Nilpotent groups and algebras · The theory of nilpotent groups and algebras is rich in many...
Transcript of Nilpotent groups and algebras · The theory of nilpotent groups and algebras is rich in many...
Nilpotent groups and algebras
V.N. Remeslennikov
Sobolev Institute of Mathematics (Omsk branch)
11 October, 2011
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CRISIS
The theory of nilpotent groups and algebras is rich in many
remarkable results. Here, a special role is played by the theory of
unipotent groups, which is closely related to the theory of
�nite-dimensional algebras over associative rings. A certain crisis of
this theory is related to 1960s and 1970s.
The theory of unipotent, and especially commutative groups over a
�eld is a beautiful completed theory. Accordiny to a results of
(1970) this is also true in the general case: over schemes of
characteristic zero the theory is trivial. Vassershtain and Dolgachev
(1974).
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Renaissance
The renewal epoch (Renaissance) of the theory began in 1980s.
And was concerned with the study of such objects by new
model-theoretic, arithmetic, geometric and asymptotic methods.
We present several examples of this process.
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1. Positive solution of Isomorphism problem for �nitely generated
nilpotent groups.
Papers
1 Grunewald F., Segal D., Some general algorithms I. Arithmetic
groups, Ann. Math., 1980, 11, # 3, p.531�583;
II Nilpotent groups, Ann. Math. 1980, 112, # 3, p.585�617.
2 Sarkisyan R.A., Algorithmic problems for linear algebraic
groups I, II, Sbornik: Mathematics, 1980, 113, # 2, 3,
p.179�216, p.400�436. (In Russian).
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2. The proof of decidability of the elementary theories of �nitely
dimensional k-algebras and k-nilpotent groups, where k is a �eld or
a local ring; the criterion of elementary equivalence for such groups
and algebras (A.G. Myasnikov, 1987�1990).
3. The development of the theory of constructive nilpotent groups
and algebras; the calculation of the algorithmic dimension (Siberian
School of Algebra and Logic, S.S. Goncharov).
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4. Gromov's theorem about the groups of polynomial growth
(almost nilpotent groups); the asymptotic methods in group theory.
5. The return to the ideas of A.I. Malcev and P. Hall; the
development of these ideas.
6. The study of �well-structured� series of groups or algebras, unlike
the study of the class of all nilpotent groups (algebras).
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The examples of the series
Example 1
Let A be a k-algebra, where k is either Z or Q, and i : k → R is an
embedding of the rings. Let A(R) = R ⊗k A. The set of algebras
{A(R)} is said to be a series, and A is a root algebra of the
series.
Example 2
It is well-known the next series of groups: GL, SL,Tr ,UT , . . .
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Purposes of the talk
1 De�ne the notion of a group series with a given root group
G , where G is �nitely generated and nilpotent.
2 De�ne �well-structured� groups and �well-structured� series.
3 De�ne the abelian deformations of groups.
4 Introduce the theorems on elementary equivalence (in the
group language) for groups of a well-structured series.
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R-groups
Let R be an associative domain. The ring R generates the category ofR-groups Gr(R). Let us add to the standard ring language Lgr new unaryoperations fr (x) for any r ∈ R and obtain the extended language Lgr(R).
De�nition 1.
An algebraic structure G of the language Lgr(R) is called an R-group if itsatis�es the next axioms:
the set G (as an algebraic structure of the language Lgr) is a group;
for g ∈ G and α ∈ R let us denote by gα the element fα(g). Thefollowing holds:
g0 = 1, gα+β = gα gβ , (gα)β = gαβ .
As the class of groups Gr(R) is a variety, the notions of R-subgroups,R-homomorphism, free R-group and nilpotent R-group are naturallyde�ned.
Example
R-modules are R-groups.
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Nilpotent R-groups
In the class of R-groups P.Hall introduced the special subclass which waslater named as the class of Hall R-groups. To de�ne this class one shouldclaim the next properties for the ring R.
De�nition 2.
