parity of the Lie superalgebra considered. In this paper, ^m|n denotes a commutative Lie superalgebra, the one with zero bracket;  denotes the center of .
  
  
  
  Atypicality and defect The irreducible finite-dimensional modules over the Lie superalgebra =(A) with Cartan matrix A over  are of the two types: generic  (Kac brightly called them “typical"),  which are in one-to-one correspondence with certain _-modules, and “atypical".  It soon became clear that different irreducible modules 
  over simple  (and “close” to simple, like  is close to ) Lie superalgebras  have different levels of atypicality, not exceeding the defined in <cit.> defect (). Kac and Wakimoto defined () only for Lie superalgebras of the form (A) over  with an even non-degenerate invariant symmetric bilinear form (NIS for short). 
  
  
  The typical irreducible (A)-modules were first described analytically by Berezin in several cases (<cit.>). Kac obtained a complete classification of typical irreducible (A)-modules using more adequate algebraic technique, see <cit.> with corrections in <cit.>. 
  
  The atypical irreducible modules were first classified and described in terms of sections of vector bundles on certain Grassmann supervarieties of dimension 0|n by J. Bernstein and D. Leites for (1|n) and (2|2n) whose defect is equal to 1, see <cit.>. 
  
  The Duflo–Serganova functor 
  Over any ground field, let  be a Lie superalgebra, let
  M be a -module given by a representation ρ. For an odd element x∈ such that x^2=0 set
  
  M_x := ρ_x/ρ_x  and  _x := _x/_x.
  
  
  It was known since long time ago that for any dg Lie superalgebra with the odd differential d, the space M_d is a _d-module.
  
  This general fact was formulated usually for the non-zero differential d of degree ± 1, see, e.g., <cit.>. In these works, the Maurer-Cartan equation allows one to add only degree ±1 elements x∈_ to the differential d.
  
  Duflo and Serganova (see <cit.>) considered a completely different case where d=0, and the -grading of  considered modulo 2 coincides with the parity of . Then, one can add any odd element x such that x^2=0 to the differential d,  and still have (d+x)^2=0. 
  
  
  Duflo and Serganova proved that the Duflo–Serganova functor
  _x : M  M_x from the category of -modules to the category of _x-modules is a tensor functor; this was a new result. 
  
  This functor helped to solve interesting problems,
  see <cit.>, where p=0.
  
  For the first applications of the DS-functor in the cases where p>2, see <cit.>.
  
  For which  was _x computed? For p=0, Duflo and Serganova (<cit.>) considered finite-dimensional Lie superalgebras  of the form (A) with indecomposable and invertible Cartan matrix A, and (a|b) for any a, b over , i.e., all simple Lie superalgebras with symmetric root system, except (a|a). 
  
  In this paper, we consider the cases of (a|a+kp) for p=0, 2,3,5, and k and a small.
  
  The Lie superalgebras considered by Duflo and Serganova for p=0, as well as (a|a+kp) for p>0, and the exceptional simple Lie superalgebras and those of the form (A) we consider here have an even NIS, see <cit.>.  
  
  The specifically super analogs of (n) — Lie superalgebras  of series   their simple subquotiets,  have an odd NIS and no Cartan matrix. For computation of the DS-homology of these Lie superalgebra, see <cit.>. The case of Poisson Lie superalgebras (0|n) and their simple subquotients '(0|n) will be considered separately.
  
  
  
  
   §.§ When _x≃_y for =(A) over 
   
  The bulk of <cit.> consists of the proof of the fact that, for the finite-dimensional Lie superalgebra =(A) with indecomposable and invertible Cartan matrix A, the following statement holds:
  [<cit.>: when _x≃_y]Theorem[<cit.>] We have
  _x≃_y⟺ ρ_x= ρ_y, where
  
  [ ρ=𝕀 in V   for  of series (V), or (V) for V≠ a|a, or (V), 
  
   in    for =(4|2; a) or (2) or (3). ]
  
  In cases (<ref>), the module given by ρ is the one of the least dimension, except for (4|2; a), where a=1, or 2 or 3 (or the values of a obtained from these under the action of S_3, see <cit.>): for these values of A, the irreducible modules of the least superdimension are (up to the change of parity) 4|2, 6|4 and 8|6, respectively. 
  
  These exceptional modules of the least dimension are lucidly (but with a typo) 
  described in <cit.>. For a=2 and 3, Duflo and Serganova proved Theorem <ref> without considering (4|2; a)-modules of the least dimension. We show that to prove that _x≃_y we should compare ranks of the operators in the adjoint module, not in the ones of the least dimension. 
  
  To show in which modules should one compute the rank of ρ(x) in order to establish if _x≃_y, we do consider these modules of least dimension, see Table (<ref>).
  
  For the Lie superalgebra (A) with a symmetrizable indecomposable Cartan matrix A, the space spanned by
  roots R over the ground field inherits the NIS (-,-) given on (A), see <cit.>. In the
  modular case, however, we have to use a different definition of roots, most lucidly described in
  <cit.>. (Note that the NIS  on (A) induces a non-degenerate symmetric bilinear form on the space spanned by roots over  as
  well.)
  
  Observe that there is no NIS on (n|n), but there is a NIS on (n|n) which has no n|n-dimensional irreducible modules; the (n|n)-module of least dimensional is adjoint.
  
  Recall that a root is called odd if the corresponding root vector is odd; β∈ R_ is isotropic if (β, β) = 0. Two roots α, β∈ R are called orthogonal if (α, β) = 0. 
  
  In addition to Theorem <ref>, Duflo and Serganova proved the following 
  
  [<cit.>: _x for =(m|n) and (a|b), where a≠ b, and (a|2b)]Theorem[<cit.>: _x for =(m|n) and (a|b), where a≠ b, and (a|2b)]
  For  =(m|n) and (a|b), where a≠ b, and (a|2b) and for 
  any set 
  
  {β_1,…,β_k}  of linearly independent mutually orthogonal isotropic roots,
  let
  x_k:=x_β_1 +…+x_β_k, and ρ=𝕀.
  
