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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