A domain R containing the subring Z is called binomial if for any λ ∈ R,n ∈ N the ring R contains the binomial coe�cient
Cn
λ =λ(λ− 1)(λ− 2) . . . (λ− n + 1)
n!
Examples
i) Z � integer numbers;
ii) �elds of a zero characteristic;
iii) Zp � p-adic numbers;
iv) the ring of polynomials over a �eld of a zero characteristic.
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Hall R-groups
De�nition 3.
Suppose R is a binomial ring. A nilpotent group G of a class m is
called a Hall R-group if for all x , y , x1, . . . , xn ∈ G and any
λ, µ ∈ R the next axioms holds:
G � is a nilpotent R-group of a class m;
y−1xλy = (y−1x)y)λ;
xλ1 . . . xλn = (x1 . . . xn)
λτ2(x)C2
λ . . . τm(x)C2
λ , where
x = {x1, . . . , xn} τi (x) is the i-th Petresco word de�ned in the
free group F (x) by
x i1 . . . xin = τ1(x)
C1λτ2(x)
C2λ . . . τi (x)
Ci
λ .
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The axioms of Hall R- groups are the universal formulas in the
language Lgr(R), hence they generates the variety of nilpotent Hall
R-groups of a class m (denoted by HNm,R).
Proposition 4.
Suppose R is a binomial ring. Then the unitriangle group UTn(R)and, therefore, all its subgroups are Hall R-groups.
For any matrix X ∈ UTn(R) and every α ∈ R de�ne
Xα =n−1∑i=0
Ciα(X − E )i ,
where E is a unity matrix.
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The series with the root subgroup
Let G be a �nitely generated torsion-free nilpotent group. The
Malcev`s base for G is a tuple
u = (u1, . . . , un)
of elements of G such that for
Gi = 〈ui , ui+1, . . . , un〉
we have
the sequence
G = G1 > G2 > . . . > Gp > Gp+1 = 1
has abelian quotients;
any factor Gi/Gi+1 is an in�nite cyclic group.
If u = (u1, . . . , un) is a Nalcev base for G , then each element of G
is uniquely represented as
g = uα11 uα22 . . . uαn
n = uα, α = (α1, . . . , αn) ∈ Zn.
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Theorem 5.
There exists so-called canonical polynomials associated with u
pi (x1 . . . , xn, y1 . . . , yn) ∈ Q[x, y], i = 1, . . . , p,
qi (x1 . . . , xn, y) ∈ Q[x, y ],
such that pi (α, β) = γi and qi (α, λ) = δi .
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De�nition 6.
Let R be a �nitely generated torsion-free group and GRu a set of all
formal products
uα = uα11 uα22 . . . uαn
n , (α1, . . . , αn) ∈ Rn.
The multiplication over GRu is de�ned by the canonical polynomials
pi qi associated with u. The obtained group is the Hall R-extension
of G .
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Proposition 7 (Ph. Hall).
If GinNm,Z is a free nilpotent group of a �nite rank , then GR is
the free nilpotent group in the variety HNm,R .
Examples:
i) UTn(R);
ii) if GΓ is partially commutative nilpotent group then GRΓ is the
same.
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The ideas of Alexey Miasnikov
Let A be an arbitrary (even non-associative) R-algebra. In 1937
Jackobson o�ered to associate with A the algebra of multiplications
M(A): 〈aL : x → ax , aR : x → xa〉 which is the subalgebra of the
associative algebra End of linear R-mappings. The centralizer of
M(A) in End was called a centroid C(A) for the algebra A. It is
clear that the ring R is canonically embedded into End and
R ⊆ C(A). One can de�ne the centroid extension of A by
AC = C(A)⊕R A.
It was the well-known idea. In 1987�1990 A. Miasnikov proposed
the following.
1 Associate with an algebra (or group) bilinear mapping
fA : M ×M → N, where M,N are R-modules well-de�ned by
A.
2 De�ne the ring P(fA) ⊇ R , and it was proved that it is a
�proper ring of scalars for the algebra (group) A�.
