ALGEBRA ISRN Algebra 2090-6293 International Scholarly Research Network 170697 10.5402/2012/170697 170697 Review Article Growth for Algebras Satisfying Polynomial Identities Regev Amitai Aljadeff E. Marko F. Mathematics Department The Weizmann Institute, 76100 Rehovot Israel weizmann.ac.il 2012 21 11 2012 2012 06 09 2012 25 09 2012 2012 Copyright © 2012 Amitai Regev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The nth codimension cn(A) of a PI algebra A measures how many identities of degree n the algebra A satisfies. Growth for PI algebras is the rate of growth of cn(A) as n goes to infinity. Since in most cases there is no hope in finding nice closed formula for cn(A), we study its asymptotics. We review here such results about cn(A), when A is an associative PI algebra. We start with the exponential bound on cn(A) then give few applications. We review some remarkable properties (integer and half integer) of the asymptotics of cn(A). The representation theory of the symmetric group Sn is an important tool in this theory.

1. Introduction

We study algebras A satisfying polynomial identities (PI algebra). A natural question arises: give a quantitative description of how many identities such algebra A satisfies? We assume that A is associative, though the general approach below can be applied to nonassociative PI algebras as well.

Denote by Id(A) the ideal of identities of A in the free algebra F{x}. If Id(A)0 then its dimension dimId(A) is always infinite, hence dimension by itself is essentially of no use here. In order to overcome this difficulty we now introduce the sequence of codimensions.

1.1. Growth for PI Algebras

Given n, we let Vn denote the multilinear polynomials of degree n in x1,,xn, so in the associative case dimVn=n!. Identities can always be multilinearized, hence the subset Id(A)Vn and its dimension give a good indication as to how many identities of degree nA satisfies. In fact, in characteristic zero the ideal Id(A) is completely determined by the sequence {Id(A)Vn}n1, but we make no use of that remark in the sequel.

To study dim(Id(A)Vn), we introduce the quotient space Vn/(Id(A)Vn) and its dimension (1.1)cn(A)=dim(VnId(A)Vn). The integer cn(A) is the nth codimension of A. Clearly cn(A) determines dim(Id(A)Vn) since dimVn is known.

The study of growth for PI algebra A is mostly the study of the rate of growth of the sequence cn(A) of its codimensions, as n goes to infinity. We have the following basic property.

Theorem 1.1 (see [<xref ref-type="bibr" rid="B129">1</xref>]).

In the associative case, cn(A) is always exponentially bounded.

This theorem implies several key properties for PI algebras. And it fails in various nonassociative cases.

Various recent results indicate that in general there is no hope to find a closed formula for cn(A). Instead, one therefore tries to determine the asymptotic behavior of that sequence, as n goes to infinity. We mention here three such results.

Recall that for two sequences of numbers, an~bn if limnan/bn=1.

(1) The asymptotics for the k×k matrices Mk(F), see Section 6.

Theorem 1.2 (see [<xref ref-type="bibr" rid="B137">2</xref>]).

When n goes to infinity, (1.2)cn(Mk(F))~[(12π)k-1(12)(k2-1)/2·1!2!(k-1)!·k(k2/2)]·(1n)(k2-1)/2·k2(n+1).

(2) The integrality theorem of Giambruno-Zaicev, see Section 10.

Theorem 1.3 (see [<xref ref-type="bibr" rid="B78">3</xref>]).

Let A be an associative PI F-algebra with char (F)=0, then the limit (1.3)limn(cn(A))1/n exists and is an integer. We denote exp(A)=limn(cn(A))1/n, so exp (A).

(3) The “1/2” theorem of Berele, see Section 11.

Theorem 1.4 (see [<xref ref-type="bibr" rid="B28">4</xref>, <xref ref-type="bibr" rid="B36">5</xref>]).

Let A be a PI algebra with 1A. Then as n goes to infinity, cn(A)~a·nt·hn, moreover, h (h is given by the previous theorem) and t(1/2), namely, t is an integer or a half integer.

1.2. Structure of the Paper

The paper reviews some of the main results about the asymptotics of codimensions. It does not contain full proofs but rather, it indicates some of the key ideas in the proofs of the main results.

We start by introducing Kemer's classification of the verbally prime T-ideals. After introducing the codimensions, two proofs of their exponential bound are reviewed. The Sn character of that space, χSn(Vn/(Id(A)Vn)) is denoted as follows: (1.4)χn(A)=χSn(VnId(A)Vn) and is called the nth cocharacter of A. Since cn(A)=degχn(A), cocharacters are refinement of codimensions, and are important tool in their study. By a theorem of Amitsur-Regev and of Kemer, χn(A) is supported on some (k,) hook. Shirshov's Height Theorem then implies that the multiplicities in the cocharacters are polynomially bounded.

We then review the proof of the Giambruno-Zaicev Theorem in the finite dimensional case, and the proof of Berele's “1/2” Theorem in the case of a Capelli identity.

The question of the algebraicity of the generating function ncn(A)·xn is examined in Section 12.

In the last section, we review a construction of nonassociative algebras where the integrality property of the exponent fails.

2. PI Algebras and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M76"><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula> Ideals 2.1. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M77"><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula>-ideals

Let F be a field. In most cases we assume that char(F)=0. We begin by studying associative F-algebras. Analogue theories exist for nonassociative algebras. Let F{x}=F{x1,x2,} denote the algebra of associative and noncommutative polynomials in the countable sequence of variables x1,x2,. The polynomial f(x1,,xn)F{x} is an identity of the F-algebra A if f(a1,,an)=0 for every a1,,anA. The algebra A satisfies polynomial identities, or in short is PI, if there exist a nonzero polynomial 0f(x)=f(x1,,xn)F{x} which is an identity of A. For example, any commutative algebra is PI since it satisfies x1x2-x2x1. Applying alternating polynomials imply that every finite dimensional algebra, associative or nonassociative is PI; see Section 3.1. The class of the PI algebras is both large and interesting! We remark that the algebra Mk(F) of the k×k matrices over F plays a basic role in PI theory.

Definition 2.1.

(1) Let Id(A)F{x} be the subset of the identities of the algebra A. (2.1)Id(A)={fF{x}f(x)=0  isanidentityofA}.

