Groups Geom. Dyn. 4 (2010), 873–900
DOI 10.4171/GGD/112
Groups, Geometry, and Dynamics
© European Mathematical Society
Nil graded self-similar algebras
Victor M. Petrogradsky
1
, Ivan P. Shestakov
2
and Efim Zelmanov
3
Abstract. In [19], [24] we introduced a family of self-similar nil Lie algebras L over fields of
prime characteristic p>0 whose properties resemble those of Grigorchuk and Gupta–Sidki
groups. The Lie algebra L is generated by two derivations
v
1
D @
1
C t
p1
0
.@
2
C t
p1
1
.@
3
C t
p1
2
.@
4
C t
p1
3
.@
5
C t
p1
4
.@
6
C /////;
v
2
D @
2
C t
p1
1
.@
3
C t
p1
2
.@
4
C t
p1
3
.@
5
C t
p1
4
.@
6
C ////
of the truncated polynomial ring KŒt
i
;i 2 N j t
p
i
D 0;i 2 N in countably many variables.
The associative algebra A generated by v
1
, v
2
is equipped with a natural Z ˚ Z-gradation. In
this paper we show that for p, which is not representable as p D m
2
C m C 1, m 2 Z, the
algebra A is graded nil and can be represented as a sum of two locally nilpotent subalgebras.
L. Bartholdi [3] and Ya. S. Krylyuk [15] proved that for p D m
2
C m C 1 the algebra A is not
graded nil. However, we show that the second family of self-similar Lie algebras introduced in
[24] and their associative hulls are always Z
p
-graded, graded nil, and are sums of two locally
nilpotent subalgebras.
Mathematics Subject Classification (2010). 17B05, 17B50, 17B66, 17B65, 16P90, 11B39.
Keywords. Modular Lie algebras, growth, nil-algebras, self-similar, Gelfand–Kirillov dimen-
sion, Lie algebras of vector fields, Grigorchuk group, Gupta–Sidki group.
1. Definitions and constructions
Let L be a Lie algebra over a field K of characteristic p>0 and let ad x W L ! L,
ad x.y/ D Œx;y for x;y 2 L, be the adjoint map. Recall that L is called a restricted
Lie algebra or Lie p-algebra [12], [26], [1] if L additionally affords a unary operation
x 7! x
Œp
, x 2 L, satisfying
i) .x/
Œp
D
p
x
Œp
for all 2 K, x 2 L;
ii) ad.x
Œp
/ D .ad x/
p
for all x 2 L;
1
The first author was partially supported by grants FAPESP 05/58376-0 and RFBR-07-01-00080.
2
The second author was partially supported by grants FAPESP 05/60337-2 and CNPq 304991/2006-6.
3
The third author was partially supported by the NSF grant DMS-0758487.