  Then,
  
  [ ρ_x_k=k   for (m|n) and (m|n), where m≠ n,
  
  2k   for (2m|2n) and (2m+1|2n),;                                                                             ]
  and, respectively,
  
  [ _x_k≃(m-k|n-k) and (m-k|n-k),
  
  (2m-2k|2n-2k) and (2m-2k+1|2n-2k).;                                                                   ]
   
  
  
  ∙ For p=0, the Lie superalgebras (n|n) is the only type of Lie superalgebras with non-invertible indecomposable Cartan matrix, see classification <cit.>. We consider the case of (n|n), the simple subquotient of (n|n), in  <ref>; the answer does not depend on p.
  
  Kac and Wakimoto define () as the maximal cardinality k of the set (<ref>). For =(A) over , the defect () is also equal to
  
  the number () of non-isomorphic DS-homology superalgebras _x.
  Clearly, () (same as def, see  eq. (<ref>)) is independent of the existence of NIS on .
  
  Let us investigate: is there a relation between the defect and the rank of ρ_x in some irreducible representation ρ apart from the one described in eq. (<ref>)? For example, for the simple Lie superalgebras (4|2; a), the values of rank of ρ_x_(), where ρ is either  or the irreducible representation σ of the least dimension, are as follows
  
  
  a   rank(_x_())   rank(σ_x_())   ()
  
  1   8   2   1 
  
  2   8   4   1 
  
  3   8   6   1 
  
  
  
  This table hints to define the defect looking at the adjoint representation instead of the one of the least dimension, but this hint is not as compelling, as the result of table (<ref>). 
  
  For the serial Lie superalgebras with Cartan matrix, both choices of ρ (the tautological and adjoint) are OK, the tautological _x-module in eq. (<ref>) is, however, of superdimension V- ρ_x_k|ρ_x_k, where ρ is the representation in tautological -module V, not in the adjoint one.
  
  For the series , for (4|2; a) for a generic, and for the two exceptional simple Lie superalgebras — (2) and (3) — the irreducible representation of the least dimension is the adjoint one, so to define the rank of x we have to take  (and be happy we do not have to consider a module of greater dimension).
  
  
  
   §.§ When _x≃_y for p>0 and =(A)?
   We want to answer the following questions:
  
  ∙ Q1) Should we take the space spanned by roots over the ground field , same as the space of weights, or should we follow the definition suggested by A. Lebedev, see <cit.>, and consider the inner product in the space spanned by the roots over , and not the one described in <cit.>?
  
  ∙ Q2) What should we take for ρ trying to imitate the approach (<ref>)? (Does the answer to the question “under what conditions on x and y we have _x≃_y?” depend on ρ?)
  
  
  The answers prove, by means of SuperLie code, the following hypothesis that guided us.
  
  
  Hypothesis
  Among all Cartan matrices of (A) 
  
  with the maximal number of 0's on its main diagonal
  select the one, denote it , whose Dynkin-Kac diagram  has the maximal number G_max of “gray” vertices not connected with each other (in terms of Cartan matrices this means that for all indices i,j such that _ii=_jj=0 there is a maximal number of pairs _ij=_ji=0). 
  Our Hypothesis consists of several parts:
  
  1) For any (A), except for (4;a),  the non-isomorphic Lie (super)algebras _x correspond to x=x_β for
  β=β_1+…+β_k, where k≤ G_max and each β_i is one of the roots corresponding to the above “grey”
  vertices not connected with “grey”
  vertices corresponding to the other β_j. Moreover,
      ((A))=((A))=G_max.
  
  
  1e) For =(4;a), we have =G_max+1 whereas ((A))=G_max.
  
  2)  If A is non-invertible, the defect () of the simple subquotient :=(A)^(1)/ is given by one of the following formulas (we were unable to find the pattern)
  
      [                                                  2a) ()=G_max-1, unless G_max=1 in which case ()=G_max=1,;                                                  2b)      ()=G_max, e.g., ((n|n+pk))=((n|n+pk)) for k≠ 0.;                                                      ]
  
  
  Observe that for  =(n|n), we have def()=()≠() in any characteristic p we verified, see  3.
  
  
  
  
  FactWhen _x≃_y For the exceptional simple Lie superalgebra  of the form either (A), or (A)^(1)/, we have _x≃_y if and only if ρ_x=ρ_y, where ρ is the adjoint
  representation of , not in the irreducible -module M of the least dimension, as
    examples show, e.g.,  (<ref>), (<ref>).
  
  
  
  
    
  Our Hypothesis is true over  Duflo and Serganova proved the statement equivalent to the claim of our Hypothesis for Lie superalgebras (A) over  with Cartan matrix A, even non-invertible, as for (a|a), see <cit.>. To see where the idea of the answer comes from, consider examples. 
  
  Series  and . To select the matrix A_m, see eq. (<ref>), we represent the supermatrices — elements of (a|b) — in the alternating format 
  
      _alt=(,,,,…, ,,…, )     if a≤ b
      
      (,,,,…, ,…, )     if a> b.
  
  Then, the elements E_i, i+1 (resp. E_i+1, i) are simple positive (resp. negative) root vectors for all i. Since the neighboring root vectors do not commute, their roots are not orthogonal to each other; take every second root. 
  So, ((a|b))=((a|b))=min(a,b). 
  
  The embeddings 
  
      (a|b)(2a|2b)(2a+1|2b)
   
  give the value of () for  of the  series, see eq. (<ref>): 
  
      (a|b)_x_k=(a-k|b-k) and (N|2n)_x_k=(N-2k|2n-2k).
  
  
  Exceptions: =(4|2; a) for a generic, (2), and (3). For them, the hypothesis describing when _x≃_y is also true; in the hypothesis, ρ is the adjoint representation, see eq. (<ref>). For =(4|2; a) and a=1,2 or 3, we can take any of the two representations: either , or the irreducible representation σ of the least dimension, see eq. (<ref>).
  
  
  
   §.§ Summary of our results
  
  We suggested a simple method for 
  determining the value of the defect of (A). We computed both () and  the Duflo-Serganova homology _x in the two cases: 
  
  (i) over  of characteristic p>0, for each exceptional Lie superalgebra (A)  with indecomposable Cartan matrix A; 
  
  (ii) for simple subquotients of the exceptional modular Lie superalgebras ^(1)(A)/, where A is not
  invertible, e.g., for (n|n+pk) the conjecture is verified for p=0, 2, 3, 5, and n and k  small.
  
  
  Hypothesis <ref> is verified directly, by means of the SuperLie package, see <cit.>. Proving Hypothesis and answering Questions in Subsection <ref> we found out the following.
  
