Motivated by the physical applications of q-calculus and of q-deformations, the aim of this paper is twofold. Firstly, we prove the q-deformed analogue of the celebrated theorem by Baker, Campbell, and Hausdorff for the product of two exponentials. We deal with the q-exponential function expq(x)=∑n=0∞(xn/[n]q!), where [n]q=1+q+⋯+qn-1 denotes, as usual, the nth q-integer. We prove that if x and y are any noncommuting indeterminates, then expq(x)expq(y)=expq(x+y+∑n=2∞Qn(x,y)), where Qn(x,y) is a sum of iterated q-commutators of x and y (on the right and on the left, possibly), where the q-commutator [y,x]q≔yx-qxy has always the innermost position. When [y,x]q=0, this expansion is consistent with the known result by Schützenberger-Cigler: expq(x)expq(y)=expq(x+y). Our result improves and clarifies some existing results in the literature. Secondly, we provide an algorithmic procedure for obtaining identities between iterated q-commutators (of any length) of x and y. These results can be used to obtain simplified presentation for the summands of the q-deformed Baker-Campbell-Hausdorff Formula.
1. Introduction
The celebrated Baker-Campbell-Hausdorff (BCH, for short in the sequel) Theorem allows the representation of the product of two exponentials in terms of a single exponential (see [1] for a comprehensive investigation of this result). The applications of the BCH Theorem range over many areas of mathematics and physics, including theoretical physics, quantum statistical mechanics, perturbation and transformation theory, the representation of time-evolution in quantum mechanics in terms of the exponential of the Hamiltonian, the study of nonclassical (i.e., coherent, squeezed) states of light, group theory, control theory, the exponentiation of Lie algebras into Lie groups, linear subelliptic PDEs, and geometric integration in numerical analysis. A quite extensive review of exponential operators and their many roles in physics was presented by Wilcox [2]. In order to motivate the main topics of the present paper (i.e., the q-deformed BCH Formula, and an algorithm for generating q-commutator identities), we first review what is known so far as the q-analogue (or q-deformation) of the BCH Theorem, along with motivations for the physical interest in this subject; see also [3] by the authors with Achilles.
The idea of q-deformation goes back to Euler in the mid eighteenth century and to Gauss in the early nineteenth century. If one defines the nth q-integer as [n]q=1+q+q2+⋯+qn-1 and, accordingly, if the q-factorial is defined as [n]q!=[n-1]q!·[n]q (where [0]q!=1), then the q-exponential is(1)expqx=∑n=0∞xnnq!.It is well known that Jackson’s q-derivative, defined by the ratio (2)Dqfx=fqx-fxxq-1satisfies Dqexpq(x)=expq(x) (see, e.g., the monograph [4] for an introduction to these topics). The reader is referred to the recent monograph [5] for an in-depth analysis and a comprehensive historical presentation of q-calculus. The advent of quantum groups, some thirty years ago [6], gave rise to vigorous renewed interest in q-deformations, although the symmetric q-integers introduced in that context, namely, {n}q≔(qn-q-n)/(q-q-1) (hence the corresponding q-exponential), differ from those defined above.
The earliest introduction of q-deformations into physics is probably due to Arik and Coon [7], who studied the q-deformed oscillator, whose creation and annihilation operators, a† and a, satisfy [a,a†]q≔aa†-qa†a=1. The conventional boson operators [a,a†]1=1 satisfy a special case of the BCH identity:(3)expαa†+βa=expαa†expβaexpαβ2,which allows transformation from normal to symmetric ordering of the boson operators. The failure to obtain q-analogue of this relation was referred to in [8] as an instance of the “q-Campbell-Baker-Hausdorff enigma.” In connection with this problem, in a very recent note [3] we addressed this “enigma,” and we announced the q-deformed BCH Theorem (which we prove here), providing improvements of existing partial results previously given in [8, 9].
The q-deformation of the BCH identity has also been considered, from very different points of view, in [10–13]; as a related topic to q-exponentiation, see [14–17] for the q-analogue of the Zassenhaus formula; as another related topic, see also the very recent works [18–20] for the q-analogue of the so-called pre-Lie Magnus expansion. q-exponentials (and their products) naturally appear in the study of the group-like elements in Hopf algebras; see [21, 22]; for the relation to Hall algebras and quantum groups, see also [23–25].