(2) T ideals. The set Id(A) is a two sided ideal in F{x}, with the additional property that it is closed under substitutions. Such ideals in F{x} are called T ideals. Thus, Id(A) is a T ideal. It is easy to show that a T ideal is always the ideal of identities of some PI algebra A.

(3) Varieties of PI algebras. Given a T-ideal IF{x}, the class of the PI algebras A satisfying IId(A) is the variety corresponding to I. We use here the language of T ideals, rather than that of varieties.

2.2. Kemer's Theory for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M111"><mml:mrow><mml:mi>T</mml:mi></mml:mrow></mml:math></inline-formula>-Ideals (See [<xref ref-type="bibr" rid="B98">6</xref>]) The Specht Problem

One of the main problems in PI theory is the Specht problem: Are T ideals always finitely generated as T ideals?

Kemer  developed a powerful structure theory for T ideals which allowed him to prove that if char(F)=0 then T ideals are indeed finitely generated. We review some of the ingredients of that theory.

Amitsur  proved that the T ideal I=Id(Mk(F)) is prime in the following sense. If fgI then either fI or gI; moreover, the only prime T ideals in F{x} are I=Id(Mk(F)).

Kemer introduced the notion of verbally prime ideals as follows.

Definition 2.2.

The T ideal I is verbally prime if it satisfies the following condition: Let f(x1,,xm) and g(xm+1,,xm+n) be polynomials in disjoint sets of variables. If (2.2)f(x1,,xm)·g(xm+1,,xm+n)IthenfIorgI. Kemer then classified the verbally prime ideals in F{x}, see Theorem 2.3 below.

2.2.1. The Algebras <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M131"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>F</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M132"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>G</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>, and <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M133"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

Let U=spanF{u1,u2,} be an infinite dimensional vector space, and let G=G(U) be the corresponding infinite dimensional Grassmann (Exterior) algebra. Then (2.3)G=span{ui1uirr=1,2,and1i1<<ir}, and G=G0G1, where (2.4)G0=span{ui1uirreven},G1=span{ui1uirrodd}. Now Mk(F) are the k×k matrices over F, while Mk(G) are the k×k matrices over G.

The Algebra <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M145"><mml:mrow><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>

We have Mk,Mk+(G). The elements of Mk, are block matrices (2.5)(ABCD), where AMk(G0), DM(G0), B is k× and C is ×k, both with entries from G1. For example: (2.6)M1,1=(G0G1G1G0).

Theorem 2.3 (see [<xref ref-type="bibr" rid="B98">6</xref>]).

The following are the three families of the verbally prime T ideals: Id (Mk(F)), Id (Mk(G)), and Id (Mk,).

The importance of the verbally prime ideals is demonstrated in the following theorem.

Theorem 2.4 (see [<xref ref-type="bibr" rid="B98">6</xref>]).

Let IF{x} be a T ideal. Then there exist verbally prime T ideals J1,,Jr such that (2.7)J1JrIJ1Jr.

3. Multilinear Polynomials

As usual, Sn denotes the nth symmetric group.

Definition 3.1.

The polynomial f(x1,,xn)F{x} is multilinear (in x1,,xn) if (3.1)f(x1,,xn)=σSnασxσ(1)xσ(n) for some coefficients ασF. Let Vn=Vn(x1,,xn) denote the vector space of multilinear polynomials in x1,,xn, so dimVn=n!. Extending the map σxσ(1)xσ(n) by linearity yields the vector space isomorphism VnFSn, where FSn is the group algebra of Sn. We will identify (3.2)FSnVn. By the process of multilinearization [8, 9] one proves the following theorem.

Theorem 3.2.

Let the PI algebra A satisfy an identity of degree d, then A satisfies a multilinear identity of degree d. Moreover, if char (F)=0 then the ideal of identities Id (A) is determined by its multilinear elements.

It follows that if A satisfies an identity of degree d then A satisfies an identity of the form (3.3)y1yd-1πSdαπyπ(1)yπ(d),απF. This fact is applied in Section 4.1 in proving the exponential bound for the codimensions.

3.1. Example: Standard and Capelli Identities

The polynomial f(x1,,xn) is alternating in x1,,xn if for every permutation πSn, f(xπ(1),,xπ(n))=sgn(π)f(x1,,xn). For example, the polynomial (3.4)Stn[x]=Stn[x1,,xn]=σSnsgn(σ)·xσ(1)·xσ(2)xσ(n) is alternating. It is called the nth Standard polynomial.

Similarly the polynomial (3.5)Capn[x;y]=σSnsgn(σ)·xσ(1)·y1·xσ(2)·y2yn-1·xσ(n), which is called the (nth) Capelli polynomial, is multilinear of degree 2n-1 and is alternating in x1,,xn.

It is rather easy to show that if dimA=d< then A satisfies both Capd+1[x] and Std+1[x], and hence every finite dimensional algebra is PI. And the same argument applies in the nonassociative case. In particular the algebra Mk(F) of the k×k matrices satisfies Stk2+1[x1,x2,,xk2+1]. The celebrated Amitsur-Levitzki Theorem  states that Mk(F) satisfies the standard identity St2k[x1,,x2k]=0. Of course, for large k the degree 2k is much smaller than k2+1.

Many infinite dimensional algebras are PI. For example, any infinite dimensional commutative algebra is PI. We remark that obviously, the free algebra F{x} itself is not PI.

4. The Codimensions

Question. How many identities are satisfied by a given PI algebra, namely, how large are T ideals?

Computing dimensions might seem useless at first sight, since if the T-ideal I is nonzero then dimI=. To answer the above question we introduce below the notion of codimensions. Given a PI algebra A, we would like to study its multilinear identities of degree n, namely, the space (4.1)Id(A)Vn(x1,,xn)=Id(A)Vn,Vn as in (3.2). Note that if y1,,yn are any n variables then Id(A)Vn(x)Id(A)Vn(y). A first step is the study of the dimensions dim(Id(A)Vn). Now (Id(A)Vn)Vn, and since dimVn=n!, clearly (4.2)dim(Id(A)Vn)=n!-dim(VnId(A)Vn). Thus, knowing dim(Id(A)Vn) is equivalent to knowing dim(Vn/(Id(A)Vn)). This leads us to introduce the codimensions cn(A) as follows.

Definition 4.1.

