Old and new questions in noncommutative ring theory · 2013-07-22 · Old and new questions in...

Introduction GS algebras Free subalgebras Open problems Infinite number of prescribed relations New open problems

Old and new questions in noncommutative ringtheory

Agata Smoktunowicz

University of EdinburghEdinburgh, Scotland, UK

e-mail: [email protected]

26 Nordic and 1st European-Nordic Congress ofMathematics, Lund, 10-14 June 2014


Outline

1 Introduction

2 GS algebras

3 Free subalgebras

4 Open problemsFamous open problems

5 Infinite number of prescribed relations

6 New open problems


Short story of Golod-Shafarevich algebras

In 1964, Golod and Shafarevich found that, provided that thenumber of relations of each degree satisfies some bounds,there exist infinitely dimensional algebras satisfying givenrelations.

Such algebras have now come to be referred to asGolod-Shafarevich algebras.

Golod-Shafarevich algebras are used to construct infinitelydimensional algebras satisfying prescribed relations (forexample nil algebras).


Definition of Golod-Shafarevich algebras

DefinitionLet K be a field and let A be a free algebra in d generators. LetI be an ideal generated by relations f1, f2, . . . with ri relations ofdegree i among f1, f2, . . ..Let R = A/I. If the series

(1− dt +∞∑

i=2

ri t i)−1

has all coefficients larger than zero then R = A/I is aGolod-Shafarevich algebra.


Golod-Shafarevich theorem

Theorem

Let HR(t) =∑∞

i=0 dim Ri t i be the Hilbert series of algebraR = A/I, with A, I as above. Golod and Shafarevich proved that

HR(t)(1− dt +∞∑

i=2

ri t i) ≥ 1

holds coefficient’s-wise.Therefore if all coefficients of (1− dt +

∑∞i=2 ri t i)−1 are larger

than zero then the algebra R = A/I is infinite dimensional.


How to check if given algebra is Golod-Shafarevichalgebra?

Let K be a field and let A be a free algebra in d generators. LetI be an ideal generated by relations f1, f2, . . . with ri relations ofdegree i among f1, f2, . . ..

Zelmanov’s survey, 1996If there is t0 > 0 such that

∑∞i=2 ri t i converges at t0 and

1− dt0 +∑∞

i=2 ri t i0 < 0 then A/I is infinitely dimensional.

Remark (A.S)If for infinitely many m, there is 0 < tm such that1− dtm +

∑mi=2 ri t i

m < 0 then algebra A/I is finitely dimensional.


Proof of remark (1)

Fix number m and let Im be the ideal of A generated by allelements from fi which have degrees not exceeding m.

If there is tm > 0 such that

1− dtm +m∑

i=2

ri tmi < 0

then by the above remark of Zelmanov A/Im is infinitelydimensional.


Proof of remark (2)

Since I is a graded ideal and

I =∞⋃

i=2

Im

it follows that A/I is infinite dimensional.

Indeed, a graded and finite dimensional algebra is nilpotent.If A/I is finitely dimensional them A(n) ∈ I for some n, andsince I is graded A(n) ⊆ In, so A/In is finitely dimensional, acontradiction.


Applications of Golod-Shafarevich algebras

General Burnside Problem (Group Theory)

Kurosh Problem (Ring Theory)

Tower of Fields Problem (Number Theory)

Recently: Topology


Active research area

Golod-Shafarevich groups.

Vaden, a student of Zelmanov, if there is less than d2

25 quadraticrelations then infinitely many subalgebras Rn of R = A/I areGolod-Shafarevich algebras.

Open QuestionIs the inverse of Golod-Shafarevich theorem true? Notgenerated in degree one-not true (Anick). In general open.Interesting results Wisilceny, recently Iyudu and Shkarin.


Can we assume something more about thesealgebras?

Golod-Shafarevich algebras are used to construct infinitelydimensional algebras satisfying prescribed relations.

Are there domains satisfying given relations?

Are there PI algebras satisfying given relations?

Are there Noetherian algebras satisfying given relations?


Golod-Shafarevich algebras and nil algebras

There are Golod-Shafarevich algebras which are nil.

Nil algebras which are commutative are nilpotent.

Nil algebras which are Noetherian are nilpotent

Nil algebras which satisfy polynomial identity are nilpotent.

