Download A family of simple Lie algebras in characteristic two

A family of simple Lie algebras in characteristic two
G. Jurman
Centre for Mathematics and its Applications
Australian National University
Canberra, ACT 0200
Australia
E-mail: [email protected]
In this paper we describe a family of simple Lie algebras defined over a field of characteristic two depending on two integer parameters. We also describe their loop algebras.
Key Words: Simple Lie algebras, Zassenhaus algebras.
1. INTRODUCTION
As reported by A.I. Kostrikin in his paper [21], the very first algebra which
distinguished modular (i.e. over fields of positive characteristic p) Lie algebra
theory from the classical one was the Witt algebra W (1 : k ), where k is a power
of p. This algebra is a generalization due to H. Zassenhaus [31] in the thirties
of an analogous structure defined by E. Witt over the integers. This algebra is
graded over the elementary abelian additive group of the field Fk and, for p > 2,
it is simple. When the characteristic is two W (1 : k ) has exactly one non-trivial
ideal, namely its derived subalgebra Zk . The simple object is called a Zassenhaus
algebra. In characteristic two, a Zassenhaus algebra admits a non-singular outer
derivation, which is quite an important feature (see [3]).
In the following years, further generalizations of the above structure led to the
construction of other families of simple Lie algebras, called algebras of Cartan type
(see [29]), namely the generalized Jacobson-Witt algebras, the special algebras,
the hamiltonian algebras and the contact algebras. Kostrikin and I.R. Shafarevich
during the sixties conjectured that, apart from the small characteristic case, every
finite-dimensional simple Lie algebra over an algebraically closed field is either
classical or of Cartan type. Recently this was proved true for p > 7 by H. Strade
The author is a member of INdAM-GNSAGA. He is grateful for his help to A. Caranti, advisor of
the doctoral dissertation [16] this work is based on. The author is also grateful to M.F. Newman for
making useful suggestions and reading various versions of this paper.
1
2
JURMAN
and R.L. Wilson in [30]. For small characteristic, the corresponding result does
not hold: in fact, several families of algebras not included in the above list have
been found, and the classification problem in the small characteristic case still
remains an open problem. Kostrikin has said that the classification of simple Lie
algebras in characteristic two is a very hard problem, due to the existence of many
simple objects. For some examples, see for instance [4], [5], [6], [14], [19], [22].
In this paper, we show a new family of simple Lie algebras in characteristic
two depending on two integer parameters, built by means of a “doubling process”
starting from Zassenhaus algebras: we will call them Bi-Zassenhaus algebras and
we denote them by B(g; h). After a background section on Zassenhaus algebras,
we give a construction for Bi-Zassenhaus algebras and we prove they are simple.
Then we investigate their second cohomology groups and central extensions: an
important topic in its own, this will be needed when dealing with the almost finite
presentation of the Bi-Zassenhaus loop algebras, described in [17]. We also show
that the family of Bi-Zassenhaus algebras does not coincide with any known family
of algebras, although occasional coincidences may occur. The following two
sections are dedicated to the construction of two further gradings of these simple
algebras: the former is used to define the algebras over the prime field, while the
latter is used to build the twisted loop algebras, which is the topic of the last section.
We then discuss some invariants of Bi-Zassenhaus loop algebras which show that
they are different from already known graded Lie algebras of maximal class; we
conclude by describing their cohomology and their central extensions. The family
of Bi-Zassenhaus loop algebras plays an important rôle in the classification [18]
of infinite-dimensional modular N-graded Lie algebras of maximal class. In the
odd characteristic case, every infinite-dimensional modular N-graded Lie algebra
of maximal class (generated by its first homogeneous component) can be built
starting from an Albert-Frank-Shalev algebra via some suitable constructions, (see
A.Caranti and M.F. Newman [9]). The family of Albert-Frank-Shalev algebras
(AFS, for short) was built by A. Shalev [28] by looping simple algebras constructed
in the fifties by A.A. Albert and M.S. Frank in [1]. It may be interesting to
observe that the Albert-Frank and the Zassenhaus algebras are isomorphic when
the characteristic is two. In characteristic two, a similar classification result holds
if we are allowed to start from either an AFS-algebra or a Bi-Zassenhaus loop
algebra. In the latter half of this paper we show the construction of the BiZassenhaus loop algebras (Bl , for short) by using a method already employed by
Caranti, S. Mattarei and Newman in [8].
A significant amount of computational work was required throughout the whole
study, although results have been proved in full generality and none of them relies
on such computations. Use of mathematical software has been indispensable: the
p-Quotient program [15] of the Australian National University and GAP [26] have
provided the examples for the construction of the algebras involved. It is perhaps
worth noting that the first suggestion of the existence of some loop algebras
different from the AFS ones came from machine computation: the following
BI-ZASSENHAUS ALGEBRAS
3
investigation led to the construction of the family of simple algebras described
here.
2. THE ZASSENHAUS ALGEBRAS
We start by recalling some facts about Zassenhaus algebras.
If k = pr for some p > 0, the Fk -vector space
W (1 : k) = he : 2 Fk i
becomes a Lie algebra by extending by bilinearity the following bracket product
among basis elements:
[e ; e ] = ( , )e+ :
In characteristic p > 2, the resulting algebra is simple, while in the remaining
case there exists a unique non-trivial ideal
Zk = he : 2 F?k i ;
where by F? we denote the non-zero elements of the field. It is immediate to check
that the algebra Zk is simple. The adjoint map of the element e0 in W (1 : k ) is
an outer derivation D for Zk .
In both cases, we will call Zassenhaus algebra the simple object.
The algebra W (1 : k ) can be differently graded by applying the basis transformation
8
>
>
>
>
>
>
<
X
e
2
F
X k
yj = , k,2,j e
>
X
2Fk
>
>
>
>
y
=
,
e ;
>
k
,
2
:
2Fk
whose inverse is
y,1 = e0 +
(1)
8
>
< 0
e = y,1 + yk,2
kX
,2
e
=
i+1 yi + yk,2 ;
>
: i=,1
and by using the identity
X
2Fk
z = ,Z=(k,1)Z (z; 0) ;
(2)
4
JURMAN
where the map delta is a generalization of the Kronecker map, and for a ring A it
is defined as
x = y in A
A (x; y) = 10 ifotherwise
;
when A is not specified, we understand it as Z.
Under the map (1), the space W (1 : k ) gets a new basis
W (1 : k) = hyi : ,1 i k , 2i
and the Lie product now reads as
[yi ; yj ] = ai;j yi+j ;
where the coefficients ai;j are defined as
+j , i+j ;
ai;j = i + jj + 1 , i + ji + 1 = ii +
1
j+1
and they satisfy the properties stated in the following lemma, which is Lemma 1.5
of [2]:
Lemma 2.1. Let i; j; l be arbitrary integers, ai;j defined as above. Then the
following identities hold:
1. ai;i = 0 and ai;j = ,ai;j ; and a,1;j = 1 for all j
2. If fi; j g 6= f,1; 0g, then ai;j = ai,1;j + ai;j ,1 .
3. ai;j ai+j;l + aj;l aj +l;i + al;i al+i;j = 0.
4. aj;l ai;j +l = ai;j aj +l;i + ai;l aj;i+l .
0.
In this new basis, the algebra W (1 : k ) is defined over the prime field and it is
graded over the integers by giving to every basis element yi its index as weight.
Furthermore, it is possible to get a grading on the additive cyclic group Ck,1 , too
by declaring y,1 and yk,2 both of weight k , 1. With respect to this new basis,
the simple ideal in characteristic two becomes
Zk = hyi : ,1 i k , 2i ;
while the derivation reads as
D = ad (y,1 + yk,2 ) :
In order to investigate the behaviour of D, we recall Lucas’ Theorem (see [23],
[20]) to evaluate binomial coefficients over fields of positive characteristic: if
5
BI-ZASSENHAUS ALGEBRAS
n
X
n
X
i
a = ai p and b = bi pi is the p-adic expansion of the two integers a
i=0
i=0
and b, then we have the equivalence
n a
a Y
i
b i=0 bi
(mod p) :
Note that we can safely ignore signs here since p = 2.