Let A be an F-algebra, then (4.3)cn(A)=dim(VnId(A)Vn) is called the nth codimension of A, and {cn(A)}n=1 is the sequence of codimensions of A.

For example, A satisfies An=0 (i.e., is nilpotent) if and only if cn(A)=0. And A is commutative if and only if cn(A)1 for all n.

Remark 4.2.

Note that if A is not PI then Id(A)=0, hence cn(A)=n! for all n. In fact, the algebra A is PI (i.e., Id(A)0) if and only if there exist n such that cn(A)<n!. This follows directly from the definition.

4.1. Exponential Bound for the Codimensions

A basic property of the codimensions sequence cn(A) in the associative case is, that it is bounded by exponential growth. Applications of this fact are given in the sequel.

Theorem 4.3 (see [<xref ref-type="bibr" rid="B129">1</xref>]).

Assume the (associative) algebra A satisfies an identity of degree d, then (4.4)cn(A)((d-1)2)n for all n.

Proof.

We sketch two proofs (both different from the original proof). Both proofs apply the notion of a d-good permutation, which we now review.

Call σSnd-bad if there are indices 1i1<<idn with σ(i1)>>σ(id). Otherwise σ is d-good, and we denote (4.5)gd(n)=|{σSnσisd-good}|. By a Shirshov-Latyshev argument [13, 14], if A satisfies an identity of degree d, then A satisfies an identity (3.3) which, by a certain inductive argument implies that cn(A)gd(n).

By an argument based on Dilworth Theorem in Combinatorics , Latyshev then showed that gd(n)((d-1)2)n, thus completing the (first) proof.

A second proof of the bound gd(n)((d-1)2)n goes as follows . First, by the RSK correspondence , (4.6)gd(n)=λn,(λ)d-1(fλ)2, where fλ is the number of standard Young taableaux of shape λ, and (λ) is the number of parts of λ. Then by the Schur-Weyl theory, (4.7)λn,(λ)d-1(fλ)2=dim(B(d-1,n)), where dimU=d-1 and B(d-1,n)EndF(Un). The (second) proof now follows since     dim(EndF(Un))=(d-1)2n.

4.1.1. Application: The <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M280"><mml:mi>A</mml:mi><mml:mo>⊗</mml:mo><mml:mi>B</mml:mi></mml:math></inline-formula> Theorem

Codimensions where introduced, in , in order to prove the following theorem.

Theorem 4.4 (see [<xref ref-type="bibr" rid="B129">1</xref>]).

If A and B are (associative) PI algebras then AB is PI.

Proof.

It is not too difficult to show that cn(AB)cn(A)·cn(B). Assume now that A satisfies an identity of degree d, and B satisfies an identity of degree h. Together with Theorem 4.3, this implies that (4.8)cn(AB)(d-1)2n(h-1)2n, hence for a large enough n, cn(AB)<n!. Finally by Remark 4.2,  R is PI if and only if there exist n such that cn(R)<n!, and the proof follows.

Remark 4.5.

For explicit identities for AB see Remark 8.2 below, which also implies the following. Again let A satisfy an identity of degree d, and B an identity of degree h. Then AB satisfies an identity of degree about e(d-1)2(h-1)2, where e=2.718.

We remark that both Theorems 4.3 and 4.4 fail in some non-associative cases.

The above results motivate the study of the following problem.

Question 1.

Given a PI algebra A, find a formula for the sequence cn(A). In most cases this seems to be too difficult, and one tries to, at least, get a “nice” asymptotic approximation of cn(A).

In Section 6 we determine such asymptotics for the algebras Mk(F). We also give such partial results for the other verbally prime algebras Mk, and Mk(G).

5. Cocharacters

Recall the identification FSnVn (see (3.2)) and let A be a PI algebra. The regular left action of Sn on Vn is as follows. Let σSn and g(x1,,xn)Vn, then (5.1)σg(x1,,xn)=g(xσ(1),,xσ(n)). This makes Vn into a left Sn module, with Id(A)Vn a submodule of Vn.

Definition 5.1.

The Sn-character (5.2)χn(A)=χSn(VnId(A)Vn) of the quotient module Vn/(Id(A)Vn) is called the nth cocharacter of A.

Cocharacters where introduced in  in the study of standard identities.

Since char(F)=0, from the ordinary representation theory of Sn [19, 20] it is well known that the irreducible Sn characters are parametrized by the partitions λ on n: (5.3)Irred(Sn)={χλλn}. Also, FSn decomposes as follows: (5.4)FSn=λnIλ, where the Iλ are the minimal two sided ideals of FSn, and there are natural bijections (5.5)χλλIλ. Here degχλ=fλ and dimIλ=(fλ)2 where, again, fλ is the number of standard Young taableaux of shape λ.

Given a tableau Tλ of shape λn, it determines the semi-idempotent eTλ=RTλ+CTλ-, and FSneTλIλ is a minimal left ideal.

The Sn-character of the regular representation FSnVn is χSn(Vn)=λnfλχλ. It follows that the nth cocharacter of A can be written as (5.6)χn(A)=λnmλ(A)χλ for some multiplicities mλ(A), and mλ(A)fλ.

Clearly, the degree of χn(A) is the corresponding codimension cn(A), and since deg(χλ)=fλ, hence (5.7)cn(A)=deg(χn(A))=λnmλ(A)fλ. For example  let G be the infinite dimensional Grassmann (Exterior) algebra, then (5.8)mλ(G)={1ifλ=(m-r,1r)0otherwise.

Remark 5.2.

Properties of the identification FSnVn imply the following characterization of Capelli identities (Section 3.1) by cocharacters . Let A be a PI algebra with χn(A)=λnmλ(A)χλ its cocharacter. Then A satisfies Capd+1[x;y] if and only if mλ(A)=0 whenever (λ)d+1.

As an application of cocharacters one can prove the following.

Proposition 5.3 (see [<xref ref-type="bibr" rid="B134">23</xref>]).

If A satisfies Cap d+1 and B satisfies Cap h+1 then AB satisfies Cap dh+1.

We remark that a proof of this result without applying cocharacters is yet unknown.

Question 2.

Given a PI algebra A, find a formula for the multiplicities mλ(A). In most cases this is too difficult, and one tries to at least get some approximate description of mλ(A).

Remark 5.4.