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Bilinear mappings as the objects of model theory
Let M, N be precise R-modules, R . A bilinear mapping
f : M ×M → N is called non-degenerate for two variables if
f (x ,M) = 0 or f (M, x) = 0 implies x = 0. We call the map f
�onto� if N generates by f (x , y), x , y ∈ M. We associate with f
multi-sorted algebraic structures, one of them is
B(f ) = 〈M,N, δ〉,
where the predicate δ describes the map f . The another one is
BR(f ) = 〈R,M,N, δ, sM , sN〉,
where sM , sN describes the action of R over the nodules M and N
correspondingly.
A. Miasnikov proved (for f = fA with some constraints) that there
exists such ring P(f ) that BP(f )(f ) is absolutely interpretated in
B(f ). Moreover, P(f ) is the maximal ring, there f is bilinear.
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The ideas of Alexey Miasnikov
1 One should replace the �nitely dimensional algebra A (or
R-group with multiplicative basis) to the bilinear mapping
fA : A/Ann(A)× A/Ann(A)→ A2,
(x +Ann(A), y +Ann(A))→ xy( [x , y ])
2 Compute P(fA) (as the inner centroid of the algebra A) and
consider A as a P(fA)-algebra.
The ideas became useful in the works of A. Miasnikov in
1987�1990, and in the latest ones.
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Elementary theory of �nite dimensional algebras andpartially commutative nilpotent groups
Montserrat Casals-Ruiz, Gustavo A. Fern�andez-Alcober,
Ilya V. Kazachkov and Vladimir N. Remeslennikov
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Scheme of the paper
1 Introduction
2 Preliminaries in Algebra
Partially commutative algebras and groupsPartially commutative monoids and groupsPartially commutative associative algebrasFree Lie algebrasPartially commutative Lie algebrasPartially commutative nilpotent algebrasCategories of R-groups over a ring R
Nilpotent R-groupsMalcev correspondencePartially commutative nilpotent R-groupsExtensions of groups and algebras and cohomology.
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3 Preliminaries in model theory
Signatures, formulas, algebraic systemsTheories and elementary equivalenceInterpretationsModel theory of bilinear mapsModel theory of �nite dimensional algebrasThe ring P(f ) of some algebrasAbelian deformations
4 Structured algebras and characterisation theorems
5 Characterisation of rings elementarily equivalent to a partiallycommutative nilpotent associative and Lie algebras
Associative algebrasLie algebras
6 Characterisation of groups elementarily equivalent to a
partially commutative nilpotent group
7 Open problems
8 References
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Remind Alexey Miasnikov`s theorem
Theorem 2.6 (Theorem 1, [1])
Let f be a non-degenerate �onto�R-bilinear map. Then the maximal
enrichment EM = (f ,P(f )) exists and is unique up to isomorphism.
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De�nition 3.4
A �nite dimensional faithful R-algebra A is called structured if
R = P(fA);
A is torsion free as a P(fA)-module;
Ann(A) < A2 and
the modules A2, A/Ann(A) and Ann(A) are free as
P(fA)-modules.
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Proposition 3.5
Let A be a structured P(fA)-algebra. Let B be a ring such that
B ≡ A. Then
1 B is a Z-algebra and P(fA) ≡ P(fB);
2 B2 is a free P(fB)-module and the ranks of B2 (as a
P(fB)-module) and of A2 (as a P(fA)-module) coincide, i.e.
rank(B2) = rank(A2);
3 Ann(B) is a free P(fB)-module and
rank(Ann(B)) = rank(Ann(A));
4 B/Ann(B) is a free P(fB)-module and
rank(B/Ann(B)) = rank(A/Ann(A)).