For k = 2r we can show that
ai;j = 0
for i + j
< ,1 or i + j > k , 3 ;
(3)
where the former case is immediate since it happens only for i = j = ,1, while
for the latter one we write i + j = k , 2 + for 0 k , 2 and compute
ai;j =
=
=
=
i+j+1 + i+j +1
i j i+j+1 + i+j +1
i i+1
i+j+2
i + 1
k+
i+1
g+h
20 i+1
0 (mod 2) ;
(mod 2)
since j k , 2 and then < i + 1 < k . Moreover, if yi is a basis element for
Zk , we have that
yi D = ai;,1 yi,1 + ai;k,2 yk+i,2 ;
in fact, we have that ai;,1 = 1 , (i; ,1), while by using the properties of
binomial coefficients and Lucas’ Theorem we can evaluate the latter coefficient as
ai;k,2 = i + kk ,, 22 + 1 , i + k ,i 2 + 1
k
+
i
,
1
k
+
i
,
1
=
k,2 , k,1 i,1
k+i,1
= k+
k,2 + k,1
6
JURMAN
+i
= kk ,
1
r
= 2r,1 + 22r,+2 +i + 1
= (i; ,1) :
This means that D cycles on the elements of the basis as
yk,3 7! yk,4 7! : : : 7! y1 7! y0 7! y,1 7! yk,3 :
3. THE BI-ZASSENHAUS ALGEBRAS
We can now begin the construction of the new family.
Let g 2, h 1 be positive integers. Define k = 2g+h and the vector space B(g; h) over Fk of dimension 2k with basis
= 2g , 1. Build
fe(i;): (i; ) 2 F2 Fk g :
Define on the basis elements a bracket operation
[e(i;) ; e(j;)] = (1+ij(,1) + 1+ij(,1) )e(i+j;+) ;
(4)
and extend it by bilinearity to the whole of B(g; h). Note that, in expanded form,
it reads
[e(0;) ; e(0;)] = ( + )e(0;+)
[e(0;) ; e(1;)] = ( + )e(1;+)
[e(1;) ; e(1;)] = ( + )e(0;+) :
Endowed with (4), B(g; h) becomes a Lie algebra over Fk . Actually, since the
characteristic of the underlying field is two, the condition [x; x] = 0 immediately
follows, so we have only to show that the Jacobi identity holds. Moreover, if at
least two elements out of three have zero first index, then the Lie product coincides
with the Zassenhaus algebra one, so the check is superfluous. Hence we are left
with only two identities to check:
J 1 = [e(0;) ; e(1;); e(1; )] + [e(1;); e(1; ); e(0;) ] + [e(1; ); e(0;) ; e(1;)]
J 2 = [e(1;) ; e(1;); e(1; )] + [e(1;); e(1; ); e(1;) ] + [e(1; ); e(1;) ; e(1;)] ;
that expand as
J 1 = ( + )[e(1;+) ; e(1; )] + ( + ) [e(0;+ ); e(0;)]
7
BI-ZASSENHAUS ALGEBRAS
+( + )[e(1; +); e(1;) ]
= (( + ) (( + ) + ) + ( + ) ( + + )+
+ ( + ) (( + ) + ))) e(0;++ )
,
= ( + )+1 + + + +1 + + + +
+ +1 + + ( + )+1 + + e(0;++ )
,
= +1 + +1 + + + +1 + + + +
+ +1 + + +1 + +1 + + e(0;++ )
= 0;
J 2 = ( + )[e(0;+) ; e(1; )] + ( + )[e(0;+ ); e(1;)] +
+( + )[e(0; +); e(1;)]
= (( + )( + + ) + ( + )( + + )+
+ ( + )( + + )) e(1;++ )
= 0:
So B(g; h) is a Lie algebra; it includes a copy of W (1 : k ) as the subalgebra
generated by the elements of the basis with zero first index. The algebra B(g; h)
has a grading over the abelian additive group of F2 Fk obtained by weighting
every basis element by its index:
B(g; h) =
M
(i;)2F2 Fk
B(g; h)(i;) =
M
(i;)2F2 Fk
Fk e(i;) :
The subspace
B(g; h) = he(i;) : (i; ) 2 F2 F?k i
is a non-trivial ideal of the algebra B(g; h): now look at it as a Lie algebra on its
own, and let I be a non-zero ideal of B(g; h). We can write any non-zero element
in I as
X
=
2, F?k
(A e(0;) + B e(1;) ) ;
where A ; B 2 Fk are not both zero for every 2 ,
then choose an index 2 , and compute
I 3 = [; e(0; )] =
=
X
2, F?k
F?k . If Card(, ) > 1,
(A ( + )e(0;+ ) + B ( + )e(1;+ ))
X
2 F?k
(A e(0;) + B e(1;) ) ;
8
JURMAN
where
Card( ) = Card(f + j 2 , , f gg) = Card(, ) , 1 ;
so we can always assume , = fg and then write as Ae(0;) + Be(1;) . If
B = 0, then the ideal I includes a basis element e(0;) . Otherwise consider the
following polynomial in Fk [t]:
f (t) = B 2 (t + ),1 + A2 :
It has degree , 1 = 2g , 2 > 0, since g > 1, so it has at most , 1 distinct
roots in Fk . As h 1, then Card(Fk , f0; g) = k , 2 > , 1 and thus there
exists 2 Fk , f0; g such that f ( ) 6= 0. Hence the linear system
Ax + B ( + ( + ) )y = 1
Bx + Ay
= 0
has an unique solution (C; D), since its discriminant is
element = Ce(0;+ ) + De(1;+ ) and compute
f ( ) 6= 0.
Define the
I 3 [; ] = 1 e(0;) + 0 e(1;) = e(0;) :
Thus I always includes an element of the basis with zero first index, say e(0;) .
But then
; 1e
I 3 e(0;) (i;+ ) = e(i; )
8(i; ) 2 F2 F?k :
So I = B(g; h) is a simple Lie algebra, of dimension d = 2(k , 1) = 2g+h+1 , 2:
we call it Bi-Zassenhaus algebra.
We now move towards the construction of the central extensions of the BiZassenhaus algebras. A first step is to describe their derivations.