The approach of codimensions and of cocharacters in the study of PI algebras applies also in the nonassociative case (though with different phenomena).

5.1. The Cocharacters of Matrix Algebras

In the case of the 2×2 matrices M2(F) there is the following nice formula for the multiplicities mλ(M2(F)) of χn(M2(F)).

Example 5.5 (see [<xref ref-type="bibr" rid="B43">24</xref>], [<xref ref-type="bibr" rid="B46">25</xref>, Theorem 12.6.5], [<xref ref-type="bibr" rid="B52">26</xref>]).

Denote mλ(M2(F))=mλ. First, if (λ)>4 then mλ=0. So let λ=(λ1,,λ4).

If λ=(n) then m(n)=1.

If λ=(λ1,λ2) with λ2>0 then m(λ1,λ2)=(λ1-λ2+1)·λ2.

If λ=(λ1,1,1,λ4) (so λ41) then m(λ1,1,1,λ4)=λ1·(2-λ4)-1.

And mλ=(λ1-λ2+1)·(λ2-λ3+1)·(λ3-λ4+1) in all other cases.

Recent results of Berele  and of Drensky and Genov  indicate that when k3, there is no hope in getting nice formulas for the cocharacter-multiplicities mλ(Mk(F)). A somewhat similar phenomena is discussed in Section 12.

5.2. Trace Identities, Codimensions, and Cocharacters

In the case of k×k matrices, the following is an extension of the previous theory of codimensions and cocharacters.

Instead of ordinary polynomials we can consider trace polynomials, namely, polynomials involving variables and traces, for example x1·x2·tr(x3·x4) is a (mixed) trace polynomial. These trace polynomials can be evaluated on the algebra Mk(F) (or on any algebra with a trace) hence yielding trace identities. For example, it can be proved that the trace polynomial (5.9)g(x1,x2)=tr(x1)tr(x2)-x1tr(x2)-x2tr(x1)-tr(x1x2)+x1x2+x2x1 is a trace identity of M2(F). This is an example of a mixed trace polynomial, while the polynomial (5.10)p(x1,x2,x3)=tr(x1)tr(x2)tr(x3)-tr(x1x3)tr(x2)-tr(x2x3)tr(x1)-tr(x1x2)tr(x3)+tr(x1x2x3)+tr(x2x1x3) is a pure trace polynomial, which is also an identity of M2(F). We then have trace identities of Mk(F), “pure” and “mixed,” hence trace codimensions cnptr(Mk(F)) and cnmtr(Mk(F)), and trace cocharacters χnptr(Mk(F)) and χnmtr(Mk(F)).

The Procesi-Razmyslov theory of trace identities , together with the Schur-Weyl theory [32, 33], imply the following formula for the pure trace cocharacters χnptr(Mk(F)) of Mk(F).

Theorem 5.6.

We have (5.11)χn ptr (Mk(F))=λn,(λ)kχλχλ.

This implies that the trace codimensions are given by the following formula: (5.12)cn ptr (Mk(F))=λn,(λ)k(fλ)2. This formula is the starting point for computing the asymptotic formula of cn(Mk(F)) given in Section 6.

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For the 2×2 matrices, Procesi  proved the following formula for cn(M2(F)).

Theorem 6.1 (see [<xref ref-type="bibr" rid="B126">34</xref>]).

We have (6.1)cn(M2(F))=1n+1(2n+2n+1)-(n3)+1-2n.

It was already mentioned that when k3, it most likely is impossible to find an exact formula for the multiplicities mλ(Mk(F)), and the same is probably true about cn(Mk(F)). Instead of giving up, one looks for the asymptotic of cn(Mk(F)).

Here we have the folowing theorem.

Theorem 6.2 (see [<xref ref-type="bibr" rid="B137">2</xref>]).

When n goes to infinity, (6.2)cn(Mk(F))~[(12π)k-1(12)(k2-1)/2·1!2!(k-1)!·k(k2/2)]·(1n)(k2-1)/2·k2(n+1).

For example, when k=2 both Theorems 6.1 and 6.2 give the same asymptotic value (6.3)cn(M2(F))~4π·1n·4n, compare with (12.13).

We review the major steps toward the proof of Theorem 6.2. First, it follows from deep results of Formanek [2, 26, 35] that the nth codimensions and the n+1-st pure trace codimensions are asymptotically equal: (6.4)cn(Mk(F))~cn+1ptr(Mk(F)). We saw that by Theorem 5.6, (6.5)cnptr(Mk(F))=λn,(λ)k2(fλ)2. Thus, (6.6)cn(Mk(F))~λn+1,(λ)k2(fλ)2. The last major step here is the computation of the asymptotic behavior of the following sum: (6.7)Sk2(2)(n)=λn,(λ)k2(fλ)2. The asymptotics, as n, of the more general sums (6.8)Sh(β)(n)=λn,(λ)h(fλ)β is given in . That asymptotic is of the form Sh(β)(n)~a·nb·rn, where a=a(β,h), b=b(β,h) and r=r(β,h)=hβ, all given explicitly in . We remark that the constant term a is evaluated by applying the Selberg integral (6.9).

Theorem 6.3 (see [<xref ref-type="bibr" rid="B145">36</xref>]).

The Selberg integral (6.9)0101i=1kuix-1(1-ui)y-11i<jk|ui-uj|2z    du1duk=j=0k-1Γ(x+jz)Γ(y+jz)Γ(1+(j+1)z)Γ(x+y+(k+j-1)z)Γ(1+z).

Together, the above steps yield the asymptotic value of Theorem 6.2.

6.1. The Other Verbally Prime Algebras

For the other verbally prime algebras (see Section 2.2.1), we quote the following partial asymptotic results.

Theorem 6.4.

(1) [37, Theorem 7] (6.10)cn(Mk,)~a·(1n)(k2+2-1)/2·(k+)2n. The constant a is yet unknown.

(2) [37, Theorem 8] Let G be the infinite dimensional Grassmann algebra, then (6.11)cn(Mk(G))~b·(1n)g·(2k2)n, where the constants b and g are yet unknown, and (k2-1)/2g(2k2-1)/2.

