aa r X i v : . [ m a t h . R A ] O c t GENERATING ADJOINT GROUPS
BE’ERI GREENFELD
Abstract.
We prove two approximations of the open problem of whetherthe adjoint group of a non-nilpotent nil ring can be finitely generated: Weshow that the adjoint group of a non-nilpotent Jacobson radical cannot beboundedly generated, and on the other hand construct a finitely generated,infinite dimensional nil algebra whose adjoint group is generated by elementsof bounded torsion. Introduction
Nil rings give rise to multiplicative groups, called adjoint groups. If R is nil (or,more generally Jacobson radical) then its adjoint group R ◦ consists of the elementsof R as underlying set, with multiplication given by r ◦ s = r + s + rs . We can thinkof R ◦ as { r | r ∈ R } with usual multiplication (1 + r )(1 + s ) = 1 + r + s + rs . Theconnections between group-theoretic properties of R ◦ and ring-theoretic propertiesof R were intensively studied (for instance, see [7, 8, 9, 20]).In this way, nil rings give rise to braces (in fact, two-sided braces, namely bracesarising from Jacobson radical rings). For more information about braces and theirconnections with nil rings see [18, 19, 20].The following is an open question posed by Amber, Kazarin, Sysak [5, 6] andthen repeated by Smoktunowicz [19]: Question 1.1.
Is there a non-nilpotent, nil algebra whose adjoint group is finitelygenerated?
Note that by [22, Theorem 4.3], if R is Jacobson radical such that R ◦ is generatedby two elements then R is nilpotent.Bounded generation is a stronger notion of finite generation: it asserts thata group is generated by a finite set { g , . . . , g n } such that every element can beexpressed as g i · · · g i n n , namely, all elements have finite width with respect to thegenerating set. Boundedly generated groups are extensively studied, and for relatedinformation and results we refer the reader to [1, 2, 11, 16, 17, 21].It is easy to see that the adjoint group of a non-nilpotent nil algebra cannot beboundedly generated; in Theorem 2.1 we prove that the adjoint of a non-nilpotentJacobson radical cannot be boundedly generated.Suppose R is a nil algebra of positive characteristic p >
0, then its adjoint group R ◦ is residually-p torsion. However, its torsion cannot be bounded, since (unlessthe algebra is nilpotent) its elements’ nilpotency indices are unbounded.In Theorem 3.2 we construct a finitely generated, infinite dimensional nil ring (infact, a Golod-Shafarevich algebra) whose adjoint group is generated by elementsof bounded torsion. Indeed, if the answer to Question 1.1 is affirmative then it The author thanks Prof. Agata Smoktunowicz for helpful discussions. evidently yields such example; in this sense, Theorem 3.2 should be seen as anapproximation of Question 1.1.Our example also results in an infinite, finitely generated brace whose adjointgroup is generated by a set of elements of bounded torsion. Note that the anal-ogous question of Question 1.1 for braces has a positive solution: for any finite,non-degenerate involutive solution (
X, r ) of the Yang-Baxter equation there is anassociated brace G ( X, r ), which is infinite, both of whose multiplicative and addi-tive groups are finitely generated and also its adjoint group is; note that the additivegroup of G ( X, r ) is a free abelian group (our two-sided braces have p-torsion addi-tive groups). For details, see [12, Theorem 4.4].2.
Boundedly generated adjoint groups
Recall that a group is boundedly generated if G = H · · · H n for some cyclicsubgroups H , . . . , H n ⊆ G . The minimal such n is called the cyclic width of G .Note that the adjoint group of an infinite dimensional nil algebra cannot beboundedly generated, since that would mean the algebra admits a Shirshov base,namely every element would be a linear combination of monomials of the form f i · · · f i n n for some f , . . . , f n ; since the algebra is nil, there are only finitely manysuch monomials, so the algebra must be finite dimensional.We now extend this result to an arbitrary Jacobson radical. Note that Jacobsonradicals need not be nil, so the above argument does not hold any longer. Theorem 2.1.
Let R be an algebra which is Jacobson radical. Then R ◦ is boundedlygenerated if and only if R is finite dimensional and nilpotent.Proof. First observe that we only have to deal with the case where the base field F is finite. Indeed, observe by Nakayama’s lemma that R/R is a non-zero F -vectorspace. Therefore the additive group of R/R , being isomorphic to ( R/R ) ◦ occursas a homomorphic image of R ◦ . Hence, if R ◦ is finitely generated then so is theadditive group of the base field F , from which it follows that F must be finite. Wehenceforth assume F = F p .Assume that R is an algebra which is Jacobson radical and R ◦ is boundedlygenerated with cyclic width m . Denote G n = ( R n +1 ) ◦ ⊳ R ◦ . Note that G n hasfinite index since R/R n +1 is finite F -dimensional. Given a finite group G denoteby exp ( G ) the minimum power α such that g α = e for all g ∈ G .Observe that exp ( R ◦ /G n ) ≤ p ( n + 1) since given f ∈ R , we have that:(1 + f ) p ⌈ log p ( n +1) ⌉ = 1 + f p ⌈ log p ( n +1) ⌉ ∈ G n (as: f p ⌈ log p ( n +1) ⌉ ∈ R n +1 and n + 1 ≤ p ⌈ log p ( n +1) ⌉ ≤ p ( n + 1).)Note that if a group G is boundedly generated of cyclic width m then for anyfinite index normal subgroup H we have that [ G : H ] ≤ exp ( G/H ) m and in partic-ular in our case: p dim F p ( R/R n +1 ) = [ R ◦ : G n ] ≤ exp ( R ◦ /G n ) m ≤ ( p ( n + 1)) m . It follows that (denoting R = R ): n X i =0 dim F p ( R i /R i +1 ) = dim F p ( R/R n +1 ) ≤ m (1 + log p ( n + 1)) < n ENERATING ADJOINT GROUPS 3 for n ≫
1, hence for some 0 ≤ i ≤ n we have that R · R i = R i +1 = R i . Since R is a finitely generated algebra which is Jacobson radical it follows by Nakayama’slemma that R i = 0. Hence R is a finite dimensional, nilpotent algebra. (cid:3) Remark 2.2.