On every Bi-Zassenhaus algebra act the two natural outer derivations D =
ad e(0;0) and E = ad e(1;0)
e(i;) D = e(i;) e(i;) E = 1+i(,1) e(1,i;) ;
and two more Ft (t = 0; 1) obtained by linear extension of the maps acting on the
basis elements as
e(i;) Ft = ie(i+t;) ;
where i + t is obviously to be taken modulo two. By using induction and equation (4), the action of the 2-powers of these and of the adjoint maps is the following,
9
BI-ZASSENHAUS ALGEBRAS
for s > 0:
e(i;) Ft2
e(i;) E 2s
e(i;) D2s
e(i;) (ad e(l; ))2s
=
=
=
=
i(i + t)e(i;) ;
2s g, e(i;) ;
2s e(i;) ;
s,
((1+l(,1) + 1+l(,1) ))2 e(i;) :
+
1
1
This implies that, for s > 0,
Ft2 = (1 , t)Ft ;
s
E
= E2 ;
D = E 2h ;
s
s
,l h
s,
s h
(ad e(l; ))2 = E 2
+ 2 (1+l(,1)) E 2 :
2g+h+s
(5)
+1
+(1
)( +1)
1
+
In particular, among all the 2-powers of D and E only g + h +1 of them represent
s
distinct derivations (for instance, the powers E 2 for 0 s g + h). Furthermore,
F1 is nilpotent and F0 is unipotent. It is actually possible to describe the whole
algebra Der(B(g; h)) of derivation of a Bi-Zassenhaus algebra. By the results
in [29, Ch.3, Prop.4.4, Th.4.5], the space Der(B(g; h)) is (F2 Fk )-graded by
Der(B(g; h))(t; ) = f 2 Der(B(g; h)): (B(g; h)(i;) ) B(g; h)(i+t;+ ) g :
The latter condition can be expressed by introducing, for every (t; ) 2 F2 Fk ,
the maps (t; ) : F2 Fk ! Fk such that (e(i;) ) = (t; ) e(i+t;+ ) . In this
notation, the Leibniz rule for derivation applied to two basis elements e(i;) , e(j; )
reads as
1+ij(,1) + 1+ij(,1) (t; )(i + j; + ) +
( + )1+j(i+t)(,1) + 1+j(i+t)(,1) (t; )(i; ) +
1+i(j+t)(,1) + ( + )1+i(j+t)(,1) (t; )(j; ) = 0 ;
(6)
where i + t and j + t are taken modulo two. When 6= 0, equation (6) in the four
cases fi = 1; j = 0; = g, fi = j = 0; = + g, fi = 1; j = 0; = = g
and fi = j = 0g gives the relations
(t; )(1; ) =
8
>
<
>
:
(t; )(0; )
when t = 0
+ (0; )
+ (t; )
when t = 1 ;
10
JURMAN
(t; )(0; ) = (t; )(1; ) = 0 ;
+ , (0; ) + (0; ) :
(t; ) (0; + ) = +
(t; )
(t; )
+
Since (t; ) (0; ) is a function from the finite field Fk to itself, it can be represented
as a polynomial in Fk [x] of degree at most k , 1. Then the above conditions
imply that (t; ) (i; ) is a multiple of 1+it(,1) + 1+it(,1) . This means that,
for 6= 0, Der(B(g; h))(t; ) = had e(t; ) i. When (t; ) = (0; 0), the relation (6)
gives
(0;0)(i + j; + ) = (0;0) (i; ) + (0;0)(j; ) :
This implies that (0;0) (0; ) is linear over F2 , while (0;0) (1; ) + (0;0) (0; )
is constant. By [25, Th.9.4.4], every Fs -linear operator over Fsn can be uniquely
represented as a polynomial
nX
,1
al xql over Fsn : thus (0;0) (0; ) is a Fk -linear
l=0
l +h,1
combination of f2 ggl=0
. Meanwhile, the above cited constant part can be
expressed by a multiple of the derivation F0 . Summarizing this, we can write
Der(B(g; h))(0;0) = hD2l : 0 l g + h , 1i hF0i. Finally, for (t; ) = (1; 0)
the equation (6) again gives linearity for (1;0) (0; ), with the additional condi-
+ (0; + ) = (1; ) + (1; ), which restricts the
(1;0)
(1;0)
+ (1;0)
possible choices of the maps (1;0) (i; ) to multiples of 1+i(,1) or i. Thus
Der(B(g; h))(1;0) = hE i hF1 i. So, the graded Lie algebra of derivations of a
Bi-Zassenhaus algebra has dimension 2g+h+1 + g + h + 1 and it is generated by
tion
Der(B(g; h)) = hE 2 ; Ft : s = 0 : : : g + h; t = 0; 1i ad B(g; h) :
s
The Lie products among its basis elements which are not immediate can be
obtained by the equations (5) and are summarized in the following table, for
s > 0:
[E; F0 ]
[E; sF1 ]
[E 2s ; F0 ]
[E 2 ; F1 ]
[F0 ; F1 ]
=
=
=
=
=
E; h
[F0 ; ad e(l; )]
D = E 2 ; [E; sad e(l; )]
0;
[E 2 ; ad e(l; )]
0;
[F1 ; ad e(l; )]
F1 :
+1
=
=
=
=
lad e(l; ) ;
1+s lg(,,1) ad e(1,l; ) ;
2 ad e(l; ) ;
lad e(1,l; ) ;
+
1
Then Der(B(g; h))(1) = Der(B(g; h))r = hE; D; F1 i ad B(g; h) for r 1,
while Der(B(g; h))(2) = hDi ad B(g; h) and Der(B(g; h))(s) = ad B(g; h)
for s 3.
By a classical result of Zassenhaus (see [29, Ch.1, Th.7.9]), the existence of an
outer derivation implies that the Killing form of the algebra is degenerate. Does
BI-ZASSENHAUS ALGEBRAS
11
B (g; h) admit any nondegenerate associative form? We first recall that, for any
Lie algebra L over a field F , a symmetric bilinear form : L L ! F is called
associative if ([x; y ]; z ) = (x; [y; z ]) for every x; y; z 2 L and nondegenerate
if its radical Rad() = fx 2 L: (x; y ) = 0 8y 2 Lg is the zero subspace. By
linearity, (0; ) = 0, so we can write
0 =
=
=
=
(0; e(j;) )
([e(i;) ; e(i;) ]; e(j;))
(e(i;) ; [e(i;) ; e(j;)])
(1+ij(,1) + 1+ij(,1) )(e(i;) ; e(i+j;+) ) ;
which yields
(B(g; h)(i;) ; B(g; h)(j;)) = 0
for 6= .
So we have to look for possible non-zero values of the form on pairs of elements
whose second indices are the same. Associativity condition tested on the sets
fe(i;) ; e(j;+); e(l;) g when either i = j = l = 0 or exactly one of them is zero
gives the equation
(e(0;) ; e(0;)) = (e(0;) ; e(0;) ) = +
+ (e(1;) ; e(1;)) ;
which has
(e(0; ) ; e(0; )) = (e(1; ) ; e(1; )) = 0 8 2 F?
as its only solution, otherwise the identity
(e(1;) ; e(1;)) + (e(0;) ; e(0;)) = + should be satisfied for every values of 2 F? , f g, which cannot occur. Then
the only possible non-zero values can be assumed for (e(0; ) ; e(1; ) ): the sole
equations coming from associativity and involving these elements come from the
sets fe(i;) ; e(j;+ ) ; e(i; ) g when either i = j = l = 1 or exactly one of them is
one; all these equations state that the product
(e(1; ); e(0; ))
must be constant as a function of . No more associativity conditions remain to
be tested, so we can say that the Bi-Zassenhaus algebra admits an unique (up to
scalar multiple) associative form defined by
(e(i;) ; e(j;)) = (i + j ) Fk (; ) ;
(7)
12
JURMAN
and extended by bilinearity and symmetry. It is nondegenerate, since for every
basis element e(i; ) the form is non-zero on the couple (e(0; ) ; e(1; ) ) and then
its radical is the zero subspace. For a given non-singular associative form on
a finitely dimensional Lie algebra L, the space SkDer (L) of the outer skew
derivations is defined as the subspace of Der(L) generated by the derivations which satisfy (a; b) + (a; b) = 0 and (a; a) = 0 for all a; b 2 L (note
that the latter equality follows form the former one when the characteristic is not
two). By using equations (7) and linearity, a check of the stated conditions on the
basis elements shows that
SkDer (B(g; h)) = hE 2 : 1 c g + hi = hD2 : 0 c g + h , 1i :
c
c
Note that E itself is not a skew derivation for the associative form . By a
standard result (see [27]), there is a isomorphism between the space of skew
derivations SkDer (L) and the second cohomology group H2 (L; F): the image of
a derivation under this map is the 2-cocycle defined by ()(a; b) = (a; b)
for every a; b 2 L. Note that can be defined over all the skew derivations,
and in that case it has the inner ones as kernel. In the case of the Bi-Zassenhaus
algebra, by such isomorphism we obtain g + h independent 2-cocycles c for
0 c g + h , 1 whose actions on the basis elements is the following:
c (e(i;) ; e(j;)) = (D2c )(e(i;) ; e(j;) )
c
= (e(i;) D2 ; e(j;) )
c
= 2 ,1 (i + j )Fk (; ) :
The introduction of g + h central elements denoted by c for 0 c g + h , 1
as a basis for the vector space
W=
g+X
h,1
c=0
Fk c
allows us to construct the central extension
b g; h) = B(g; h) W ;
B(
where the Lie operation [ ; ] is defined as follows:
[ a; b] = [a; b] +
g+X
h,1
c=0
c (a; b)c :
The sequence
b g; h) ! B(g; h) ! 0
0 ! W ! B(
BI-ZASSENHAUS ALGEBRAS
13
is the universal covering of the Bi-Zassenhaus algebra (see [13]).