  FactWhere do roots live, which representation determines defect
   Over , the space of roots should be considered over , as it was defined by A. Lebedev, see <cit.>, not over the ground field as the space of weights. For motivations, see Subsections <ref>, <ref>.
  
   To get the correct answer to the question “when _x≃_y?”, we have to take ρ=, not the irreducible representation σ of the least dimension, see, e.g., table (<ref>), and especially table (<ref>): the same rank of σ_x and σ_y might correspond to _x≄_y, whereas rank σ_x≠rank σ_y might correspond to _x≃_y.
  
  
  
  
  
  
  § THE EXCEPTIONAL CASES
  
  
  
  By N(A) we denote (A) corresponding to the Nth Cartan matrix A as listed in <cit.>; recall that A/a|B means that (A)= A|B and ^(1)(A)/=a|B. The odd root vectors are boxed and isotropic roots are underlined. The multiplication tables in _x are obtained with the aid of SuperLie package. After several cases illustrating our answers, we refer the reader to <cit.> for bulky lists of roots and the corresponding root vectors.
  
  
  Notation ⊕_c, see <cit.> This notation is needed to describe
  the following Lie superalgebras or the corresponding DS-homologies 
  
  [              (2,3), (2,6), and (3,3)                             for p=3,; (4; α),  (6, 6),  (7, 6), and (8, 1)                             for p=2.;                                      ]
  This notation describes the case where  and
   are nontrivial central extensions of the Lie (super)algebras 
  and , respectively, and ⊕_c
  — a nontrivial central extension of ⊕ (or, perhaps,
  a more complicated semidirect sum , where  is an ideal) with 1-dimensional center
  spanned by c — is such that the restriction of the extension of
  ⊕ to  gives  and that to  gives .
  
  Consider the 4 Lie superalgebras (A), where p=2, listed in eq. (<ref>) in more details (see <cit.>). Then, (A)_ is of the form
  
  (B)⊕_c(2)≃(B)⊕(X^+,X^-),
  where the
  matrix B is not invertible (so (B) has a grading element d
  and a central element c), and where X^+, X^- and c span the
  Heisenberg Lie algebra (2). The brackets are:
  
      [                                                [^(1)(B), X^±]=0;; [d,X^±]=X^±   for (6, 6), (7,6), and (8,
      1); 
      α X^±   for (3; α);                                                     [X^+,X^-]=c. ]
  
  
  
  The odd part of (A) (at least in two of these four cases)
  consists of two copies of the same (B)-module N, the
  operators _X^± permute these copies, and _X^±^2=0,
  so each of the operators maps one of the copies to the other, and
  this other copy to zero.
  
  
  Notation ⊕_c^d Recall the definition and examples of double extensions, see <cit.>. Actually, in examples in eq. (<ref>), and _x for (7,6), we have double extensions  and  of the Lie (super)algebras 
  and , respectively, and ⊕_c^d is a double extension of ⊕ by means of a central
  element c and an outer derivation d such that the restriction of the extension of ⊕ to  (resp. )
  gives  (resp. ). 
  
  For p≥ 5: (4|2; a) for a≠ 0, -1, (2), and (3) The answer is the same as for p=0, namely: ()=1 and _x is given by table (<ref>); verified for p=5, 7, 11:
  
      (4|2;a)   (2)   (3)
      _x   ^1|0   (2)   (3)
      _x   8   14   16
  
  
  Each of the other exceptional Lie superalgebras (A) with indecomposable Cartan matrix A exists only in characteristics 2, 3 and 5. The two Lie superalgebras 3(2,3) and 1(3,3) (indigenous to p=3) have “the same”[Pretending that the elements of the Cartan matrix are integers, not elements of ; we do the same  describing analogs of Serre relations, see <cit.>.] Cartan matrix as 3(2) and 6(3) (existing for p=0 and any p>3), respectively. Each of the other exceptional superalgebras (A) has no analogs among other exceptions, except for two pairs (2;5)↔(2;3) and
  (5;5)↔(5;3), versions of which we consider one after the other for clarity. Although the algebras in these pairs have “the same” Cartan matrices for p=5 and 3, respectively, their structures are different (same thing with Lie superalgebras and their desuperizations for p=2, e.g., ↔). Let π_i be the ith fundamental weight of the Lie algebra , let R(∑π_i) denote the representation of  with highest weight ∑π_i and the corresponding -module.
  
  (2;5) of 10|12 We have (2; 5)_ = (4) and 
  (2; 5)_ = R(π_1 + π_2)  is an irreducible the  (2; 5)_-module. We consider the following Cartan matrix and basis elements:
  
  [  0 -1; -2  1;    ]   
  the root vectors   the roots
  
  x_1, x_2   α_1,  α_2
  
  x_3=[x_1, x_2], 
  x_4=[x_2, x_2]   α_1+α_2,  2α_2
  
  x_5=[x_2, [x_1, x_2]]   α_1+2α_2
  
  x_6=[[x_1, x_2], [x_2, x_2]
  ]   α_1+3α_2
  
  x_7=[[x_1, x_2],[x_2, [x_1, x_2]]], x_8=[[x_2, x_2],[x_2, [x_1, x_2]]]   2α_1+3α_2,  α_1+4α_2
  
  x_9=[[x_1, x_2],[[x_1, x_2], [x_2, x_2]
   ]]   2α_1+4α_2
  
  x_10=[[x_2, [x_1, x_2]
   ], [[x_1, x_2], [x_2, x_2]]]   2α_1+5α_2
  
   
   
  
  
  
  [        x       _x       _x rank  _x;      x_1      0|2     ^0|2       10;          ]
  We get the same answer as for x_1 in eq. (<ref>) for x_7, x_8, x_10, and x_1+x_10.
  
  To find out which of the four homological linearly independent root vectors x's in table (<ref>) are orthogonal to one another (to compute the defect) we have to
  decide do we consider the inner product in the space generated by the set of roots R over the ground field  or over : unlike weights, which are considered over , the space of roots should be considered over  for several reasons as A. Lebedev taught us, see <cit.>; here is one more reason. 
  