7. Shirshov's Height Theorem (See [<xref ref-type="bibr" rid="B146">13</xref>]) 7.1. The Theorem

A powerful tool in the study of PI algebras is Shirshov's Height Theorem, which we now quote. We consider the alphabet x1,,x of letters; W(x1,,x) is the set of all words (i.e., monomials) in x1,,x; U(d,) the subset of the words of length ≤d. We consider finitely generated PI algebra A=F{a1,,a}.

Theorem 7.1 (Shirshov's Height Theorem).

Consider a PI algebra satisfying the identity (3.3): (7.1)y1yd-1πSdαπyπ(1)yπ(d). There exists h=h(d,) large enough such that any finitely generated algebra A=F{a1,,a} that satisfies the identity (3.3), satisfies the following condition.

Modulo Id(A), F{x1,,x} is spanned by the elements (7.2){u1k1uhkhuiU(d,),anykj}.

7.2. Application: Bounds on the Cocharacters 7.2.1. The <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M456"><mml:mo stretchy="false">(</mml:mo><mml:mi>k</mml:mi><mml:mo>,</mml:mo><mml:mi>ℓ</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> Hook Theorem

Denote by H(k,;n) the partitions of n in the (k,) hook: (7.3)H(k,;n)={(λ1,λ2,)nλk+1},H(k,)=n(H(k,;n)). Let χn be Sn characters, n=1,2,. We say that χn is supported on H(k,), and denote χnH(k,), if for all n, (7.4)χn=λH(k,;n)mλχλ. Similar terminology applies when H(k,) is replaced by another family of subsets of partitions. We have the following theorem.

Theorem 7.2 (see [<xref ref-type="bibr" rid="B98">6</xref>, <xref ref-type="bibr" rid="B10">38</xref>]).

Let A be any (associative) PI algebra, then there exist k, such that its cocharacters χn(A) are supported on the (k,) hook H(k,).

Explicitly, let A satisfy an identity of degree d, and let k,e·(d-1)4-1, where e=2.718 is the base of the natural logarithms. Then χn(A)H(k,), namely, (7.5)χn(A)=λH(k,;n)mλ(A)·χλ.

7.2.2. An Application of Shirshov's Theorem

We remark that the proof of Theorem 7.2 applies the exponential-bound of Theorem 4.3. Theorem 7.2 is the first step towards proving the following polynomial bound.

Theorem 7.3 (see [<xref ref-type="bibr" rid="B22">39</xref>, <xref ref-type="bibr" rid="B31">40</xref>]).

For any PI algebra A, all its cocharacter-multiplicities mλ(A) are polynomially bounded. There exist a constant C and a power p such that for all n and λn, mλ(A)C·np.

The proof of Theorem 7.3 also applies Shirshov's Height Theorem 7.1, as well as a 2 version of the Schur-Weyl theory [32, 33, 41].

8. Explicit Identities

Amitsur  proved that any PI algebra satisfies a power of a standard identity, namely, an identity of the form Stuv[x]=(Stu[x])v, where Stu[x] is the uth standard polynomial. Amitsur's proof, which applies Structure Theory of Rings , yields a bound on the index u but not on v. A recent proof of Amitsur's theorem  applies the identification FSnVn, together with the exponential bound on cn(A) and yields a combinatorial proof of that theorem, a proof which gives bounds on both u and v. Moreover, the same arguments yield explicit identities in various other cases, for example in the AB case.

Theorem 8.1 (see [<xref ref-type="bibr" rid="B10">38</xref>, <xref ref-type="bibr" rid="B132">43</xref>]).

Let A be a PI algebra satisfying cn(A)αn, and let u and v be integers satisfying u,ve·α2, then St uv[x1,,xu] Id (A). In particular if A satisfies an identity of degree d, and u,ve·(d-1)4, then St uv[x1,,xu] Id (A) (of degree uv which is about e2(d-1)8.)

In fact, with these u and vA satisfies the power of the Capelli identity: ( Cap u[x;y])v Id (A).

Remark 8.2.

Let A satisfy an identity of degree d and B an identity of degree h. Denote α=(d-1)2(h-1)2 then cn(AB)αn. Thus, if u,ve·((d-1)(h-1))4 then AB satisfies the identities Stuv[x1,,xu] and (Capu[x;y])v. Note that here the degree of Stuv[x1,,xu], for example, is about e2·((d-1)(h-1))8.

Also, AB satisfies some identity of degree n, where n is about eα=e((d-1)(h-1))2. Indeed, let eα<n, then the classical inequality (n/e)n<n! implies that αn<n!. Thus, if e(d-1)2(h-1)2<n then, for that n, (8.1)cn(AB)((d-1)2(h-1)2)n<n! and by Remark 4.2  AB satisfies an identity of degree n where n is about e((d-1)(h-1))2.

9. Nonidentities for Matrices: The Polynomial <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M540"><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">[</mml:mo><mml:mi>x</mml:mi><mml:mo>;</mml:mo><mml:mi>y</mml:mi><mml:mo stretchy="false">]</mml:mo></mml:math></inline-formula>

Usually, lower bounds for codimensions are harder to obtain than upper bounds. Given a PI algebra A, a lower bound for cn(A) can be obtained by the following technique. Find a polynomial p=p(x1,,xn)Vn which is a nonidentity of A, namely, pId(A). In addition, with the identification VnFSn=λnIλ (see (5.4)), verify that pIμ for some μn. Then cn(A)fμ. This follows since 0FSnpIμ, so JμFSnp for some minimal left ideal JμIμ with JμId(A)=0, hence fμ=dimJμcn(A). We now construct such a polynomial p via the polynomial Lk(x;y), when A=Mk(F) [44, Definition  2.3].

Corresponding to the sum 1+3++(2k-1)=k2, construct the monomial Nk(x;y) of degree 2k2: (9.1)Nk(x;y)=(x1)(y1)(x2x3x4)(y2y3y4)(x5x9)(y5y9). For example, N3(x;y)=(x1)(y1)(x2x3x4)(y2y3y4)(x5x9)(y5y9). Now alternate the x's and alternate the y's to obtain Lk(x;y): (9.2)Lk(x;y)=σ,πSk2sgn(σ)sgn(π)Nk(xσ(1),,xσ(k2);yπ(1),,yπ(k2)). Let Tμ be the conjugate tableau of Tμ, and (9.3)Tμ=134591326781418. Then Lk(x;y)=CTμ- and in that sense Lk(x;y) corresponds to the tableau Tμ where μ=(2k2). It is not difficult to show that Lk(x;y) takes central values on Mk(F).