Furthermore, note that E and F1 satisfy only the first identity for being skew:
hence the two symmetric maps
E (e(i;) ; e(j;)) = (E )(e(i;) ; e(j;)) = i(,1) (1 + i + j )Fk (; )
and
F (e(i;) ; e(j;)) = (F1 )(e(i;) ; e(j;) ) = i(1 + i + j ) Fk (; ) ;
1
extended by bilinearity satisfy the 2-cocycle identity, but not the further condition
(a; a) = 0 for every a 2 B(g; h).
Now we are interested in extending the Bi-Zassenhaus algebra by its outer
derivation E , so we have to extend consistently the 2-cocycles. By a result
of [11], given a perfect Lie algebra H and an outer derivation of H , all the
cocycles of H hi are obtained by extending the cocycles of H and setting
([a; b]; ) as (a; b) + (a; b) for every 2 H 2 (H; F) and a; b 2 H . This
b on the central extension H W defined
is equivalent to introduce a derivation as
ub = u +
X
2H 2 (H;F)
(u; ) ;
(8)
where the central elements s span W . Now, since
c ([e(i;) ; e(j;) ]; E ) = c (e(i;) E; e(j;)) + c (e(i;) ; e(j;) E )
c
= 2 ,1 (1+i(,1) + 1+j(,1) )(1 + i + j )Fk (; )
= 0;
we obtain that the extended cocycles satisfy c (a; E ) = 0 for all 0 c g + h , 1
and a 2 B(g; h). In this way we have obtained the cohomology of the extension
b g; h) hE
b i, where the extended derivation E
b defined as above coincide with
B(
E.
b g; h)hE
bi ?
Is it even possible to extend the associative form to the whole of B(
Carrara proved in [11] that it is a general fact that (W; W ) = (B(g; h); W ) = 0;
b ) = 0. Finally,
moreover, the above equations c (a; E ) = 0 imply (B(g; h); E
the equations
([[e(1;) ; e(0;) ] ; Eb) = (e(1;) ; [ e(0;) ; Eb] )
when runs over Fk give the homogeneous linear system
g+X
h,1
c=0
2c ,1 (zc ; Eb) = 0 ;
14
JURMAN
whose matrix has maximal rank g + h, so its only solution is the trivial one, which
b ) = 0). Then in this presentation the only possible extension of implies (W; E
is always zero out of B(g; h).
To end with, we take a look at the isomorphism problem for the Bi-Zassenhaus
algebras, following a method employed in [6]. For any Lie algebra L, define
Mn (L) as the Fk -vector space generated by
n
X
i=0
i
(ad L)2 .
Due to the charac-
i
teristic, a spanning set for Mn (B(g; h)) is given by all the maps (ad e(l; ) )2
for i between zero and n, whose expression is in equation (6). If we define, for
x1 ; x2 ; n 2 N, the two functions fn (x1 ; x2 ) = Cardfi 2 f1; 2g: xi > tg and
n (L) = dim Mn (L) , dim Mn,1(L), we have that
n (B(g; h)) = f3 (g , 1; h)
n,1
for n = 1 ,
for n > 1 .
By using this result, we can discuss possible isomorphisms in the Bi-Zassenhaus
family or with the four families of known algebras which can assume the same
dimension, namely the special Cartan type S , Brown’s G2 , Kaplansky’s G and
the non-alternating Hamiltonian P .
Bi-Zassenhaus algebras B.
Since n (B(a + 1; b)) = n (B(b + 1; a)), two Bi-Zassenhaus algebras of parameters (g1 ; h1 ), (g2 ; h2 ) are not isomorphic if g1 + h1 6= g2 + h2 or h1 6= g2 , 1
or h2 6= g1 , 1.
Open problem: Is B(a + 1; b) ' B(b + 1; a) ?
Special algebras S (see [29, Ch.4]).
dim S(3 : (m1 ; m2 ; m3 )) = 2m1+m2 +m3 +1 , 2, for mi > 0. We know that
S(3 : (1; 1; 1)) 6' B(2; 1) since for such values of the parameters the special
algebra is restrictable, i.e. (ad x)2 2 ad S(3 : (1; 1; 1)) for any element x.
Open problem: Is B(g; h) ' S(3 : (m1 ; m2 ; g + h , (m1 + m2 ))) for (g; h) 6=
(2; 1) ?
Brown’s algebras G2 (see [6]).
dim G2 (2 : (m1 ; m2 )) = 2m1 +m2 +2 , 2, for m1 m2 > 0.
1 ; m2 )
n (G2 (2 : (m1 ; m2 ))) = 2f +(mf1 (;m
m
n 1 2)
for n = 1 ,
for n > 1 .
Now we discuss the possibility that n (B(g; h)) = n (G2 (2 : (m1 ; m2 ))); by
symmetry, we can suppose h + 1 g 2. To obtain the same dimension and
to satisfy the above identity for n = 1, we have (m1 ; m2 ) = (g + h , 2; 1) with
g + h 4. If g + h = 4, then 2 (G2 (2 : (2; 1))) = 0 6= 1 = 2 (B(2; 2)), so
we can suppose g + h 5. Moreover, if g 3, then 2 (B(g; h)) = 2 > 1 2 (G2 (2 : (g + h , 2; 1))). Thus we must choose g = 2, but then h (B(2; h)) =
BI-ZASSENHAUS ALGEBRAS
15
1 6= 0 = 2 (G2 (2 : (h; 1))). Then Bi-Zassenhaus algebras and G2 algebras are
never isomorphic.
Kaplansky’s algebras G (see [19]) and non-alternating Hamiltonian algebras
P (see [22]).
dim G(n) = dim P(n; 1) = 2n , 2 and G(n) ' P(n; 1) for every n 4 ([22,
Th.5.12]). Brown proved in [6] that 1 (G(n)) = n , 1, so there is no BiZassenhaus algebra isomorphic to a Kaplansky’s or non-alternating Hamiltonian
algebra for n > 4.
Open problem: Is (P(4; 1) ')G(4) ' B(2; 1) ?
4. A DIFFERENT PRESENTATION
In order to be able to build a loop algebra, a cyclic grading of the simple algebra
is needed: this will be shown in the next section. An intermediate step to take
is to find a grading which allows the simple algebra to be defined over the prime
field. To do this, we resemble the construction for Zassenhaus algebras.
Define a new basis
fy(t;j): (t; j ) 2 F2 f,1; 0; ; k , 2gg
B(g; h), where the y(a;j) are related to the e(a;) by the formulas (1), for
a = 0; 1. By using the identity (2), it can be shown that the Lie product in the new
for
basis is
[y(s;i) ; y(t;j) ] = b(s;i);(t;j) y(s+t;i+j+st(1,)) ;
where the coefficients read now as
i + j + st(2 , + A(i; j )(k , 1))+i + j + st(2 , + A(i; j )(k , 1));
i+1
j+1
and the function A(i; j ) is
i+j ,2
A(i; j ) = 10 for
otherwise :
Note that this is the compact form of the following expanded product, where
bi;j = b(1;i);(1;j) :
[y(0;i) ; y(0;j) ] = ai;j y(0;i+j)
[y(1;i) ; y(0;j) ] = ai;j y(1;i+j)
[y(1;i) ; y(1;j) ] = bi;j y(0;i+j+1,) :
Now we investigate some properties of the coefficients: first of all, they lie
in f0; 1g and so the Bi-Zassenhaus algebra is defined over the prime field F2 .