One mathematical motivation for the study of q-deformations concerns the fact that they allow a more refined view of the features of the systems involved. Thus, the characterization of the irreducible representations of the special unitary group SU(n) requires the specification of the eigenvalues of all the n-1 Casimir operators, whereas the fundamental Casimir operator is sufficient for characterizing the irreducible representations of the corresponding quantum groups [26]. This is a consequence of the fact that the eigenvalues of the latter are polynomials in q rather than integers. An encyclopedic treatment of q-deformed special functions is provided by Vilenkin and Klimyk [27].
In physical applications one can identify two types of motivations for the study of q-deformations. On the one hand, q-deformations are invoked as generalizations of fundamental theories (see Wachter [28]). On the other hand, they are proposed as models or approximations of more complicated Hamiltonians. Thus, the Arik-Coon q-oscillator mentioned above has been used to mimic anharmonicities in molecular vibrations [29]. Composite bosons and quasi-bosonic elementary excitations such as excitons have also been modelled by q-deformed boson operators [30] that were shown to exhibit Bose-Einstein condensation in two dimensions [31].
Nowadays, as described in the recent monograph by Ernst [5], the interest in q-calculus deserves no further motivation, due to its wide applications, in addition to the physical contexts described above, to many branches of pure mathematics as well: from analytic number theory to noncommutative geometry, from combinatorics to hypergeometric function theory.
After this short introduction on the physical interest in q-deformation and q-exponentiation, we now focus on the q-analogue of the BCH Theorem and on the identities between q-commutators. Let us denote by [a,b]q≔ab-qba the so-called q-commutator (also abbreviated as q-mutator) of a and b. If x and y are any symbols, by an iterated q-commutator centered at [y,x]q (see also Definition 2) we mean any arbitrarily long polynomial in x,y of the form(4)βz1∘βz2∘⋯∘βzny,xq,where z1,…,zn may be any of x and y and where βz denotes indifferently a left or a right q-commutator operator, that is, any of the maps adqz and adq∗z defined by(5)a⟼adqza≔z,aq,or a⟼adq∗za≔a,zq.For example, [[[y,[x,[y,x]q]q]q,y]q,x]q is an iterated q-commutator centered at [y,x]q, whereas [x,y]q or [y,[x,x]q]q are not centered at [y,x]q.
In a very recent note [3], we announced the following result: if expq is as in (1), then(6)expqxexpqy=expqzqx,y,where the formal power series zq(x,y) can be represented by infinitely many summands, each of which is an iterated q-commutator centered at [y,x]q. In [3] we also provided, without proof, an explicit expression for these summands; we prove these results in the present paper (see Theorem 1).
Our formula (6) is consistent with the 1953 result by Schützenberger [32] (see also Cigler [33]), ensuring that(7)expqxexpqy=expqx+yiffy,xq=0.Whereas from (7) it follows that the series zq(x,y) satisfying (6) is of the form zq(x,y)=x+y+zq′(x,y), where zq′(x,y) is an infinite sum of polynomials each containing the factor [y,x]q, it does not follow the fact that zq′(x,y) is a series of iterated q-commutators centered at [y,x]q. About this question, which we now answer, we recall that
it was posed and (only) partially solved in the 1995 work [8] by the second-named author and Duchamp;
in 1983, Reiner [9] showed that zq′(x,y) is a series of right-nested q-commutators of x and y: by the latter we mean any expression of the form (see also (5))(8)adz1∘adz2∘⋯∘adznz0,
where z0,z1,…,zn may be any of x or y;
a crucial tool in our arguments is provided by the following identity:(9)ab=11-q2a,bq+qb,aq,
transforming products into q-mutators.
According to Reiner’s result recalled in (b), the innermost q-commutator may be (and often will be) any of (10)x,yq,y,xq,x,xq,y,yq,but this expansion does not imply Schützenberger’s result, where only [y,x]q is expected. To this extent our result improves Reiner’s result. Broadly speaking, we renounce Reiner’s (left- or right-) nested presentation, in favor of presentation with centered [y,x]q and with iterated commutators (in the above sense), consistently with Schützenberger [32]. Incidentally, due to its relevance in this context, we provide a result playing a role analogous to that of the Dynkin-Specht-Wever Lemma, characterizing the q-commutators centered at [y,x]q.