For k=2 and k=3 it was verified that Lk(x;y)Id(Mk(F)), and it was conjectured that for all kLk(x;y)Id(Mk(F)) namely, that Lk(x;y) is a nonidentity of Mk(F) . This conjecture was verified by Formanek .

Theorem 9.1 (see [<xref ref-type="bibr" rid="B53">35</xref>]).

The plolynomial Lk(x;y), which corresponds to the rectangle μ=(2k2), is a nonidentity of Mk(F). Hence Lk(x;y) is a central polynomial.

By Young's rule it follows that (9.4)Lk(x;y)=pμ+λ2k2,(λ)k2+1pλ, where pμIμ and pλIλ, see (5.4). Since Mk(F) satisfies the Capelli identity Capk2+1, it follows that for all λ with (λ)k2+1, pλId(Mk(F)). And since Lk(x;y)Id(Mk(F)), hence pμId(Mk(F)), where μ is of the 2×k2 rectangular shape μ=(2k2). Thus, also pμ is a central polynomial for Mk(F)). The fact that μ is a rectangle plays an important role in proving lower bounds for codimensions, since the following is applied: two or more rectangles of same height can be glued together horizontally, while two or more rectangles of same width can be glued together vertically.

For cn(Mk(F)), Theorem 9.1 and further results of Formanek [2, 26, 35] imply that the ordinary cocharacters and the trace cocharacters are nearly equal, hence they have same asymptotic. Since the trace codimensions are much easier to handle than the ordinary codimensions (see Theorem 5.6), this fact allows the computation—in the next section—of the exact asymptotic of cn(Mk(F)). Also, the fact that Lk(x;y) is a nonidentity of Mk(F) has applications in proving lower bounds for various other types of codimensions.

10. The Giambruno-Zaicev Theorem: <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M607"><mml:mtext>exp</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>∈</mml:mo><mml:mi>ℤ</mml:mi></mml:math></inline-formula>

For most PI algebras A it seems hopeless to find a precise, or even asymptotic, formula for the codimensions cn(A). We therefore ask a much more restricted question.

Question 3.

Given the associative PI algebra A, what can be said about the asymptotic behavior of the codimensions cn(A)?

As a first step we have the remarkable integrality property given by Theorem 10.1 below. See also Theorem 11.1 and its relation to Theorem 10.1.

Theorem 10.1 (see [<xref ref-type="bibr" rid="B78">3</xref>]).

Let A be an associative PI algebra (with char(F)=0), then the limit (10.1)limn(cn(A))1/n exists and is an integer.

We denote exp(A)=limn(cn(A))1/n, so exp(A).

10.1. Review of the Proof When <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M617"><mml:mtext>dim</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo><</mml:mo><mml:mi>∞</mml:mi></mml:math></inline-formula>

When A is finite dimensional, the number exp(A) can be calculated as follows. We may assume that F is algebraically closed. First, by a classical theorem of Wedderburn and Malcev [45, Theorem 3.4.3], A=BJ, where J=J(A) is the Jacobson radical of A, and B is semisimple. Thus B=B1Br where Bj are simple, namely, BjMkj(F). Consider now all possible nonzero products of the type (10.2)Bi1JBi2JJBiq0(q1), where the Bij are distinct, and for such nonzero products let (10.3)h=maxdim(Bi1Biq). Then h=ki12++kiq2, and h is the limit in Theorem 10.1: (10.4)limn(cn(A))1/n=h, Namely, exp(A)=ki12++kiq2.

For example, consider the algebra of upper block triangular matrices (10.5)A=UT(k1,k2,k3)=(B1J1J20B2J300B3), where for 1i3, BiMki(F) and Ji are rectangular matrices of the corresponding sizes. Then the Wedderburn-Malcev decomposition of A is A=B1B2B3+J where B1B2B3 is the semisimple part and J=J1+J2+J3 is the Jacobson radical. Notice that the matrix units ei,j in A satisfy (10.6)0e1,1·e1,k1+1·ek1+1,k1+1·ek1+1,k1+k2+1·ek1+k2+1,k1+k2+1B1J1B2J3B3B1JB2JB3, so B1JB2JB30 and also, this is the maximal such nonzero product. It follows that here (10.7)h=h(A)=dim(B1B2B3)=k12+k22+k32.

10.1.1. The Upper Bound

In the general case the exponent h=exp(A)=limn(cn(A))1/n is given by (10.3). To prove this, one first proves the following upper bound.

Lemma 10.2.

There exist constants a1, g1 such that for all n, cn(A)a1·ng1·hn.

Let Par(n) denote the partitions of n. Given h, define the subsets NSK(h,n)Par(n) by (10.8)NSK(h,n)={λ=(λ1,λ2,)nj>hλjK}, so NSK(h,n) is nearly a strip of height h (with at most K cells below the h row). The proof of Lemma 10.2 follows by showing that the cocharacters χn(A) are supported on such nearly a strip NSK(h,n), and by the polynomial bound mλ(A)a·nb on the multiplicities in the cocharacters, see Theorem 7.3. Thus cn(A)anbfλ, where the sum fλ is supported on such nearly a strip NSK(h,n). Similar to the estimates in , such sum (10.9)λNSK(h,n)fλ is bounded by a-·nb-·hn, and the proof of Lemma 10.2 follows.

10.1.2. The Lower Bound

Here we prove the folowing lemma.

Lemma 10.3.

There exist constants a2>0 and g2 such that for all n, a2·ng2·hncn(A).

A key ingredient in proving the lower bound is the polynomial Lk(x;y) which is a nonidentity for Mk(F), see Section 9. We already noted that Lk(x;y) has a component pμIμ which is central (nonidentity) for Mk(F), where μ is the 2×k2 rectangle (2k2), see Theorem 9.1.

We start with a single matrix algebra B=Mk(F), with the corresponding central polynomials Lk(x;y) and pμ(1). Rectangles of the same height (width) can be glued horizontally (vertically). Gluing horizontally (2k2) to itself w times yields the 2w×k2 rectangle μ(w)=((2w)k2), with a corresponding w-power (pμ(1))w=pμ(w), which is central and nonidentity for Mk(F), and pμ(w)Iμ(w). Thus fμ(w)cn(Mk(F)). For that μ(w)=((2w)k2)n, n=2wk2, the asymptotic of fμ(w) then yields the lower bound (10.10)a-2·ng-2·(k2)ncn(Mk(F)) for some constants a-2>0 and g-2.