  Since (α_1, 2α_1+5α_2)≡ 0 5, the only possible pair of mutually orthogonal isotropic candidates: x_1 and x_10; but since _x for x=x_1+x_10 is the same as for x=x_1, see table (<ref>), we conclude that ()=1. Therefore, we have to 
  
  consider the inner product in the space of roots over .
  In this case, (α_1, 2α_1+5α_2) = - 10 ≠ 0. Thus x_1 and x_10 are not orthogonal.
  
  
  1(5;5) of = 55|32 We have _ = (11) and _ =R(π_5) as the _-module.
  
  [       x      _x      _x rank _x;     x_3      15     (4)      32;         ]
  We get the same answer as for x_3 in eq. (<ref>) for the other homological elements of the same rank
  
      x_4,x_5,x_6,x_7, 
      x_15, 
      x_17, 
      x_18, 
      x_19, 
      x_20, 
      x_26, 
      x_27, 
      x_28, 
      x_34, 
      x_35, 
      x_39.
  
  Hypothesis <ref> is confirmed, ((5;3))=1.
  
  For p=3 
  (2;3) of 10|8 
  We have _ = (4) and 
  _ = R(2π_2) is an irreducible _-module. We consider the following Cartan matrix and basis elements
  
  [  0 -1; -2  1;    ]   
  the root vectors   the roots
  
  x_1, x_2   α_1,  α_2
  
  x_3=[x_1, x_2],  x_4=[x_2, x_2]   α_1+α_2,  2α_2
  
  x_5=[x_2, [x_1, x_2]]   α_1+2α_2
  
  x_6=[[x_1, x_2],[x_2, x_2]]   α_1+3α_2
  
  x_7=[[x_2, x_2],[x_2,
  [x_1, x_2]]
   ]   α_1+4α_2
  
  x_8=[
  [x_1, x_2],[[x_1, x_2],
  [x_2, x_2]] ]   2α_1+4α_2
  
  
  
  
  [        x       _x       _x  rank _x; x_1, x_7      2|0     ^2|0        8;          ]
  
  
  Since (α_1, α_1+4α_2)=-8, it follows that ()=1, Hypothesis <ref> is confirmed.
  
  
  7(5;3) of = 39|32 
  We have _ = (9)⊕(2) and _ = R(π_4)⊠𝕀 as the _-module.
  
  
  Let M be the irreducible -module with the highest weight (0,0,0,0,1), then M = 18|16; see <cit.>. The answer in eq. (<ref>) allows to consider M as well as the adjoint module.
  
  [                  x                 _x                 _x                 _x               _M x;                x_1               15|8              (1|4)                 24                 10; x=x_4+x_5, x_1+x_3                7|0                (3)                 32                 16;                    ]
  We get the same answer as for x_1 in eq. (<ref>) for the other homological elements of the same rank
  
      x_2, 
      x_7, 
      x_10, 
      x_13, 
      x_15, 
      x_16, 
      x_20, 
      x_21, 
      x_24, 
      x_26, 
      x_27, 
      x_29, 
      x_30, 
      x_31, 
      x_33.
  
  Hypothesis <ref> is confirmed, ((5;3))=2.
  
  
  
  1(1,6) of 21|14 For =(1,6), we have _ = (6) and _ = R(π_3) as the _-module.
  
  [                     x                    _x                    _x                    _x; x_3,  x_9, x_12, x_15                   7|0                   (3)                    14;                       ]
  Since (α_3, 2α_1+4α_2+α_3)=-6≡ 0 3, the only possible pair of mutually orthogonal isotropic candidates: x_3 and x_15; otherwise ()=1. However, rank _x_3+x_15=rank _x_3. Therefore, we have to obey the rule (<ref>). Hypothesis <ref> is confirmed.
  
  
  
  
  2(2,3) of 12/10|14 For =(2,3), we have _ = (3) ⊕(2) and _ = (3)⊠𝕀 as the _-module. Clearly, (^(1)(2,3)/)_ = (3) ⊕(2).
  
  
  
  
  For =(2,3), we have
  [                  x                 _x                 _x                 _x;                x_1                2|4 (1|1)⊕_c(1|1)⊕^1|0                 10;            x_1+x_2                1|3               ^1|3                 11;                    ]
  Hypothesis <ref> is confirmed, ()=2.
  
  For =^(1)(2,3)/, we have
  [       x      _x      _x      _x;     x_1     0|4    ^0|4      10; x_1+x_2     0|4    ^0|4      10;         ]
  Hypothesis <ref> is confirmed, ()=1.
  
  2(2,6) of 36/34|20 For =(2,6), we have _ = (6) and _ = R(π_3) as the _-module. Clearly, (^(1)(2,6)/)_ = (6).
  
  = 2(2,6)
  
  [              x             _x             _x             _x; x_2, x_3, x_4,           16|0    (3)⊕_c^d(3)             20;                ]
  
  
  
  = 2^(1)(2,6)/
  
  [              x             _x             _x             _x; x_2, x_3, x_4,           14|0        (3)⊕(3)             20;                ]
  We get the same answer for the other homological elements of the same rank. Hypothesis <ref> is confirmed, ()=()=1.
  
  
  
  7(3,3) of 23/21|16 
  Let _7:=R(π_3). We have 
  
      (3,3)_=((7) z) d and 
      (3,3)_=(_7)_+⊕ (_7)_-;
  
   the action of d — the outer derivative of (3,3)^(1) — separates the identical (7)-modules _7 by acting on these modules as the scalar multiplication by ±1, as indicated by subscripts, z spans the center of (3,3). 
   
  For = (3,3), we have 
  [              x             _x             _x             _x;           x_1,            9|2 (3)⊕_c^d (1|1)             14;        x_1+x_4            7|0            (3)             16;                ]
  Hypothesis <ref> is confirmed, ()=2.
  
  For = ^(1)(3,3)/, we have 
  [        x       _x       _x       _x;     x_1,      7|2 (3)⊕^0|2       14; x_1+x_4,      7|2 (3)⊕^0|2       14;          ]
  Hypothesis <ref> is confirmed, ()=1.
  
  2(3,6) of 36|40 For =(3,6), we have _ = (8) and _ = R(π_3) as the _-module.
  
  [          x         _x         _x         _x;        x_1      10|14 ^(1)(2,3)/         26;    x_1+x_4        0|4       ^0|4         36;            ]
  We get the same answer for the other homological elements of the same rank. Hypothesis <ref> is confirmed, ((3,6))=2.
  