16
JURMAN
Moreover, they are obviously symmetric in
following identity:
(s; i), (t; j ).
They also satisfy the
b(s;i);(t;j) = 0 for i + j + st(1 , ) < ,1 or i + j + st(1 , ) > k , 3 : (9)
When st = 0 the result directly follows from the analogous one (3) proved above
for the ai;j , so we can assume s = t = 1. When i + j < , 2, we have
bi;j =
=
i+j+2+k,,1 + i+j +2+k,,1
i+1
j +1
i + j + 2 + 2g+h , 2g + i + j + 2 + 2g+h , 2g
i+1
j+1
g+h
g
2 ,2
i+j+2 + i+j +2
(mod 2)
0
i
+1
j
+1
1 i +i +j +1 2 + i +i +j +1 2
(mod 2)
0 (mod 2) :
In the latter case, when i + j + 1 , > k , 3, two subcases may occur; either
i + j + 1 , = k , 2, and then
, 1 = 1 + 1 0 (mod 2)
bi;j = ki +, 11 + kj +
1
i + j + 1 , k , 1, so we can write i + j + 2 , = k +
0 k , 1, so that < i + 1 , < i + 1 < k and we get
or
for some
2, + i+j+2,
bi;j = i + ji +
+1
j +1 i
+
j
+
2
,
i
+
j
+2,
=
+
i + 1 i+1,
k
+
k
+
= i+1 + i+1,
k0
+
(mod 2)
i+1
i+1,
0 (mod 2) :
Then we take a closer look on bi;,1 and bi;k,2 . When , 1 i k , 3 then
bi;,1 = i +i +1 ,1 + i + 10 , = 1 ;
17
BI-ZASSENHAUS ALGEBRAS
while
bi;k,2 = i +i +k ,1 + i +k k,,1 = (i; , 1)+ (i; , 1) 0 (mod 2) ;
the identity above is a direct application of Lucas’ Theorem for the latter binomial
coefficient, while for the former one it comes from the fact that, if i = + for
some 0 k , 3 , , then
i + k , = i + k , 2g+h
i+1
k,1,
0
2g+h , 2g 0 (mod 2):
When ,1 i , 2, we have
bi;,1 = i +i +k ,1 + i + k0 , g+h
g
= 2 ,i 2+ 1+ i + 1 + 1
g+h
g 1
2 0, 2 ii +
+ 1 + 1 (mod 2)
1 + 1 (mod 2) ;
and
bi;k,2 = i +i +k ,1 + i +k k,,1 g+h
g 1 2g+h , 2g i + 1 2 0, 2 ii +
+ 1 + 2g+h , 2g 2g , 1 (mod 2)
1 + 0 (mod 2) ;
As a consequence, bi;k,2 = 1 , bi;,1 for every index i. By using the properties
above, we can see that in the new presentation, the Bi-Zassenhaus algebra is
generated by the basis elements whose second index is not k , 2. In view of
property (9), nothing changes if we present B(g; h) in a slightly different way: let
Gk be the ring isomorphic to Z=(k , 1)Z where we choose as representatives the
set f[n]: ,1 n k , 3g; then we have
B(g; h) = hy(s;[i]) : (s; [i]) 2 F2 Gk i :
This notation is useful when dealing with the outer derivations; in the new basis,
D = ad (y(0;,1) + y(0;k,2) ) and E = ad (y(1;,1) + y(1;k,2) ). Since
y(0;[i])E = ai;,1 y(1;[i,1]) + ai;k,2 y(1;[i+k,2]) = (ai;,1 + ai;k,2 )y(1;[i,1])
y(1;[i])E = bi;,1 y(1;[i,]) + bi;k,2 y(0;[i,+k,1]) = (bi;,1 + bi;k,2 )y(0;[i,]) ;
18
JURMAN
and the behaviour of the involved coefficients has been shown above, we can
write
y(s;[i])E = y(1,s;[i,1+s(1,)]) :
(10)
An alternative equivalent construction of E can be described in terms of D in the
old notation as
y(s;[i]) E = y(1,s;[i]) D1+s(1,) :
On the elements of the basis, E cycles as follows:
y(1;[,1]) 7! y(0;[,1,]) 7! y(1;[,2,]) 7! y(0;[,2,2]) 7! 7! y(1;[]) 7! y(0;[0]) 7! y(1;[,1]) :
Moreover, the derivations Ft now acts as y(s;[i]) Ft = sy(s+t;[i]) .
By translating the expression (7) for the nondegenerate associative form in
terms of the new basis by means of the transformations (1) and the property (2),
we can see that now is defined over the prime field and it reads as
(y(s;[i]) ; y(t;[j])) = (s + t)Z=(k,1)Z (2k , 5 , ([i] + [j ]); 0)
= (s + t)([i] + [j ]; k , 4) :
Similarly, the new expression for the cocycles is the following:
c (y(s;[i]) ; y(t;[j]) ) = (s + t)Z=(k,1)Z ([i] + [j ]); 2c , 3)
= (s, + t)
([i] + [j ]; 2g+h + 2c , 4) + ([i] + [j ]; 2c , 3) ;
now, a[i];[j ] = 1 when [i] + [j ] = 2c , 3 for 1 c g + h, while it is
zero for c = 0 and when [i] + [j ] = 2g+h + 2c , 4 for c > 1; then the map
gc (y(r;[l]) ) = r([l]; 2c , 3) is a coboundary for 1 c g + h, and hence we
can take as representatives of H 2 (B(g; h); F2 ) the following 2-cocycles, which
we denote as c :
c (y(s;[i]) ; y(t;[j ]) ) = (s + t) ([i] + [j ]; rc )
where
c=0
rc = 2,g2+h + 2c , 4 for
for 1 c g + h , 1 :
b g; h) the product
So we can define, on the extension B(
[ y(s;[i]) ; y(t;[j]) ] = [y(s;[i]) ; y(t;[j])] +
g+X
h,1
c=0
c (y(s;[i]) ; y(t;[j ]) ) c :
BI-ZASSENHAUS ALGEBRAS
Here the map E
19
= E acts on the pair y(s;[i]) ; y(t;[j]) as
,
(1 + s + t) (1 , s)(([i] + [j ]; ,2) + ([i] + [j ]; 2g+h , 3)) +
+ s(([i] + [j ]; , 3) + ([i] + [j ]; 2g+h + , 4)) ;
whereas the derivation F1 acts now as y(s;[i]) F1 = sy(1,s;[i]) , so that the map
1
1 in the new presentation has the following expression:
F = F
F1 (y(s;[i]) ; y(t;[j ]) ) = s(1 + s + t) ([i] + [j ]; 2g+h , 4) :
Now we study the extension of the cocycles c to the derivation E , by using the
definition already employed in the previous sections, which in the new presentation
has the following shape:
b(s;[i]);(t;[j]) c (y(s+t;[i]+[j]+st(1,)) ; E ) = c (y(1+s;[i,1+s(1,)]) ; y(t;[j]) )
+ c(y(s;[i]) ; y(1+t;[j,1+t(1,)]) )
= (1 + s + t)
(11)
(([i , 1 + s(1 , )] + [j ]; rc )
+ ([i] + [j , 1 + t(1 , )]; rc )) ;
where rc is ,2 for c = 0 and 2g+h + 2c , 4 for 1 c g + h , 1. Moreover, we
have that the two delta functions are not zero for the following values of (s; [i]),
(t; [j ]):
(0; [0]), (0; [,1]) (for c = 0): in this case only one of the two deltas is not
zero so we get a non-zero right-hand side of (11), while the coefficient in the
left-hand side is a,1;0 = 1;
(1; [ , 1]), (1; [,1]) (for c = 0): the same argument applies here and the
coefficient is b[,1];[,1] = 1; furthermore, note that for any other values of [i]; [j ]
whose sum is , 2 the left-hand side coefficient vanishes, giving a trivial identity;
(1; [2c , 1]), (0; [,1]) (for c > 0): same behaviour here; the coefficient is
a,1;2c ,1 = 1;
(0; [w]), (0; [2g+h + 2c , 3 , w]) with 2c w 2g+h , 3 (for c > 0): in
this case and its symmetric both the delta functions are not zero and so we have a
zero right-hand side; the coefficient a[i];[j ] on the left-hand side is zero by (3);
(1; [w]), (1; [2g+h + 2g + 2c , 5 , w]) with 2c + 2g , 2 w 2g+h , 3
(for c > 0): also here we get a trivial identity, since both the deltas are non zero
and the coefficient b vanish by (9);
(1; [w]), (1; [2g + 2c , 4 , w]) with ,1 w 2c , 2 (for 1 c g): in
this case and its symmetric
of the delta functions is non-zero, while the
, c ,1 only
, one
c
coefficient b[i];[j ] = 2w+1
+ 2g +22 c,,13,w = 1 + 0 = 1;
20
JURMAN
(1; [w]), (1; [2g + 2c , 4 , w]) with 2c , 1 w 2g , 3 (for ,1 c c ,1 +
here both the deltas have value one and the coefficient b[i];[j ] = 2w+1
2c ,1 = 0 + 0 = 0;
2g +2c ,3,w
(1; [w]), (1; [2g + 2c , 4 , w]) with ,1 w 2g , 3 (for c > g): these
values and
symmetric
make
, their
,
only one of the deltas different from zero, while
c ,1
c
b[i];[j] = 2w+1
+ 2g +22 c,,13,w = 1 + 0 = 1.