In general it is not possible to assure that there is a domain, ora polynomial identity algebra or Noetherian algebra satisfyinggiven number of relations of each degree, even small (infinitedimensional algebra).


New result

Theorem (A.S., J.Algebra, 2013)

Given set of homogeneous relations and a finite number ofrelations and less than 2n/8 relations of degree n, and relationsare of scarce degrees. Then we can construct prime, gradedalgebras with linear growth satisfying the prescribed relations.

RemarkBy Warfield-Small theorem, prime, finitely generated algebraswith linear growth are Noetherian algebras which satisfypolynomial identity.

RemarkPolynomial Identity algebras are ’close’ to commutativealgebras.

RemarkNoetherian algebras are ’close’ to domains.


Precise formulation

Let K be an algebraically closed field, and let A be the freenoncommutative algebra generated in degree one byelements x , y .Let ξ be a natural number. Let I denote the ideal generatedin A by homogeneous elements f1, f2, . . . , fξ ∈ A.Suppose that there are exactly ri elements amongf1, f2, . . . , fξ with degrees larger than 2i and not exceeding2i+1.Assume that there are no elements among f1, f2, . . . , fξ withdegree k if 2n − 2n−3 < k < 2n + 2n−2 for some n.Denote Y = {n : rn 6= 0}. Suppose that for all n ∈ Y ,m ∈ {0} ∪ Y with m < n we have

23n+4r32m < rn < 22n−m−3

.


Assertion

Then A/I contains a free noncommutative graded subalgebra intwo generators.

These generators are monomials of the same degree. Inparticular, A/I is not Jacobson radical.

Moreover, A/I can be homomorphically mapped onto a graded,prime, Noetherian algebra with linear growth which satisfies apolynomial identity.


General construction

Let K be a field and A be a free K -algebra generated in degreeone by two elements x , y .

Suppose that subspaces U(2m),V (2m) of A(2m) satisfy, forevery m ≥ 1, the following properties:

1. V (2m) is spanned by monomials;

2. V (2m) + U(2m) = A(2m) and V (2m) ∩ U(2m) = 0;

3. A(2m−1)U(2m−1) + U(2m−1)A(2m−1) ⊆ U(2m);

4. V (2m) ⊆ V (2m−1)V (2m−1),where for m = 0 we set V (20) = Kx + Ky , U(20) = 0.


Ideal E

We define a graded subspace E of A by constructing itshomogeneous components E(k) as follows.Given k ∈ N, let n ∈ N be such that 2n−1 ≤ k < 2n.Then r ∈ E(k) precisely if, for all j ∈ {0, . . . ,2n+1 − k}, we have

A(j)rA(2n+1 − j − k) ⊆ U(2n)A(2n) + A(2n)U(2n).

More compactly,

E(k) = {r ∈ A(k) | ArA∩A(2n+1) ⊆ U(2n)A(2n) +A(2n)U(2n)}.(1)

Set then E =⋃

k∈N E(k).


Ideal E

Lemma

The set E is an ideal in A.

Moreover, if all sets V (2n) are nonzero, then algebra A/Eis infinite dimensional over K .


More on construction

1. If we have given relation fi of degree < 2n, then we canassure that elements of degree 2n+1 in AfiA is a subset of

U(2n)A(2n) + A(2n)U(2n)

2. This imply that relation fi = 0 holds in A/E

3. If we have finite number of relations, then from some point wewill have no restrictions on sets U(2n), V (2n). We can assumeV (2n) are generated by one element from some point-then A/Eis a PI (polynomial identity) algebra with linear growth.


More on construction

1. Since we have no restrictions on sets V (2n) and U(2n) fromsome point, we can assume that V (2n+1) = V (2n)V (2n) for alln > N.

2. Then elements of V (2N) will generate free subalgebra inA/E .

3. The construction of sets U(2n) and V (2n) for n < N willassure that all prescribed relations hold in algebra A/E .


New result

TheoremLet K be a field and A be a free K -algebra generated in degreeone by two elements x , y. Suppose that subspacesU(2m),V (2m) of A(2m) satisfy properties 1-4 above, andmoreover that there is n such that

dim V (2n) = 2, V (2m+1) = V (2m)V (2m)

for all m ≥ n.Then, the algebra A/E contains a free noncommutative algebrain 2 generators, and these generators are monomials of thesame degree.