An expansion of the series zq′(x,y) up to fourth order in terms of nested q-commutators that depend both on [y,x]q and [x,y]q was obtained by the second-named author and Solomon [34], and it was claimed that the dependence on [x,y]q can be eliminated (consistently with Schützenberger [32]) by means of some (unspecified) operator identities. In the present paper we determine these operator identities.
As a byproduct, we exhibit the expansion of zq(x,y) up to degree 4 in terms of [y,x]q-centered q-mutators only:(11)zqx,y=x+y-12qy,xq-q3q!2q·x,y,xqq+y,xq,yq-q24q!2q·y,xq,yq,yq+y,y,xq,yqq+x,y,xqq,xq+x,x,y,xqqq+14q!2q3·q5qy,x,y,xqqq-q23qx,y,y,xqqq-q3y,y,xq,xqq-q2q2+3q+1x,y,xq,yqq+summands of length≥5 in x,y.Incidentally, we observe a striking novelty of the q-BCH series compared to the classical undeformed BCH series: the latter does not contain summands with three x and one y or three y and one x (this is due to the properties of the Bernoulli numbers; see [1]), whereas the q-BCH series does. In the formal limit as q→1 in the above expansion, these summands disappear, due to the skew symmetry of the classical commutator. In a forthcoming study, we shall investigate higher degrees, and we shall also consider computational issues and implementation using the computer algebra system Reduce [35].
The ultimate goal (to which we shall devote future investigations) will be the analysis of the formal limit as q→1 of our expansion, which would eventually provide a brand new proof of the classical undeformed BCH Theorem, a problem which seems highly nontrivial, since it is interlaced with the identities holding true among q-commutators. Finally, we hope that an understanding of the q-BCH Formula will shed light on q-Zassenhaus and q-Magnus expansions as well.
As an application (see the Appendix), we show that our explicit q-mutator expansion is convergent in any Banach algebra (when q<1); see Theorem A.1: in particular, this is true in any matrix algebra or (more generally) in any finite dimensional associative algebra. This parallels the classical undeformed case, where it is possible to use the Dynkin expansion [36], to give a domain of convergence for the BCH series. Furthermore, we hope that this convergence result may be useful to shed light on the q-analogue of the classical undeformed passage from the Lie algebra to the Lie group multiplication.
Although we crucially use the underlying (free) associative structure of Kq⟪x,y⟫ (the algebra of the formal power series in x,y with coefficients in K(q)) to obtain a closed formula for the q-BCH series, the proof of the convergence of the latter is obtained only by using the estimate (12)a,bq≤Ma·b,for some constant M>0, and this suggests that our presentation of the q-BCH series in terms of q-mutators may be of relevance for the study of other contexts, with nontrivial commutation identities. Despite the lack of nontrivial relations in K(q)⟪x,y⟫, the analysis in the free associative setting is intended as a first step towards a future comprehension of structures with nontrivial relations, like quantum groups or, more generally, Hopf algebras. To the best of our knowledge, even in the free associative setting, the analysis of the q-commutator-form of the q-deformed BCH series, along with its local convergence in Banach algebras, appears here for the first time.
As happening for the classical BCH Formula (starting, e.g., from Dynkin’s expansion [36]), in order to get the above simplified expansion starting from our general formula for zq(x,y) in Theorem 1, one has to take into account the linear dependency relations among the q-mutators of the same bidegree in x,y. In this paper we furnish an algorithm to obtain these identities for any bidegree.
Our procedure for generating q-commutator identities is fully described in Section 3; here we anticipate the main tools: along with identity (9) (which transforms the left and right multiplications into q-commutations), we shall combine the following identities:(13)a,bq,aq=a,b,aqq,a,c,bqq+b,c,aqq=b,cq,aq+a,cq,bq,a,c,bqq-a,cq,bq=qq2+1c,aq,bq-c,bq,aq+b,a,cqq-a,b,cqq,holding true for any a,b,c. Clearly, if we repeatedly apply to these identities any choice of (14)adqx,adqy,adq∗x,adq∗y(see the notation in (5)), we obtain new identities between q-commutators. In Section 3 we shall investigate an algorithm to obtain the smallest number of independent (linear) identities existing among the generators (4) of the [y,x]q-centered q-mutators of a fixed bidegree (i,j) in x and y.