In the general case we are given Bi1JBi2JJBiq0 as in (10.2). To each Bij corresponds the nonidentity polynomial Lkij(x;y), with the corresponding rectangular tableaux (2kij2), all with the same width =2. These tableaux can be glued vertically, thus yielding the rectangular tableau ρ=(2h), where h=j(kij)2. To that tableau there corresponds a polynomial which is essentially the product of the polynomials Lkij(x;y), with the corresponding component which is the product of the corresponding polynomials pμ, hence that polynomial is central nonidentity of A, and similarly for powers of these polynomials. Similar to the above case of a single matrix algebra B=Mk(F), the asymptotic of fρ(w)=f((2w)h) then yields the lower bound (10.11)a2·ng2·hncn(A) for some constants 0<a2 and g2. This proves the lower bound.

Corollary 10.4.

Putting together the lower and the upper bounds, Theorem 10.1 then follows.

This completes our review of the proof of Theorem 10.1. For extensions of this theorem, see Remark 11.2.

11. Berele's “1/2” Theorem (See [<xref ref-type="bibr" rid="B28">4</xref>])

Applying Theorem 10.1, Berele proved the following remarkable theorem.

Theorem 11.1 (see [<xref ref-type="bibr" rid="B28">4</xref>], see also [<xref ref-type="bibr" rid="B36">5</xref>]).

Let A be a PI algebra with 1A. Then as n goes to infinity, cn(A)~a·nt·hn, where, h (given by Theorem 10.1) and t(1/2), namely, t is an integer or a half integer.

Based on various examples, that theorem was conjectured for some time. It was first proved in  under the hypothesis that A satisfies a Capelli identity, and later in general in . Here is an outline of the proof in the Capelli case.

Proof .

Recall that cn(A)=mλ(A)fλ, where mλ(A) is the multiplicity of χλ. If A satisfies a Capelli identity then there exists an integer k such that mλ(A)0 only for partitions λ with at most k parts (see Remark 5.2). The proof is now based on investigating the asymptotics of the mλ(A) and fλ, and we proceed to do so.

Let (11.1)Uk(A)=F{x1,,xk}Id(A)F{x1,,xk} be the universal PI algebra for A in k generators. This algebra has an grading by degree and a corresponding Poincaré series Pk(t). It also has a finer k grading by multidegree with corresponding Poincaré series P(t1,,tk). Belov  proved that Pk(t) is (the Taylor series of) a rational function with coefficients in (see also , Theorem 9.44). It is not difficult to adapt that proof to show that P(t1,,tk) is also a rational function with integer coefficients. The proof also implies that the denominator of this rational function can be taken to be a product of terms of the form (1-t1a1tkak). We call such a rational function “nice.”

The Poincaré series P(t1,,tk) is related to the cocharacters χn(A)=mλ(A)χλ via (11.2)P(t1,,tk)=n=0λnmλ(A)Sλ(t1,,tk), where Sλ is the Schur function of λ.

Quite a lot is known about Taylor series of nice rational functions, see for example . If F(t1,,tk) is any nice rational function with Taylor series b(n1,,nk)t1n1tknk, then the coefficients b(n1,,nk) can be described using a finite set of polynomials. Namely, k can be partitioned into regions R1,,Rm with corresponding polynomials q1,,qm such that for each i, (11.3)c(n1,,nk)=qi(n1,,nk)for(n1,,nk)Ri. Using properties of Schur functions it can be shown that an analogue formula holds for the mλ(A). Hence we may write (11.4)cn(A)=i{qi(λ)fλλRi}. The regions Ri, in general, are defined by linear inequalities and modular linear equations. Here is a simple example to make this more clear. Let F(t1,t2)=(1-t12)-1(1-t2)-1(1-t1t2)-1. Then b(n1,n2) is given by (11.5)b(n1,n2)={n22+1ifn1n2,n2iseven,(n2+1)2ifn1n2,n2isodd,n12+1ifn1<n2,n1iseven,(n1+1)2ifn1<n2,n1isodd. Now back to the general case. There exists a number hk called the essential height of the cocharacter defined to be the largest number such that there exist nonzero multiplicities mλ(A) in which λ=(λ1,,λk) and λh can be taken arbitrarily large. Giambruno and Zaicev proved that h=limn(cn(A))1/n. This number, denoted exp(A), is also important for the proof of Theorem 11.1. Let v1 be the vector in k whose coordinates are h ones followed by k-h zeros. Then in (11.4) certain summands will be on regions of the form (11.6){v0+α1v1++αdvdαi,,αd}L, where L is an integer lattice. Let R1,,Rp be the regions of this form. Then in the computation of cn(A) these terms dominate and we can refine (11.4) to (11.7)cn(A)=i{qi(λ)fλλRi}. The rest of the proof closely immitates the computation of . The main theorem is that if R' is as above and q(x1,,xk) is a polynomial then q(λ)fλ summed over partitions of n in R' will be asymptotic to a constant—times n to an integer power—times hn. Hence, the cocharacter cn(A) will be a sum of such terms.

There remains the problem that the powers of n might not be equal, and the way around this difficulty is to use the fact that if 1A then the cocharacter sequence is Young derived, see . Namely, the Poincaré series P(t1,,tk) can be written as (1-t1)-1(1-tk)-1g(t1,,tk), where g has all the nice properties of P.

We conclude this section with the following general remark.

Remark 11.2.

Recent works extended the above theorems of Sections 10 and 11 to graded polynomial identities and to PI algebras with the action of Hopf algebra, see, for example, .

12. Algebraicity of Some Generating Functions

As we show below, Theorems 10.1 and 11.1 are related to the question of whether or not the generating function of the codimensions is algebraic. We begin with the following definition.

Definition 12.1 (see [<xref ref-type="bibr" rid="B150">60</xref>, <xref ref-type="bibr" rid="B154">61</xref>]).

(1) Given the sequence an, then F(x)=n0anxn is its corresponding ordinary generating function. In particular CA(x)=n0cn(A)xn is the generating function of the codimensions of the algebra A.