Recent results on free subalgebras

Anick proved that finitely presented monomial algebras withexponential growth always contain free noncommutativesubalgebras.

Recently Bell and Rogalski proved that quotients of affinedomains with Gelfand-Kirillov dimension two over uncountable,algebraically closed fields contain free noncommutativesubalgebras in two generators.


Recent results on free subalgebras

An open question by Anick asks whether all division algebras ofexponential growth contain free noncommutative subalgebrasin two generators.

Recently Bell and Rogalski answered this question in theaffirmative in the case of algebras over C-the field of complexnumbers.

Related questions concerning Golod-Shafarevich groups havealso been studied. In particular, Zelmanov proved that a pro-pgroup satisfying the Golod-Shafarevich condition contains afree non abelian pro p-group.


Makar-Limanov Question and nil rings

Theorem (A.S., 2009, Advances in Mathematics)There are nil algebras such that when we extend their fields ofscalars, they will contain free noncommutative algebras in twogenerators.

In particular, there are finitely generated algebras which doesnot contain free subalgebras in free generators, which afterextending fields of scalars will contain free subalgebras in twogenerators.

This answered a question of Makar-Limanov.


Completely different situation

The situation in case of domains is completely different!

Theorem (Recent results by Bell and Rogalski)

Let R be a finitely generated algebra over uncountable field,which is a domain.

If Q does not contain a free noncommutative subalgebra in twogenerators, then there is a subfield F of Q and an element x inQ which is left algebraic over F .


More on free subalgebras

Theorem 1 shows that if we have a finite number of relationsand less than 2n/8 relations of degree n, and relations are ofscarce degrees then free algebra subject to these relationscontains free noncommutative subalgebras.

Therefore, there are no nil or Jacobson radical algebras whichare factor of free algebras by the the ideal generated by ri

relations of degree i , if ri satisfy assumptions of Theorem 1(A.S., 2013, J.Algebra).


Big open questions

Question 1Is there a finitely presented nil algebra?

Finitely presented algebra is a factor of a free algebra by afinitely generated ideal.

RemarkZelmanov noted that Question 1 is related to the big openquestion in Group theory, namely to the Burnside Question forfinitely presented groups.


Big problems

Previously we only assumed that our prescribed relations arehomogeneous.The case when prescribed relations are not homogeneous isrelated to some main open problems in noncommutative ringtheory, group theory and noncommutative algebraic geometry.

Kurosh problem for domainsIs a finitely generated algebraic algebra A which is a domainalways finite dimensional ?

Latyshev problemIs a finitely generated ring, which is a division ring necessarilyfinite?


Some known results

Wedderburn’s theoremFinite division rings are commutative.

Jacobson’s theoremAlgebraic division algebras over finite fields are commutative.

The Jacobson,Kaplansky, Levitzki, Shirshov resultAlgebraic algebras of bounded degree are finite dimensional.

Amitsur’s resultFinitely generated division algebras over uncountable fields arealgebraic.


Noncommutative singularities

Recently many open questions have arisen about algebrassatisfying a prescribed number of relations in the area ofresolutions of noncommutative singularities.

The following question related to equivalences of the derivedcategory of 3-folds in algebraic geometry was posed byDonovan and Wemyss.


The question of Donovan and Wemyss

Suppose that F is the formal free algebra in two variables, andconsider two relations f1, f2 such that if we write both f1 and f2as a sum of words, each word has degree two or higher.

Denote I to be the two sided ideal generated by f1 and f2.

Is it true that if F/I is finite dimensional, it cannot becommutative?


Answer (and a slightly more general result)

Lemma (A.S.)Let K be a field.Let F be either the free associative K - algebra on the set offree generators X = {x1, x2, . . . , xn} over the field K or F be theformal free power series algebra over K in n variablesx1, . . . , xn.Let d ≤ n(n−1)

2 + 1.Consider d relations f1, f2, . . . , fd ∈ F such that if we write eachof the f1, . . . , fd as a sum of words then each word has degreetwo or higher.If F/I is finite dimensional then it cannot be commutative.


Recent results

Theorem (A.S, L. Bartholdi, 2013)It is possible to construct algebras with polynomial growthsatisfying given relations if the numbers of relations of eachdegree is small enough.