g, ):
For all other values of the indices, the right-hand side of the equation is zero,
while, if 6= 2c , 2 with ,1 2g+h , 4, we have that b(0;[,1]);(0;[ +1]) =
a,1; +1 = 1 and a0;2g+h ,3 = 1. All the above can be summarized by the position
c (y(s;[i]) ; E ) = (1 + s) ([i]; 2c , 2) for 0 c g + h , 1 ;
b defined as in (8). Note that, in the
which leads to the extended derivation E
previous presentation, the 2-cocycles c differ to the c by a coboundary and read
as follows:
c
c
c (e(i;) ; e(j; ) ) = (i + j )2 ,1 Fk (; ) + ( + )2 ,1 ;
their extension to the derivation E then becomes
c
c (e(i;) ; E ) = (1 + i)2 ,1 :
As in the previous presentation (regardless of the chosen cocycles) the associative
b and to the central elements
form can only be trivially extended to the derivation E
W : we already know that (W; W ) = (B (g; h); W ) = 0; we need to complete
the extension by using the two more relations of associativity (see [11])
([[a; b] ; Eb) = (a; [ b; Eb] ) ;
b E
b] ) :
([[a; Eb] ; Eb) = (a; [ E;
If (s; [i]) and (t; [j ]) satisfy [ y(s;[i]) ; y(t;[j ]) ]
equations above reads as follows:
= c , then s + t = 1 and the first
(c ; Eb ) = (y(s;[i]) ; y(t;[j]) E )
= E (y(s;[i]) ; y(t;[j]) )
= (1 + s + t)([i] + [j ]; ,2 + s( , 1))
= 0:
Applying this result in the second equation
(aE; Eb) +
g+X
h,1
c=0
c (a; E ) (c ; Eb ) = 0 ;
BI-ZASSENHAUS ALGEBRAS
b ) = (B(g; h); E
b)
we get that (W; E
b g; h) hE
b i.
extended to B(
21
= 0, and hence can be only trivially
5. A CYCLIC GRADING
We can now introduce a third grading for the Bi-Zassenhaus algebras. In the
notations above, consider the following map, where the bar denotes the remainder
modulo two:
: f1; 2; : : :; 2k , 2g !
F2 Gk
m
7! (m; [2g,1 (m , m) , m]) :
The map is bijective, its inverse being the map
,1 (s; [i]) = 2v , s ;
where
v = 2h(s , [i]) (mod k , 1) ;
furthermore, when ,1 (m) + (n) + mn(1 , ) k , 3, the map satisfies
the property
(m) + (n) + mn(1 , ) = (m + n) ;
where we used the fact that, in the integers, m + n = m + n + 2mn; in the case
n = 1, we have that
(m) , 1 + m(1 , ) + C (k , 1) = (m + 1) ;
where C = 1 if the second component of (m) is less than , 1. These properties,
together with the (9), allow to define a new basis for the Bi-Zassenhaus algebra
um = y(m) 1 m 2k , 2 ;
such that
[um; un ] = b(m);(n) um+n
and for which, by (10), we have
um E = um+1 ;
and then B(g; h)i E = B(g; h)i+1 cyclically. Moreover, the derivation F1 acts
now as um F1 = mum+2h+1 ,1 , where the indices are to be taken modulo the
22
JURMAN
dimension
reads as
d = 2k , 2 of the algebra.
In the basis
fumg, the associative form
(um ; un) = (m + n)((m + n; 2h+2 + 1) + (m + n; 2g+h+1 + 2h+2 , 1)
= (m + n; 2h+2 + 1) + (m + n; 2g+h+1 + 2h+2 , 1) ;
while the cocycles become
(1)
(2)
c (um ; un ) = (m + n; rc ) + (m + n; rc ) ;
where, for i = 1; 2, we set
8
<
2h+1 + 1 + (i , 1)d
g+h+1
h+2 c+h+1 , 1 + (i , 1)d
c = : 2h+2 +c,2g+1 , 2
2 ,2
+ 1 + (i , 1)d
r(i)
for c = 0
for 1 c g , 1
for g c g + h , 1 ;
Note that we can write every c (um ; un ) as (1 , b(m);(n) ) c (um ; un ) +
b(m);(n) c (um ; un ), where the second addendum is a coboundary; hence, we
can take
c (um ; un ) = (1 , b(m);(n) )
(m + n; rc(1) ) + (m + n; rc(2) )
(12)
as representatives of H 2 (B(g; h); F2 ) in this presentation. Again we denote by c
the central element linked to the cocycle c .
The map E has the form
E (um ; un ) = (m + n; 2h+2 ) + (m + n; 2g+h+1 + 2h+2 , 2) ; (13)
while F1 is now
,
F1 (um ; un ) = m (m + n; 2h+1 + 2) + (m + n; 2g+h+1 + 2h+1 ) :(14)
Furthermore, the extension of the cocycles to the derivation E gives
c (um ; E ) = (m; rc(1) , 1) for 0 c g + h , 1 ;
(15)
b g; h) hE
b i,
while again the associative form admits only a trivial extension to B(
where E is defined as usual.