Note that in fact, Theorem 2.1 requires a weaker assumption on R ◦ than being boundedly generated: if R is a Jacobson radical whose adjoint group isa polynomial index growth (PIG) group, namely group in which the index of anysubgroup is polynomially bounded by its relative exponent, then R is nilpotent. Formore information and results on PIG groups, see [10, 13, 14, 17] . Adjoint group generated by a uniformly torsion set
Let F be a countable field of characteristic p >
0. Consider the free algebra A = F h x, y i . Observe that A = L ∞ n =0 A ( n ) is graded by deg( x ) = deg( y ) = 1.Denote A + = L ∞ n =1 A ( n ). For a ∈ A , let v ( a ) = max { d | a ∈ L i ≥ d A ( i ) } . Lemma 3.1.
Let a ∈ A + and write a = a + · · · + a n a sum of homogeneouselements. For every m ≥ there exist some homogeneous elements h , . . . , h k andan element b ∈ A + with v ( b ) ≥ m such that: a + b = (1 + h ) · · · (1 + h k ) . Proof.
Assume on the contrary that b ∈ A + is such that 1+ a + b = (1+ h ) · · · (1+ h k )for some homogeneous h , . . . , h k with d = v ( b ) maximal. Write b = b + · · · + b m asum of homogeneous elements. By maximality of d we get that d = min { deg( b i ) | ≤ i ≤ m } .Note that:(1 + h ) · · · (1 + h k )(1 − b ) · · · (1 − b m ) = (1 + a + b )(1 − b ) · · · (1 − b m ) =(1 + a + b )(1 − b + c ) = 1 + a + ( c + ac + bc − ab − b ) . with: c = X S ⊆{ ,...,m } ; | S | > Y i ∈ S b i . Observe that v ( ab ) , v ( b ) , v ( c ) > d and hence v ( c + ac + bc − ab − b ) > d , contra-dicting the maximality of d . (cid:3) Theorem 3.2.
There exists a finitely generated, graded, infinite dimensional nilalgebra R whose adjoint group R ◦ is generated by a set of elements of boundedtorsion.Proof. Enumerate A + = { f , f , . . . } . Using Lemma 3.1 pick b with v ( b ) ≥ f + b = (1 + h , ) · · · (1 + h ,k ) for some homogeneous elements h , , . . . , h ,k . Write b = b , + · · · + b ,n a sum of homogeneous elements withdeg( b ,d ) = d . Next use Lemma 3.1 to choose b with v ( b ) ≥ n +1 and 1+ f + b =(1 + h , ) · · · (1 + h ,k ) for some homogeneous elements h , , . . . , h ,k . Again, write b = b ,n +1 + · · · + b ,n a sum of homogeneous elements with deg( b ,d ) = d , andinductively proceed, each time constructing b l +1 with v ( b l +1 ) ≥ n l + 1 such that1 + f l +1 + b l +1 = (1 + h l +1 , ) · · · (1 + h l +1 ,k l +1 ) for suitable homogeneous elements h l +1 , , . . . , h l +1 ,k l +1 .Let I ⊳ A + be the ideal generated by { b i,j | i ≥ } . Note that I is generated by atmost one homogeneous element of each degree, starting from 14. BE’ERI GREENFELD
Now let J be the ideal generated by h p α (with α = ⌈ log p ⌉ ) for all homogeneouselements h ∈ A + . Observe that J is generated by 2 d elements of degree p α d ≥ d for d ≥ A + / ( I + J ) is a Golod-Shafarevich algebra. Indeed, let r n denotethe number of degree n generators of I + J . Then for all τ > X n> r n τ n ≤ ∞ X d =1 d τ d + X n =3 τ n = 2 τ − τ + τ − τ . Setting f ( τ ) = 1 − τ + P n> r n τ n , we get that: f (0 . ≈ − . . . < , hence A + / ( I + J ) is a Golod-Shafarevich algebra. Note that the semigroup ( A + / ( I + J )) ◦ is generated by the set { f | f homogeneous } , whose elements are torsion of order ≤ p α . Now by [23] it follows that A + / ( I + J ) admits a homomorphic image whichis an infinite dimensional and nil algebra, say R . Then R satisfies the claimedproperties. (cid:3) Further questions
We conclude with two open questions related to Question 1.1:
Question 4.1.
Is there an infinite dimensional, finitely generated nil algebra withpolynomial growth whose adjoint group is finitely generated?
In [3] it is proved that every elementary amenable subgroup of the adjoint groupof a nil ring is locally nilpotent.
Question 4.2.
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