Theorem (A.S, 2009)Images of some Golod-Shafarevich algebras have alwaysexponential growth.

Theorem 1 and Theorem 2 answered questions by Zelmanov.


Questions on growth of algebras

Question 1Can we construct algebras with small growth satisfying givenrelations, provided that the number of relations of each degreeis not too big?

Question 2Can we construct algebras with finite Gelfand-Kirillovdimension?

Question 3Which functions are growth functions of algebras?

Question 4Why should it interest us?


Gelfand-Kirillov dimension

1. Gelfand-Kirillov dimension was introduced by Gelfand andKirillov to solve open question on Lie algebras.

2. Gelfand-Kirillov dimension has applications innoncommutative algebraic geometry.

3. Gromov theorem says that if G is a finitely generated groupand the group algebra K [G] has finite Gelfand-Kirillovdimension then G is nilpotent by finite. Hence torsion group offinite Gelfand-Kirillov dimension is finite.

4. Growth of groups have been extensively studied and thereare some basic open questions in this area.


Positive results

Theorem (A.S, L. Bartholdi, Quarterely J.Math., 2013)It is possible to construct algebras with polynomial growthsatisfying given relations if the numbers of given relations ofeach degree is small enough.

Theorem (A.S, Glasgow J.Math., 2009)

Images of some Golod-Shafarevich algebras have alwaysexponential growth.

Theorem 1 and Theorem 2 answered questions by Zelmanov.


Golod-Shafarevich groups versus algebras

Results on Golod-Shafarevich groups and algebras often mirroreach other but proofs are different.

GroupsErshov-there exists Golod-Shafarevich groups without infiniteimages of polynomial growth (in fact there existsGolod-Shafarevich algebras satisfying property tau).

Algebras

(A.S.): Generic Golod-Shafarevich algebras with exponentialnumber of defining relations of scarce degrees have all infinitelydimensional modules, and homomorphic images of exponentialgrowth.


Homomorphic images

RemarkIf the number of defining relations of a GS algebra growspolynomially with respect to the degree, then there arehomomorphic images of polynomial growth (A.S+L.B).


Golod-Shafarevich groups versus algebras

Theorem (Zelmanov, 2000)Golod-Shafarevich groups contain non-abelian freepro-p-groups.

Related results on free subgroups of Golod-Shafarevich groupswere obtained by Kassabov.

Theorem ( A.S., 2012)Finitely presented algebras with a limited number of definingrelations contain free subalgebras.


Question 1

Which growth functions are possible for growth of algebras?

RemarkGrowth function should not depend of the generating set.Therefore, by analogy with group theory it is assumed that twofunctions f (x),g(x) are equivalent if there is number C suchthat f (x) < g(Cx) and g(x) < f (Cx).


Question 1

Theorem (A.S+Bartholdi, Quarterly J.Math, 2013)Let f be submultiplicative and increasing, that is,f (m + n) ≤ f (m)f (n) for all m,n, and f (n + 1) ≥ f (n).Then there exists a finitely generated algebra B whose growthfunction v(n) satisfies

f (2n) ≤ dim B(2n) ≤ 22n+3f (2n+1).

Furthermore, B may be chosen to be a monomial algebra.

RemarkThis implies that any sufficiently regular function that grows atleast as fast as nlog n can be growth function of an algebra.


New interesting questions

Question 2Which assumptions to add to generalize the Golod-Shafarevichtheorem in the case of ungraded relations?

Question 3When algebra generated by one relation is a domain?

Question 4When algebra generated by small number of relations can bemapped onto a domain? What assumptions about generatingrelations should we add?


More open problems

Question 5 (Amberg, Kazaring, 1998)

Let A be a finitely generated nil algebra which is not nilpotent.Can the adjoint group of A be finitely generated?

Question 6 (Bartholdi, 2011)What is growth of an adjoint group of an algebra?

Grigorchuk showed that there exist semigroups of growthstrictly between polynomial and exponential.

One of the tantalizing open problems is the existence of groupsof intermediate growth strictly between polynomial andexp(n1/2).

Thank you very much!

Please answer the posed questions, if possible!

Old and new questions in noncommutative ring theory · 2013-07-22 · Old and new questions in...

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