  
  6(4,3) of 24|26 For =(4,3), we have _ = (6) ⊕(2) and _ = R(π_2) ⊠𝕀 as the _-module.
  
  [                       x                      _x                      _x                      _x; x_1, x_2, x_3, x_4, x_8                     6|8                   (2|2)                      18;  x_12, x_17, x_20, x_22                                                ;        x_1+x_3, x_2+x_4                     0|2                    ^0|2                      24;                         ]
  We get the same answer for the other homological elements of the same rank. Hypothesis <ref> is confirmed, ()=2.
  
  
  13(8,3) of 55|50 We have _ = (4) ⊕(2) and _ = R(π_4) ⊠𝕀 as the _-module.
  
  [                  x                 _x                 _x                 _x; x_2, x_3, x_4, x_5              21|16         ^(1)(2,3)/                 34;   x_2+x_5, x_3+x_5                7|2           (3)⊕^0|2                 48;                    ]
  We get the same answer for the other homological elements of the same rank. Hypothesis <ref> is confirmed, ()=2.
  
  
  2(4,6) of 66|32 We have _ = (12) and _ = R(π_5) as the _-module.
  
  [    x   _x   _x   _x;  x_3 34|0  (6)   32;      ]
  We get the same answer for the other homological elements of the same rank. Hypothesis <ref> is confirmed, ()=1.
  
  
  
  
  
  4(6,6) of 78|64 We have 
  
      (6,6)_= (13) and (6,6)_ = _13:=R(π_6) as the _-module.
  
  
  [                x               _x               _x         rank  _x;              x_1            34|20       ^(1)(2,6)/               34; x_1+x_3, x_1+x_4             14|0          (3)⊕(3)               48;                  ]
  We get the same answer for the other homological elements of the same rank. 
  By Hypothesis <ref>, ()=2, and we are done.
  
  
  5(8,6) of 133|56 We have _ = (7) and _ = R(π_1) as the _-module.
  
  [        x       _x       _x rank  _x;      x_2     77|0 ^(1)(6)/       56;          ]
  We get the same answer for the other homological elements of the same rank. Hypothesis <ref> is confirmed, ()=1. (Recall that (6)=79/77|0 for p= 3, whereas (6)=78 for p≠ 3.)
  
  
  
  
  
   §.§ p=2
   Numbering of vertices of the Dynkin diagram of  type superalgebras goes along the string, starting from the end-vertex of the short branch, the end-vertex connected by an edge with the branching point is the last one.
  
  
  (3;α), where α≠ 0, 1; =10/8|8 The roots of (3;α) are the same as those of (4|2;α) of =9|8 with the same division into even and odd ones.
  
  We consider the following Cartan matrix and the corresponding
  positive root vectors 
  
  [ 0 1 0; 1 0 α; 0 α 0 ]
     
  the root vectors   the roots
  
  x_1,  x_2, x_3,   α_1,  α_2, α_3
  
  x_4=[x_1, x_2],  x_5 =[x_2,  x_3],   α_1+α_2,  α_2+α_3,
  
  x_6=[x_3, [x_1, x_2]]   α_1+α_2+α_3
  
  x_7=[[x_1, x_2], [x_2, x_3]]   α_1+2α_2+α_3
  
  
  
  
  In what follows, let M be an irreducible highest weight -module of the least dimension. 
  
  
  = (3;a), let M  be an irreducible module of M = 4|4, see <cit.>. We have
  
  
   x     _x     _ x    _M x
  
   x_1     ^4|2     5     2
  
  x_2     ^2|0     7     3
  
  x_3     ^4|2     5     2
  
  x_1+x_3     ^2|0     7     4
   
  
  
  
  For =^(1)(3;α)/, the highest weight of M is (0,1,0) (with respect to the
  maximal torus of (3;α)) and M = 8|6, see <cit.>. 
  
  
   x     _x     _ x    _M x
  
   x_1     ^2|2     6     8 
  
   x_2     ^0|0     8     6 
   
   x_3     ^2|2     6     4 
  
   x_1+x_3     ^2|2     6     6 
  
   
  
  
  We see that _M x does not give a conclusive information as to what _x is, unlike _ x. Hypothesis <ref> item 1)  is confirmed: ()=2, item 2) is NOT confirmed ()=2.
  
  
  
  (4;α), where α≠ 0, 1;
  =18|16 The roots of (4;α) are the same as those of (4;α), but divided into even and odd ones. We consider the following Cartan matrix and the
  corresponding positive root vectors
   [ 0 α 0 0; α 0 1 0; 0 1 0 1; 0 0 1 0 ]
    
  x_1, x_2  x_3,  x_4,   α_1,  α_2,  α_3,  α_4
  
  x_5=[x_1, x_2], x_6=[x_1, x_3],  x_7=[x_3, x_4],   α_1+α_2,  α_1+α_3,  α_3+α_4
  
  x_8=[x_3, [x_1,x_2]], x_9=[x_4, [x_1, x_3]],   α_1+α_2+α_3,  α_1+α_3+α_4 
  
  x_10=[[x_1,x_2], [x_1, x_3]], x_11=[[x_1, x_2],[x_3, x_4]]   2α_1+α_2+α_3,  α_1+α_2+α_3+α_4
  
  x_12=[[x_1, x_2], [x_4,[x_1, x_3]]],   2α_1+α_2+α_3+α_4
  
  x_13=[[x_3, [x_1, x_2]],[x_4, [x_1, x_3]]],   2α_1+α_2+2α_3+α_4
  
  x_14=[[x_4, [x_1,x_3]], [[x_1, x_2], [x_1,x_3]]]   3α_1+α_2+2α_3+α_4
  
  x_15=[[[x_1, x_2], [x_1, x_3]],[[x_1, x_2], [x_3,x_4]]]   3α_1+2α_2+2α_3+α_4
   
  
  
  
  Let M be the irreducible -module with highest weight (0,0,0,1), then M = 16|16, see <cit.>.
  Let V be the tautological (3)-module.
  