(2) The function F(x) is algebraic if there exist polynomials P0(x),,Pr(x) such that (12.1)Pr(x)Fr(x)++P1(x)F(x)+P0(x)=0. Algebraicity or nonalgebraicity of the generating function F(x)=n0anxn is an indication of the complexity of the sequence an.

Example 12.2 (see [<xref ref-type="bibr" rid="B126">34</xref>], [<xref ref-type="bibr" rid="B46">25</xref>, Theorem 12.6.8]).

For the 2×2 matrices M2(F) we have (12.2)cn(M2(F))=1n+2(2n+2n+1)-(n3)+1-2n. This implies that (12.3)CM2(F)(x)=1x2(1-2x-1-4x)-x3(1-x)4+11-x-11-2x, which is clearly algebraic.

Note that when k=1, M1(F)=F and cn(F)=1, hence (12.4)CF(x)=11-x which is algebraic.

12.1. Nonalgebraicity of Some Generating Functions

We quote here a classical theorem of Jungen , see also .

Theorem 12.3 (see [<xref ref-type="bibr" rid="B95">62</xref>]).

Let f:,  F(x)=n0f(n)xn, and assume that as n goes to infinity, (12.5)f(n)~b·n-g·an, where b and a are complex constants and g is a real number. For F(x) to be algebraic it is necessary that g be rational; and if g>0 then g must also be non-integral.

Applying this theorem to the codimensions of matrices we deduce the following theorem.

Theorem 12.4.

Let (12.6)CMk(F)(x)=n0cn(Mk(F))·xn be the generating functions of the (ordinary) codimensions of Mk(F). If k3 and k is odd then CMk(F)(x) is not algebraic.

Proof.

By Theorem 6.2(12.7)cn(Mk(F))~b·n-g·k2n, where g=(k2-1)/2. The proof now follows by Theorem 12.3, since g=(k2-1)/2 is an integer when k is odd.

Obviously, Example 12.2 and Theorem 12.4 motivate the following conjecture.

Conjecture 12.5.

If k3 then the generating function (12.8)CMk(F)(x)=n0cn(Mk(F))·xn is not algebraic.

For the other verbally prime algebras (see Section 2.2.1), recall from Theorem 6.4 the following partial asymptotic results: (12.9)cn(Mk,)~a·(1n)(k2+2-1)/2·(k+)2n, where the constant a is yet unknown. Also, (12.10)cn(Mk(G))~b·(1n)g·(2k2)n, where the constants b and g are yet unknown, and (k2-1)/2g(2k2-1)/2.

Corollary 12.6.

If k( mod 2) then the generating function CMk,(x) of the codimensions cn(Mk,) in not algebraic.

Proof .

Indeed, k( mod 2) implies that k2+2-1 is even, and the proof follows from Theorems 12.3 and 6.4.

Example 12.7 (see [<xref ref-type="bibr" rid="B37">64</xref>]).

We apply here a theorem due to Kemer  which says that the algebras GG and M1,1 have the same identities, hence the same codimensions. Now (12.11)c0(M1,1)=1,cn(M1,1)=12(2nn)+n+1-2n,n=1,2, see , and hence (12.12)CM1,1(x)=12+121-4x+x(1-x)2+11-x+11-2x, which is clearly algebraic. Note that (12.13)cn(M1,1)~12π·1n·4n, compare with (6.3) of Section 6. See also .

Conjecture 12.8.

If k,1 and (k,)(1,1) then CMk,(x) is not algebraic.

13. Nonassociative <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M850"><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:math></inline-formula> with <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M851"><mml:mtext>exp</mml:mtext><mml:mo stretchy="false">(</mml:mo><mml:mi>A</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> a Non Integer

As remarked before, the notions of codimensions and cocharacters also apply to nonassociative PI algebras. Here, in the most general case, the free associative algebra F{x}=F{x1,x2,} is replaced by the free algebra Fx1,x2,=Fx. Now A is a nonassociative PI algebra and again, Id(A)Fx are the identities of A. Since we are dealing now with nonassociative polynomials, hence different parenthesizes in a monomial yield different monomials. It follows that in the nonassociative case, the space Vn of the multilinear polynomials of degree n is now of dimension  dimVn=Cn·n!, where Cn is the nth Catalan number (which counts the number of different parenthesizes of a monomial of degree n). The definition of the codimensions cn(A) is, formally, the same as that in the associative case.

Definition 13.1.

(13.1) c n ( A ) = dim ( V n Id ( A ) V n ) ,

compare with Definition 4.1.

Similarly for the cocharacters, the action of Sn on Vn is again given by (5.1), and one introduces cocharacters precisely as in the associative case, see Definition 5.1. However, some phenomena here are rather different from those in the associative case. For example, a counter-example to Theorem 4.4, hence also to Theorem 4.3, was given in .

A counter-example to Theorem 10.1 in the nonassociative case was first constructed in . Recently, Giambruno et al.  constructed a family of nonassociative PI algebras A such that for every 1<α<2 there is such an algebra A for which the limit limn(cn(A))1/n exists and is equal to α. We briefly describe that construction.

13.1. The Algebra <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M872"><mml:mi>A</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>m</mml:mi><mml:mo>,</mml:mo><mml:mi>w</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula>

Let w=w1w2 be an infinite word on the alphabet {0,1}. Given an integer m2, define the sequence {ki}i1=Km,w by (13.2)ki=wi+m. Construct the algebra A=A(m,w)=A(Km,w) as follows. Its basis is (13.3){a,b}Z1Z2, where (13.4)Zi={zj(i)1jki},i=1,2,, and multiplication is given as follows: (13.5)z1(i)a=z2(i),z2(i)a=z3(i),,zki-1(i)a=zki(i),i=1,2,,zki(i)b=z1(i+1),i=1,2, and all other products are zero.

Let w=w1w2 be an infinite word. The notion of the complexity of w is classical. For each n, complexityw(n) is the number of distinct subwords of w of length n. The algebra A=A(m,w) depends on the integer m and on the complexity of the word w, and the following theorem is proved.

Theorem 13.2 (see [<xref ref-type="bibr" rid="B69">68</xref>]).

Given 1α2, we can choose m2 and a word w such that (13.6)limn(cn(A(m,w)))1/nexistsandisequaltoα.