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De�nition 3.6
A structured algebra A is called well-structured if
the modules A/A2 and A2/
Ann(A) are free;
the algebra A, viewed as a P(fA)-module admits the following
decomposition A = A/A2 ⊕ A2/
Ann(A)⊕ Ann(A);
Let U = {u1, . . . , uk}, V = {v1, . . . , vl} andW = {w1, . . . ,wm} be basis of the free modules A
/A2,
A2/Ann(A) and Ann(A), respectively. Then the structural
constants of A in the basis U ∪ V ∪W are integer. In other
words,
xy =k∑
s=1
αxysus +l∑
s=1
βxysvs +m∑s=1
γxysws , (1)
where x , y ∈ U ∪ V ∪W and αxys , βxys , γxys ∈ Z for all x , yand s.
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De�nition 3.7
Let A be a well-structured P(fA)-algebra. We de�ne the ring
QA = QA(S , s), which we sometimes refer to as an abelian
deformation of A, as follows.
Let S be a commutative unital ring of characteristic zero. Let
K , L, and M be free S-modules of ranks rank(A/A2),
rank(A2/Ann(A)) and rank(Ann(A)), correspondingly.
The ring QA, as an abelian group, is de�ned as an abelian
extension of M by K ⊕ L via a symmetric 2-cocycle. More
precisely, let x1, y1 ∈ K , x2, y2 ∈ L and x3, y3 ∈ M. Set
(x1, x2, x3)+(y1, y2, y3) = (x1+ y1, x2+ y2, x3+ y3+ s(x1, y1)),
where s ∈ S2(K ,M) is a symmetric 2-cocycle.
The multiplication in QA is de�ned on the elements of the
basis of K , L and M using the structural constants of A and
extended by linearity to the ring QA.
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Characterisation Theorem
Theorem 3.1
Let A be a well structured R-algebra where R is a binomial ring and
B be a ring. Then
B ≡ A if and only if B ' QA(S , s)
for some ring S , S ≡ R and some symmetric 2-cocycle
s ∈ S2(QA/QA2,Ann(QA)).
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Characterisation of rings elementarily equivalent to a partiallycommutative nilpotent Lie algebra
Theorem 3.12
Let R is an integral domain of characteristic zero. Then R-group
(R-algebra) is well-structured in the following cases:
1 G is a nilpotent of class c free R-group (R-algebra Lie);
2 G is UT(n,R);
3 G is a nilpotent of class c directly indecomposable partial
commutative R-group (R-algebra Lie).
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Characterisation Theorem
If G and R are above we have the following result
Theorem 6.3
Let G = N∆,c(R) and H be a group (ring Lie) so that H ≡ G .
Then H is a QN∆,c(R) group over some ring S such that S ≡ R as
rings.
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References
A. G. Myasnikov, De�nable invariants of bilinear mappings,
(Russian) Sibirsk. Mat. Zh. 31 (1990), no. 1, 104-115; English
translation in Siberian Math. J. 31 (1990), no. 1, 89�99.
A. G. Myasnikov, Elementary theories and abstract
isomorphisms of �nite-dimensional algebras and unipotent
groups, Dokl. Akad. Nauk SSSR, 1987, v.297, no. 2, pp.
290-293.
A. G. Myasnikov, Elementary theory of a module over a local
ring, (Russian) Sibirsk. Mat. Zh. 30 (1989), no. 3, 72-83, 218;
English translation in Siberian Math. J. 30 (1989), no. 3,
403-412 (1990)
A. G. Myasnikov, The structure of models and a criterion for
the decidability of complete theories of �nite-dimensional
algebras, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989),
no. 2, 379�397; English translation in Math. USSR-Izv. 34
(1990), no. 2, 389�407.31 / 32
References
A. G. Myasnikov,The theory of models of bilinear mappings,
(Russian) Sibirsk. Mat. Zh. 31 (1990), no. 3, 94-108, 217;
English translation in Siberian Math. J. 31 (1990), no. 3,
439-451.
F. Oger, Cancellation and elementary equivalence of �nitely
generated �nite-by-nilpotent groups, J. London Math. Society
(2) 44 (1991) 173-183.
M. Sohrabi, On the Elementary Theories of Free Nilpotent Lie
Algebras and Free Nilpotent Groups, PhD Thesis, 2009,
Carleton University.
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