  
  
  
   x     _x     _ x    _M x    _x
  
   x_1     8|6     10     10    (3|1)
  
   x_3, x_4     8|6     10     8     (3|1)
  
   x_14     8|6     10     16     (3|1)
  
   x_2, x_8, x_9     4|2     14     12
       _x^(3) = 0, _x/_x^(1)=, _x^(1)/_x^(2)=^2|2
  
   x_12     4|2     14     14    _x^(3)=0, _x/_x^(1)=, _x^(1)/_x^(2)=^2|2
  
   x_1+x_3     2|0     16     14     ^2|0 
  
   x_1+x_4     2|0     16     14     ^2|0 
  
   x_2+x_4     2|0     16     14     ^2|0 
  
   
  Hypothesis <ref> is NOT confirmed: ()=3. We see that _M x does not give a conclusive information as to what _x is, unlike _ x.
  
  
  
  (6,1) of = 46|32 We have
  _≃(2; 10)⊕ z and _ is a
  reducible module of the form R(π_4)⊕ R(π_5) with the
  two highest weight vectors: y_5 and 
  
      x_36=[[[x_4,x_5],[x_6,[x_2,x_3]]],
      [[x_3,[x_1,x_2]],[x_6,[x_3,x_4]]]].
   
  
  
  Let M be the irreducible -module with highest weight (1,0,0,0,0,0), then M = 16|11, see <cit.>. The Cartan matrix of (6,1) we consider is that of (6) with the vector of parities of simple
  roots being 111100. 
  
      x   _x   _x   _M x   _x
      x_1,x_2,x_3,x_4,x_10,x_11,x_12,x_13,x_17,
          24|10   22   6   (5|1)
      x_19,x_20,x_25,x_27,x_29,x_31,x_34            
      x_1+x_3,  x_2+x_4   14|0   32   10   (4)
  
  Hypothesis <ref> is confirmed, ()=2. 
  
  (6,6) of = 38|40 
  In this case, (B)≃(6), see eq. (<ref>). The module
  _ is irreducible with the highest weight vector
  
      x_35=[[[x_3,x_6], [x_4,[x_2,x_3]]], [[x_4,x_5], [x_3,[x_1,x_2]]]]
        of weight (0,0,1,0,0,1).
  
  
  
  Let M be the irreducible -module with highest weight (0,0,0,0,0,1), its dimension is M = 15|12, see <cit.>. The Cartan matrix of (6,6) we consider is that of (6) with the vector of parities of simple
  roots being 111111. 
  
      x   _x   _ x   _M x   _x
      x_1   16|18   22   6   (3|3)
      x_1+x_3   6|8   32   10   (2|2)
      x_1+x_3+x_5   0|2   38   12   ^0|2
  
  We get the same answer for the other elements of the same rank. Hypothesis <ref> is confirmed, ()=3.
  
  
  (7,1) of =80/78|54 The Cartan matrix of (7,1) is that of (7) with the parities of simple roots 1111001. 
  
      x   _x   _ x   _x
      x_1   46|20   34   ^(1)_ΠΠ(2|10)
      x_1+x_3   28|2   52   ^(1)_Π(8) ⊕_c^d (1|1)
      x_1+x_3+x_7   26|0   54   _Π^(2)(8)/
  
  
  We get the same answer for the other elements of the same rank. Hypothesis <ref> is confirmed, ()=3.
  
  Consider :=(^(1)(7,1))/. We see that ()=2, hypothesis <ref> is confirmed.
  
      x   _x   _ x   _x
      x_1   44|20   34   ^(2)_ΠΠ(2|10)/
      x_1+x_3   26|2   52   _Π^(2)(8)/⊕^0|2
      x_1+x_3+x_7   26|2   52   _Π^(2)(8)/⊕^0|2
  
  
  (7,6) of =70/68|64  The Cartan matrix of (7,6) we consider is that of (7) with the vector of parities of simple
  roots being 0101010. 
  
      x   _x   _ x   _x
      x_2   36|30   34   ^(1)(6)
      x_2+x_4   18|12   52   ^(1)(4)⊕_c^d(2)
      x_2+x_4+x_6   8|2   62    solvable = 1|2,
               _x^(i)=7|0   if i=1,
      
       1|0   if i=2,
      
       0|0   if i=3,
  
  We get the same answer for the other elements of the same rank. Hypothesis <ref> is confirmed, ()=3. 
  
  
  Consider :=(^(1)(7,6))/. We see that ()=3, hypothesis <ref> is confirmed.
  
      x   _x   _ x   _x
      x_2   34|30   34   ^(2)(6)/
      x_2+x_4   16|12   52   ^(2)(4)/⊕^2|0
      x_2+x_4+x_6   6|2   62   ^6|2
  
  
  (7,7) of =64/62|70  The Cartan matrix of (7,7) we consider is that of (7) with the vector of parities of simple
  roots being 1111111.  
  
      x   _x   _ x   _x
      x_1   30|36   34   _ΠΠ^(1)(6|6)
      x_1+x_3   12|18   52   _ΠΠ^(1)(4|4)⊕_c^d(1|1)
      x_1+x_3+x_5   2|8   62    solvable,
               =1|2, _x^(i)=1|6   if i=1,
      
       1|0   if i=2,
      
       0|0   if i=3,
      x_1+x_3+x_5+x_7   0|6   64   ^0|6
  
  We get the same answer for the other elements of the same rank. Hypothesis <ref> is confirmed, ()=3.
  
  Consider :=^(1)(7,7)/. We see that ()=3, hypothesis <ref> is confirmed.
  
  
      x   _x   _ x   _x
      x_1   28|36   34   _ΠΠ^(2)(6|6)/
      x_1+x_3   10|18   52   _ΠΠ^(2)(4|4)/⊕^0|2
      x_1+x_3+x_5   0|8   62   ^0|8
      x_1+x_3+x_5+x_7   0|8   62   ^0|8
  
  
  
  (8,1) of =136|112 We have (see eq. (<ref>)) 
  (B)≃(7). The Cartan matrix of 𝔢(8,1) we consider is that of (8) with the vector of parities of simple
  roots being 11001111. 
  
      x   _x   _ x   _x
      x_1   78|54   58   ^(1)(7,1)/
      x_1+x_5   44|20   92   ^(2)_ΠΠ(2|10)/
      x_1+x_5+x_7   26|2   110   ^(2)_Π(8)/⊕^0|2
      x_1+x_5+x_7+x_8   6|2   120   ^6|2
  
  We get the same answer for the other elements of the same rank. By Hypothesis <ref>, ()=4. 
  