In order to show the efficiency of our procedure for generating q-commutator identities (and the nontriviality of the dependency relations among q-mutators of a fixed bidegree), we close the introduction by showing, as an example, the set of 15 independent identities obtained with our algorithm for the 24 generators of the [y,x]q-centered q-mutators of bidegree (2,3) in x,y (i.e., of degree 2 in x and degree 3 in y): denoting the generators by (15)b1=adqx∘adqy∘adqyy,xq,b2=adqy∘adqx∘adqyy,xq,b3=adqy∘adqy∘adqxy,xq,b4=adq∗x∘adqy∘adqyy,xq,b5=adq∗y∘adqx∘adqyy,xq,b6=adq∗y∘adqy∘adqxy,xq,b7=adqx∘adq∗y∘adqyy,xq,b8=adqy∘adq∗x∘adqyy,xq,b9=adqy∘adq∗y∘adqxy,xq,b10=adq∗x∘adq∗y∘adqyy,xq,b11=adq∗y∘adq∗x∘adqyy,xq,b12=adq∗y∘adq∗y∘adqxy,xq,b13=adqx∘adqy∘adq∗yy,xq,b14=adqy∘adqx∘adq∗yy,xq,b15=adqy∘adqy∘adq∗xy,xq,b16=adq∗x∘adqy∘adq∗yy,xq,b17=adq∗y∘adqx∘adq∗yy,xq,b18=adq∗y∘adqy∘adq∗xy,xq,b19=adqx∘adq∗y∘adq∗yy,xq,b20=adqy∘adq∗x∘adq∗yy,xq,b21=adqy∘adq∗y∘adq∗xy,xq,b22=adq∗x∘adq∗y∘adq∗yy,xq,b23=adq∗y∘adq∗x∘adq∗yy,xq,b24=adq∗y∘adq∗y∘adq∗xy,xq,we have the following 15 independent relations among them: (16)b6=b9,b18=b21,b7=b13,b10=b16,b7+b8=b4+b5,b19+b20=b16+b17,b14+b15=b8+b9,b17+b18=b11+b12,q2+1b14-b9=qb21-b20+b3-b2,q2+1b17-b12=qb24-b23+b6-b5,q2+1b7-b5=qb11-b10+b2-b1,q2+1b19-b17=qb23-b19+b14-b13,1-q+q2b20-b3=qb9+b15+b21-b14-b8-b2,1-q+q2b23-b6=qb12+b18+b24-b17-b11-b5,1+q4b22+q1+q2b19+b16+b10+b1+2q2b13+b7+b4=1+q4b2+q1+q2b23+b14+b8+b5+2q2b20+b17+b11.
2. Method: The q-Deformed BCH Formula
Notation. We fix the algebraic setting we work in: A=K(q)⟪x,y⟫ will denote the associative algebra of the formal power series in two noncommuting indeterminates x and y, with coefficients in K(q), which is the field of the rational functions in the symbol q over a field K of characteristic 0. (We recall that whereas K[q] usually denotes the ring of the polynomials in the indeterminate q, by K(q) one identifies the field of the quotients of the ring K[q].) The associative multiplication in A is the usual Cauchy product of formal power series. The notation K(q)〈x,y〉 will stand for the associative algebra of the polynomials in x and y over K(q). From now on, we introduce on A the bilinear map(17)a,bq≔ab-qba,a,b∈A.We say that [a,b]q is the q-mutator (shortcut of “q-commutator”) of a and b.
Given i,j∈N∪0, Ai,j denotes the set of the homogeneous polynomials in A with degree i with respect to x and degree j with respect to y. We say that any element of Ai,j has bidegree (i,j) (with respect to x,y, resp.). We also set, for any n∈N∪{0},(18)An≔spanAi,j:i+j=n. Thus, for example,(19)A0=Kq,A1=spanKqx,y,A2=spanKqx2,xy,yx,y2.Obviously, we have the direct sum/product decompositions(20)An=⨁i+j=nAi,j,∀n≥0,A=∏i,j≥0Ai,j,A=∏n≥0An.The typical element of A is therefore(21)p=p0+p1+⋯≕∑n=0∞pnwhere pn∈An.Finally, we denote by I the two-sided ideal in A generated by [y,x]q and we set(22)Ii,j≔I∩Ai,j.We can consider the quotient of A modulo I, denoted as usual by (23)AI=aI:a∈A.We also use the standard notation for the equivalence a~b modulo I: (24)a≡bmodIor a≡Ib.A/I is an associative algebra with the obvious operations. As I is a two-sided ideal generated by a homogeneous polynomial of bidegree (1,1), then I=∏i,j≥0Ii,j. Notice that, obviously,(25)Ii,0=0=I0,jfor every i,j∈N∪0.