6. THE BI-ZASSENHAUS LOOP ALGEBRAS
Now we can construct the twisted loop algebra through the following standard
technique. Use the derivation E to extend the simple Bi-Zassenhaus algebra, then
tensor it with the polynomial ring in one variable over the prime field:
(B(g; h) F2 E ) F2 [t] ;
BI-ZASSENHAUS ALGEBRAS
23
the Bi-Zassenhaus loop algebra Bl (g; h) is the subalgebra generated by u1 t and
E t. The cyclic grading of the simple algebra and the behaviour of the derivation
show that Bl (g; h) is an infinite-dimensional, N-graded algebra of maximal class
generated by its first homogeneous component, since
Bl (g; h) =
1
M
i=1
Bl (g; h)i ;
where
Bl (g; h)1 = F2 (u1 t) + F2 (E t)
Bl (g; h)i = F2 (ui ti ) for i > 1 :
We now want to show that Bi-Zassenhaus loop algebras are not isomorphic to
AFS-algebras. To this aim, we investigate their two-step centralizers, i.e. the one
dimensional subspaces
Ci = CBl (g;h) (Bl (g; h)i )
1
for i 2 ;
since by Theorem 3.2 in [8] a graded Lie algebra of maximal class is determined
up to isomorphism by its sequence fCi g of two-step centralizers. Moreover,
by Lemma 3.3 of the same paper, the first centralizer C2 occurs in contiguous
subsequences, called constituents, separated by isolated occurrences of a different
centralizer. Following the notation of [9], suppose that in the sequence of two-step
centralizers we have a pattern of the form
Ci 6= C2 ; Ci+m 6= C2 ;
Ci+1 = Ci+2 = : : : = Ci+m,1 = C2 ;
then (Ci+1 ; Ci+2 ; : : : ; Ci+m ) is a constituent of length m. By [8, Theorem 5.5],
a two-step centralizer different from the first one will first occur in class 2q for
some power q of the characteristic of the underlying field; the number q is called
the parameter of the algebra, while by convention we say that the length of the first
constituent is 2q . As a consequence of the previous theorem, when the algebra has
only two distinct two-step centralizers, the sequence of the constituents lengths
settles its isomorphism type. By definition,
[u1 ; um] = b(1;[1]);(m)um+1 = b(1;[1]);(m)um E ;
and then the two-step centralizers are exactly
Cm = F ((u1 , b(1;[,1]);(m)) E ) t) ;
(16)
and since b(;);(;) 2 f0; 1g the loop algebra Bl (g; h) has only two distinct twostep centralizers, generated by x = u1 t and y = (u1 , E ) t in the standard
24
JURMAN
notations. In particular, by definition we have that
b(1;[,1]);(m) = ab [,1];[,g2,g, m]
[,1];[2 (1,m),1]
1
1
when m 2 2Z
when m 2 2Z + 1 ;
and then, from previous calculations, that
8
>
<
1 , Gk ([,1]; [,2g,1m])
b(1;[,1]);(m) = > 1 ,
:
X
,2
for m 2 2Z
Gk ([w]; [2g,1 (1 , m) , 1])
w=,1
for m 2 2Z + 1 ;
Thus, to express explicitly the two-step centralizers in (16) in terms of m we have
to solve the following equations in Gk :
[,2g,1 m] = [,1]
[2g,1 (1 , m) , 1] = [w]
for m 2 2Z, and
for m 2 2Z + 1, w
2 f,1; ; , 2g :
The solutions are respectively, for n 2 N0 ,
m = 2q + nd and
m = 2q( , w) , 1 + nd ;
(17a)
(17b)
where q = 2h is called the parameter of the loop algebra and denotes the first
occurrence of the second two-step centralizers F2 x; remember that we use d to
denote the dimension 2k , 2 of the Bi-Zassenhaus algebra. The sequence of the
two-step centralizers (16) reads then
Cm = FF2 xy
2
for m as in (17b)
otherwise :
(18)
Then the constituent lengths sequence can be constructed: if we denote exponentially patterns’ repetition, it reads as
2q; 2q , 1; (2q,1; 2q , 12 )1 :
Note that this implies that two Bi-Zassenhaus loop algebras with different parameters are never isomorphic.
In the paper [8] two techniques are described to build more algebras of maximal
class starting from a given one L: they are called inflation and deflation and
the obtained algebras are respectively indicated by L" and L# . For a complete
theoretical approach to them, we refer to the cited work. For our purposes, we
can rely on the results listed in the Propositions 3.3–3.6 in [9], where p is the
25
BI-ZASSENHAUS ALGEBRAS
characteristic of the field (in our case p = 2). Moreover, we recall the constituent
lengths sequence of the Albert-Frank-Shalev algebra:
AFS(a; b; n; p) = 2q; (qr,2; 2q , 1; (qr,2 ; 2q)s,1 )1 ;
where q = pa , r = pb,a and s = pn,b . Since an algebra is inflated if and only
if all constituents have length a multiple of the p, neither the Bi-Zassenhaus loop
algebras nor the Albert-Frank-Shalev algebras are inflated: actually, they are the
only non-inflated ones [18]. Given any algebra of maximal class L, its inflation
L" can be obtained by multiplying all its constituents lengths by p: then, if we
denote by q " = q h+1 the parameter of the inflated algebra, we have
Bl (g; h)" = 2q" ; 2q" , 2; (2(q"),1 ; 2q" , 2)1 :
Repeated inflations (say, times) will change the original lengths 2q; 2q , 1 of
the constituent into 2ph+ ; 2ph+ , p . When an algebra L of maximal class is
inflated, deflation has the effect to divide all constituent lengths by p: this follows
from the fact (L" )# = L; the converse is true only for inflated algebras. When L
is not inflated, the effect of deflation can be read from the sequence of two-step
centralizers by using [9, Proposition 3.3]: if for i 2 all the two-step centralizers
Cip ; Cip+1 ; : : : ; C(i+1)p,1
coincide with C2 = Fy , then Ci# = Fy , otherwise if one of them equals Fu 6= Fy ,
then Ci# = Fu. So, for i 4, write down the two-step centralizers of Bl (g; h)
obtained in (18), and for every l 2 set
C# =
l
Fy
Fx
when C2l = C2l+1
otherwise :
= Fy
Grouping them in constituents, we get two different results, depending on the
parameter of the Bi-Zassenhaus loop algebra: if h = 1 the sequence of constituent
lengths of the deflated algebra reads as
4; (2,1 ; 3)1 ;
while when h > 1, if we let q #
= q=2 = 2h,1 , we get
2q# ; 2q# , 1; (2(q#) ; 2q# , 1) :
Both the above patterns are recognisable as AFS algebras, for a suitable choice of
the parameters: in particular, if we understand AFS(0; b; n; 2) = AFS(b; n; n; 2)
we can write in every case
Bl (g; h)# = AFS(h , 1; h; h + g; 2) :
26
JURMAN
Note that we also have that (see [8, Proposition 7.3])
AFS(h; h + 1; h + g; 2)# = AFS(h , 1; h; h + g; 2) ;
so we have two non-isomorphic loop algebras whose deflation coincide: this is
a consequence (see [7]) of the isomorphism , in characteristic two, between the
Zassenhaus algebra Zk
Zk = he : 2 F?k i; [e ; e ] = ( + )e+
and the Albert-Frank algebra AF
a
b
b
a
AF(a; b; k) = he0 : 2 F?k i; [e0 ; e0 ] = (2 2 + 2 2 )e0+
b,a
= he00 : 2 F?k i; [e00 ; e00 ] = ( + )2 e00+
given by
(e00 ) = e b,a :
2
The last part of the paper is devoted to describe the cohomology of the loop
algebras; first we deal with the cocycles coming from the simple algebras.
b g; h) hE
b i) F2 [t] and the subalgebra B
e (g; h) generated
Consider the loop (B(
b
by u1 t and by E t. Here the product is given, for u; v 2 B(g; h) hE i and
a; b 2 N, by the following expression:
[ u ta ; v tb ] = [u; v] +
g+X
h,1
c=0
c (u; v ) c ta+b ;
where the value c (u; v ) is given by (12) and (15). In the paper [17], we employ a
e (g; h) by introducing the elements
different notation for the central elements in B
nf (c) = c trc +nd ;
(1)
where f (0) = 1 and f (c) = g + h + 1 , c for 1 c g + h , 1; in this notation,
the weight of ns11 is less than the weight of ns22 if either n1 < n2 or n1 = n2 and
s1 < s2 .