  
  (8,8) of =120|128 In the -grading
  with the Cartan matrix with the parities of simple roots
  00000001 and with e_8^±=± 1 and e_i^±=0 for
  i≠ 8, we have _0=(8). 
  
  The Lie algebra _ is isomorphic to
  _Π^(2)(16) d, where
  d=E_6,6+…+E_13,13, and _ is an irreducible
  _-module with the highest weight element
  x_120 of weight (1,0,…,0) with respect to h_1,…,h_8;
  _ also possesses a lowest weight vector. The Cartan matrix of (8,8) we consider is that of (8) with the vector of parities of simple
  roots being 11111111. 
  
      x   _x   _ x   _x
      x_1   62|70   58   ^(1)(7,7)/
      x_1+x_3   28|36   92   _ΠΠ^(2)(6|6)/
      x_1+x_3+x_5   10|18   110   _ΠΠ^(2)(4|4)/⊕^0|2
      x_1+x_3+x_5+x_7   0|8   120   ^0|8
  
  
  We get the same answer for the other elements of the same rank. By Hypothesis <ref>, ()=4.
  
  
  
  § (N|N) FOR P=0, 2,3,5 (CHECKED FOR N=2,3, 4)
  
  This case was not considered in <cit.>. For it, the Hypothesis on the value of defect  is also true; the form of the answer differs from that in eq. (<ref>). We consider the alternating format (so all Chevalley generators are odd).
  
  For any x∈{x_1, x_3}, we have
  
  [                                     _x,  _x _x_1 + x_3,  _x_1+x_3;                 (2|2)         5,  _x =(1|1)       7, _x_1+x_3 = 0;                 (2|2)   6, _x = (1|1)≃(0|2)   7,  _x_1+x_3 = ^0|1;                 (2|2)         6,  _x = ^0|2   6,  _x_1+x_3 = ^0|2;                       ]
  
  
  For any x∈{x_1, x_3, x_5}, and y=x_1+x_3, and z=x_1+x_3+x_5, we have
  
  [                         _x,  _x         _y,  _y         _z,  _z;           (3|3)   9,  _x =(2|2)  15, _y = (1|1)       17, _z =0;           (3|3)  10, _x = (2|2) 16,  _y = (0|2)   17, _z = ^0|1;           (3|3)  10,  _x =(2|2)  16,  _y = ^0|2   16, _z = ^0|2;                 ]
  
  
  For any x∈{x_1, x_3, x_5, x_7}, and y=x_1+x_3, and z=x_1+x_3+x_5, and v=x_1+x_3+x_5+x_7, we have
  
  [                         _x,  _x         _y,  _y         _z,  _z          _v, _v;           (4|4)  13,  _x =(3|3)  23, _y = (2|2)  29, _z = (1|1)        31, _v=0;           (4|4)  14, _x = (3|3) 24,  _y = (2|2)  30, _z = (1|1)     31, _v=^0|1;           (4|4)  14,  _x =(3|3) 24,  _y = (2|2)   30, _z = ^0|2   30, _v = ^0|2;                 ]
  
  
  
  
  
  
  
  § (N|N + PK) FOR P=2,3,5 AND SMALL NON-ZERO N, K
  
  
  
  
  
   §.§ p=2
  
  For x = x_1, and y = x_1 + x_3, we have
  [                     _x,  _x       _y,  _y;         (2|4)    10,  (1|3)      16,  (2);         (2|4)    10,  (1|3) 16,  ^2|0≃(2);         (2|6)    14,  (1|5)      24,  (5);         (2|6)    14,  (1|5)      24,  (4);         (2|8)    18,  (1|7)      32,  (6);         (2|8)    18,  (1|7)      32,  (6);               ]
  
  
   
  For x = x_1, y = x_1 + x_3, and z=x_1+x_3+x_5 we have
  [                     _x,  _x       _y,  _y       _z,  _z;         (3|5)    14,  (2|4)    24,  (1|3)      30,  (2);         (3|5)    14,  (2|4)    24,  (1|3) 30,  ^2|0≃(2);         (3|7)    18,  (2|6)    32,  (1|4)      42,  (4);         (3|7)    18,  (2|6)    32,  (1|5)      42,  (4);         (3|9)    22,  (2|8)    50,  (1|7)      54,  (6);         (3|9)    22,  (2|8)    40,  (1|7)      54,  (6);               ]
  
  
  For x = x_1, y = x_1 + x_3, z=x_1+x_3+x_5, and v=x_1+x_3+x_5+x_7 we have
  [                     _x,  _x       _y,  _y       _z,  _z       _v,  _v;         (4|6)    18,  (3|5)    32,  (2|4)    42,  (1|3)      48,  (2);         (4|6)    18,  (3|5)    32,  (2|4)    42,  (1|3) 48,  ^2|0≃(2);         (4|8)    22,  (3|7)    40,  (2|6)    54,  (1|5)      64,  (4);         (4|8)    22,  (3|7)    40,  (2|6)    54,  (1|5)      64,  (4);        (4|10)    26,  (3|9)    48,  (2|8)    66,  (1|7)      80,  (6);        (4|10)    26,  (3|9)    48,  (2|8)    66,  (1|7)      80,  (6);               ]
  
  
  
  
  
  
   §.§ p=3
  
  
  For x = x_1, and y = x_1 + x_3, we have
  [                 _x,  _x     _y,  _y;       (2|5)  12,  (1|4)    20,  (3);       (2|5)  12,  (1|4)    20,  (3);       (2|8)  18,  (1|7)    32,  (6);       (2|8)  18,  (1|7)    32,  (6);      (2|11) 24,  (1|10)    44,  (9);      (2|11) 24,  (1|10)    44,  (9);             ]
  
  
  
  For x = x_1, y = x_1 + x_3, and z=x_1+x_3+x_5, we have
  [                 _x,  _x     _y,  _y     _z,  _z;       (3|6)  16,  (2|5)  28,  (1|4)    36,  (3);       (3|6)  16,  (2|5)  28,  (1|4)    36,  (3);       (3|9)  22,  (2|8)  40,  (1|7)    54,  (6);       (3|9)  22,  (2|8)  40,  (1|7)    54,  (6);      (3|12) 28,  (2|11) 52,  (1|10)    72,  (9);      (3|12) 28,  (2|11) 52,  (1|10)    72,  (9);             ]
  
  
  For x = x_1, y = x_1 + x_3, z=x_1+x_3+x_5, and v=x_1+x_3+x_5+x_7, we have
  [               _x,  _x    _y,  _y    _z,  _z    _v,  _v;      (4|7) 20,  (3|6) 36,  (2|5) 48,  (1|4)   56,  (3);      (4|7) 20,  (3|6) 36,  (2|5) 48,  (1|4)   56,  (3);     (4|10) 26,  (3|9) 48,  (2|8) 66,  (1|7)   80,  (6);     (4|10) 26,  (3|9) 48,  (2|8) 66,  (1|7)   80,  (6);            ]
  
  
  
  
  
   §.§ p=5
  
  For x = x_1, we have
  [             _x,  _x;     (1|6)  12,  (5);     (1|6)  12,  (5);    (1|11) 22,  (10);    (1|11) 22,  (10);           ]
  
  
  For x = x_1, and y = x_1 + x_3, we have
  [                 _x,  _x     _y,  _y;       (2|7)  16,  (1|6)    28,  (5);       (2|7)  16,  (1|6)    28,  (5);      (2|12) 26,  (1|11)   48,  (10);      (2|12) 26,  (1|11)   48,  (10);             ]
  
  
  For x = x_1, y = x_1 + x_3, and z=x_1+x_3+x_5, we have
  [               _x,  _x    _y,  _y    _z,  _z;      (3|8) 20,  (2|7) 36,  (1|6)   48,  (5);      (3|8) 20,  (2|7) 36,  (1|6)   48,  (5);            ]
  
  
  For x = x_1, y = x_1 + x_3, z=x_1+x_3+x_5, and v=x_1+x_3+x_5+x_7, we have
  [               _x,  _x    _y,  _y    _z,  _z    _v,  _v;      (4|9) 24,  (3|8) 44,  (2|7) 60,  (1|6)   72,  (5);      (4|9) 24,  (3|8) 44,  (2|7) 60,  (1|6)   72,  (5);            ]
  
  
  
  
  
  
  
  § COMMENTS
  
  
  Two types of Lie superalgebras The set of simple -graded Lie (super)algebras of finite dimension or of polynomial growth and their deformations is a disjoint (at least, if p>3) union of two subsets: 
  
  
    (S)with a symmetric set of roots relative the maximal torus,
  
  i.e., with every root α there is a root -α of the same multiplicity;
   
  
    (N)with a non-symmetric set of roots.
  
  
  Some (or rather MOST) of the methods used to investigate Lie (super)algebras of type (S) rely on the existence of Casimir elements (in most cases, just one (degree-2) Casimir suffices) and symmetry of the root lattice. 
  
  In the study of Lie (super)algebras of type (N) — Lie (super)algebras of vector fields with polynomial or formal coefficients (briefly referred to as vectorial Lie (super)algebras) and modules over them, these Casimirs are completely or partly absent[Such as the possibility to use the center of the universal enveloping algebra U() which is trivial. However, in <cit.>, Serganova showed that, at least for =(n), the superalgebra U(), whose center is trivial, should be replaced by U():=U()/(U()), whose center is sufficiently big, where (A) is the radical of the algebra A. It is very tempting to investigate applicability of Serganova's idea to other Lie (super)algebras  with trivial center of U(). And with non-trivial centers as well.], and hence can not be used. So, the problems of representation theory of the non-symmetric" (in particular, vectorial) Lie (super)algebras seem to be much more difficult than for the symmetric Lie (super)algebras.
  
  Fortunately, over , the description of irreducible continuous (with respect to the natural (x)-adic topology) modules over simple vectorial Lie algebras is very simple — modulo representation theory of finite-dimensional simple Lie algebras; superization is similar, see <cit.>. 
  
  Two types of homological elements Another tool for the study of Lie (super)algebras is  certain homology relative an element x. Here we considered an odd x such that x^2=2.
  
  For (N)-type simple modular Lie algebras for p>2, one considers the sandwich elements x, i.e., such that (_x)^2=0. S. Kirillov proved that the normalizer of the sandwich subalgebra[In a totally different setting and over , the term sandwich algebra is used in <cit.>, causing confusion.] is the maximal subalgebra for p>3, see <cit.>. 
  
  On inhomogeneous ad-homological elements when p=2 We say that a non-zero x∈ is ad-homological if (_x)^2=0. For =(2|6),
   inhomogeneous (with respect to parity) elements x=x_1+x_3+x_5 and y=x_1+x_3+x_5+x_7 are ad-homological. We get _x = (2) and _y = 0, where __x = 30, and __y = 32. The meaning of ad-homological elements and their homology (when p=2) is unknown. Observe that  nobody computed the homology  corresponding to sandwiches; their meaning is also unknown.   
  
  
  Both sandwiches and DS-homology lead to what is called support varieties. There are several non-equivalent definitions of these varieties: compare <cit.> with <cit.>. 
  
  Other definitions of the defect In <cit.>, there is given another definition of defect def(), equivalent to the above one () for the simple (relatives of) Lie superalgebras  with Cartan matrix over , but with a wider range of application:
  
  def():= Krull dim(H^(,_;))  for the trivial -module .
  
  
  In <cit.>, it is mentioned that this definition is applicable to Lie superalgebras without Cartan matrix, such as (n), (n) and their (not necessarily simple) relatives, and (n|n). 
  
  We observe that, moreover, the invariant (<ref>) is meaningful for any /2-grading of any Lie superalgebra and even for /2-graded Lie algebras. A priori, this cohomology is a supercommutative superalgebra whose Krull dimension was recently defined, as on cue, see <cit.>. 
  
  
  For =(n) and (n), and their relatives, def() was computed in <cit.>, but _x was not computed for any x. For (n), and its simple subquotient, _x is computed in <cit.>. 
  
  The third definition of defect —  — is given by formula (<ref>).
  
  
  
  
  Acknowledgements. For the possibility to conduct
  difficult computations of this research we are grateful to M. Al Barwani, Director of the High Performance
  Computing resources at NYUAD. We thank E. Herscovich for helpful comments. D.L. was partly supported by the grant AD 065 NYUAD.
  A. K. was partly supported by the QuantiXLie Centre of Excellence, a project cofinanced by the Croatian Government and European
  Union through the European Regional Development Fund — the Competitiveness and Cohesion Operational Programme (KK.01.1.1.01.0004).
  
  
  
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