As we have seen above, the associative form admits only a trivial extension
b g; h) hE
b i; furthermore, if we define the additional loop cocycle in the
to B(
standard way by multiplying the associative form either by the residue map (if
a = 1, see [13]) or its analogue (if a = 2, see [10]), we obtain
(u ta ; v tb ) (ta ; tb ) = (u ta ; v tb ) a Z=2a Z (a + b; 0) = 0 ;
27
BI-ZASSENHAUS ALGEBRAS
since vanishes on elements whose weights have even sum, and the parity of
e g; h) is the same of those in B(
b g; h).
elements in B(
e
Note that the centre of B(g; h) is infinite-dimensional, and the quotient of
e g; h) over its centre is isomorphic to the algebra of maximal class Bl (g; h); so
B(
we can apply the translation to Lie algebras of a classical group-theoretical result
due to B.H. Neumann (see [24, pp. 52-53]):
Theorem 6.1. Let be any epimorphism from a finitely generated Lie algebra
Le to a finitely presented algebra L. Then ker() is finitely generated as an ideal
e.
in L
e g; h) ! Bl (g; h) is the homomorphism induced by the canonical
If : B(
b g; h) to B(g; h), then ker() = W = Z (B(
e g; h)), so Bl (g; h)
surjection from B(
cannot be finitely generated. So we can state as a direct consequence of the above
theorem that
Theorem 6.2. The Bi-Zassenhaus loop algebras are not finitely presented.
In fact, a stronger result (proved in [17]) holds, analogous to the one proved by
Carrara in [12] for AFS-algebras:
Theorem 6.3. For every Bi-Zassenhaus loop algebra Bl (g; h), there exists a
finitely presented graded Lie algebra M (g; h) such that
M (g; h)=Z2 (M (g; h)) = Bl (g; h):
Now we conclude by introducing the loop cocycles: this will give a construction
of the covering algebra M (g; h).
First, look at the maps E , F1 : they are not cocycles of the simple algebra,
since they do not vanish on the pairs (a; a). The natural way to turn them into
cocycles of the loop algebras is to define
A (v ta+d; w tb+d ) = A (v; w) F ( + ; 1) ;
2
b g; h) of weight a and b respectively and A is
where v and w are elements of B(
either E or F1 . Now we show if they can be consistently extended to the whole
e g; h).
of B(
First we deal with F1 . When v and w are elements of B(g; h) we have the
expression (14); actually, we can get a simpler expression by adding a “coboundary” b(m);(n) ( (m + n; 2h+1 + 2) + (m + n; 2g+h+1 + 2h+1 )): in this way
the coefficient in front at the sum of the two delta maps is m + b(m);(n) ; this
coefficient is always zero if m + n = 2g+h+1 + 2h+1 , while it is always one when
m + n = 2h+1 + 2, but in the case m = 1; n = 2h+1 + 1 and its symmetric.
28
JURMAN
Hence we have
,
F (um tm+d ; un tn+d) = (m + n; 2h+1 + 2)
,(m; 1)(n; 2h+1 + 1) ,(n; 1)(m; 2h+1 + 1)
F ( + ; 1) ;
1
(19)
2
which means that F1 is not zero only on the elements (um tm+ad ; un n
t +bd ) such that a + b is odd and (um ; un ) is one of the 2h+1 , 1 couples
f(2; 2h+1); (3; 2h+1 , 1); ; (2h+1; 2)g. Now we use the cocycle identity
A ([[a1 ; a2 ] ; a3 ) + A([[a2 ; a3 ] ; a1 ) + A ([[a3 ; a1 ] ; a2 ) = 0
(20)
to define the above maps on the remaining elements. Since B(g; h) is perfect
e g; h) is a commutator (see [11]), we have that
and any central element of B(
(1)
a
+
d
r
+
d
b g; h); the result comes from
F1 (v t ; c t c
) = 0 for any v 2 B(
(1)
r
+
d
c
evaluating equation (20) with a3 = c t
and a1 ; a2 satisfying [ a1 ; a2 ] =
v ta+d . Now choose a1 = um tm+d, a2 = up tp+d and a3 = Eb t.
(1)
Consider first the case when [ a; b] = c trc +nd , so that b(m);(p) = 0. When
n is even the cocycle equation gives F1 (c trc(1) +nd ; Eb t) = 0, while when
n is odd it reduces to
F (c trc +nd; Eb t) = F (up+1 tp+1+d ; um tm+d )
+F (up tp+d; um+1 tm+1+d ) :
(1)
1
1
1
(i)
Since m+p = rc , the right-hand side of the previous equation vanishes when 1 c g + h , 1. If c = 0, we know that the only couples of elements whose product
is 0 is (y(0;,1) ; y(1;,1) ) and its symmetric, that is, (m; p) = (1; 2h+1 ). With these
values, the right-hand side of the above equations is one, so we can summarize
(1)
b t) = (c; 0)F (n; 1).
what we found by saying that F1 (c trc +nd ; E
2
Finally, when b(m);(p) = 1 we find the last non-trivial case
F (um+p tm+p+(+)d; Eb t) = F (up+1 tp+1+d; um tm+d )
+F (up tp+d ; um+1 tm+1+d )
= 0;
1
1
1
e g; h).
by equation (19). Then F1 is a non-trivial cocycle of B(
We can carry on a similar argument for E as well, but we get a different result.
First we reach a simpler form for this map on elements in B(g; h) by splitting
its expression (13) into (1 , b(;)) E + b(;) E and getting rid of the second
29
BI-ZASSENHAUS ALGEBRAS
addendum, which is a coboundary: in terms of the y(s;[i]) presentation, since
a[i];[j] = 1 for [i] + [j ] = 2g+h , 3 and b[i];[j] = 1 for [i] + [j ] = 2g+h + , 4,
it is equivalent to express the map E as
E (y(s;[i]) ; y(t;[j ]) ) = (1 + s + t) ([i] + [j ]; ,2 + s( , 1)) ;
in this form,
,
E (um tm+ad ; un tn+bd ) = (1 , b(m);(n) ) (m + n; 2h+2 )
,
+ (m + n; 2g+h+1 + 2h+2 , 2) (21)
F (a + b; 1)
X
=
ZZ ((m; n); (M; N ))
2
(M;N )2M
F (a + b; 1) ;
2
where M is the set
M = f(2h+1 ; 2h+1); (1; 2h+2 , 1); (2h+2 , 1; 1);
(2h+1( , ) , 1; 2h+1( + 3) , 1); 0 , 3g :
b g; h).
As in the previous case, E (v ta+d ; c trc +d ) = 0 for any v 2 B(
(1)
Moreover, from the cocycle identity (20) and from expression (21) it follows
b t) can be different from zero only when v = u2h+2 ,1 ;
that E (v ta+d ; E
furthermore, note that, when m + n = 2h+2 , 1 for 1 m; n 2k , 2
we have that b(m);(n) = 1 if and only if either (m; n) = (M , 1; N ) or
(m; n) = (M; N , 1) for (M; N ) 2 M: when this is the case, then exactly
one of E (um+1 ; un ), E (um ; un+1 ) is one. By using these results, if we
g+h+1 ,2+d
apply the cocycle equation (20) to the elements a1 = u2g+h+1 ,2 t2
,
h
+2
2
,
1+
d
b
a2 = u2h+2 ,1 t
and a3 = E t first, and then to a1 = u2h+1 (,),2 h+1 ( , ),2+d
h+1
2
b t for
t
, a2 = u2h+1 (+3),1 t2 (+3),1+d and a3 = E
+ even and 0h+2 , 3 we get a contradiction, since we obtain that
E (u2h+2 ,1 t2 ,1+(++1)d; Eb t) should be one in the first case and zero
e g; h).
in the second: then E cannot be extended to the whole of B(
e
Now we can construct a central extension M (g; h) of B(g; h) by defining the
Lie product as
[[u ta ; v tb ]] = [ u ta ; v tb ] + F (u ta ; v tb ) ! ta+b ;
where ! ta+b are central elements lying in classes 2h+1 + 2 + (2n + 1)d; in
h
the paper [17] we define n! = ! t2 +2+(2n+1)d . In particular, note that the
h
e g; h), while in M (g; h) it satisfies
element 0 t2 +1+nd is always central in B(
1
+1
+1
[[0 t2
nd ; Eb t]] = F (n; 1) ! t2h+1 +2+nd ;
2
h+1 +1+
30
JURMAN
so it lives in Z2 (M (g; h)) for odd n’s.
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