Since the lowest order term in the q-exponential series (1) is 1, there exists the inverse map of expq, say logq (called the q-logarithm), defined on the set 1+A+, where A+ is the set of the formal power series in x,y whose zero-degree term is null. We use the notation(26)logq1+p=∑n=1∞λnpnp∈A+,where the coefficients λn∈K(q) are given by the recurrence formula:(27)λ1=1,λn=-∑i=1n-1λi∑j1+⋯+ji=n1j1q!⋯1jiq!n≥2.Thus, the unique series zq(x,y) closing the identity (6) is(28)zqx,y≔logqexpqxexpqy,referred to as the q-Baker-Campbell-Hausdorff series, shortly, q-BCH series. Therefore, an explicit expression for zq(x,y) in terms of polynomials is(29)zqx,y=∑n=1∞λn∑i1,j1,…,in,jn≠0,0xi1yj1⋯xinyjni1q!j1q!⋯inq!jnq!,where λn’s are as in (26).
Starting from the (tautological) identity yx=qxy+[y,x]q, any monomial xi1yj1⋯xinyjn in (29) can be rewritten, modulo I, by moving any x on the left and any y on the right. Namely, one has(30)xi1yj1⋯xinyjn=qi2j1+⋯+inj1+j2+⋯+jn-1xi1+⋯+inyj1+⋯+jn+an element of I.Since, as we prove in Section 4 (starting from (7)), we have zq(x,y)=x+y+zq′(x,y) with zq′(x,y)∈I, by means of (30) one can write zq(x,y) as a series whose summands (other than x+y) are polynomials in I; that is, they contain the factor [y,x]q.
Now, by means of the rearranging identity (9), we can write any element of I in terms of iterated q-mutators of x and y centered at [y,x]q. An explicit example will clarify this: from (30) we have xyxy3=qx2y4+elementofI; explicitly (by applying four times (9) in the last four equalities) (31)xyxy3-qx2y4=xy,xq+qxyy3-qx2y4=xy,xqy3=11-q2x,y,xqq+qy,xq,xq·y3=11-q22x,y,xqq,yq+qy,x,y,xqqq+qy,xq,xq,yq+q2y,y,xq,xqq·y2=11-q23x,y,xqq,yq,yq+qy,x,y,xqq,yqq+qy,x,y,xqqq,yq+q2y,y,x,y,xqqqq+qy,xq,xq,yq,yq+q2y,y,xq,xq,yqq+q2y,y,xq,xqq,yq+q3y,y,y,xq,xqqq·y=11-q24x,y,xqq,yq,yq,yq+qy,x,y,xqq,yq,yqq+qy,x,y,xqq,yqq,yq+q2y,y,x,y,xqq,yqqq+qy,x,y,xqqq,yq,yq+q2y,y,x,y,xqqq,yqq+q2y,y,x,y,xqqqq,yq+q3y,y,y,x,y,xqqqqq+qy,xq,xq,yq,yq,yq+q2y,y,xq,xq,yq,yqq+q2y,y,xq,xq,yqq,yq+q3y,y,y,xq,xq,yqqq+q2y,y,xq,xqq,yq,yq+q3y,y,y,xq,xqq,yqq+q3y,y,y,xq,xqqq,yq+q4y,y,y,y,xq,xqqqq.This methodology can be applied to any summand in (29). For example, if we use the bidegree notation(32)zqx,y=x+y+∑i,j≥1Qi,jx,y,where Qi,j(x,y) has degree i with respect to x and degree j with respect to y, we can readily obtain the associative presentation of Q1,3(x,y):(33)Q1,3x,y=14q!2q·q3-q+1·xy3+q3-q+1·y3x+q2q3-2q2+2q-1·y2xy+q2q3-2q2+2q-1·yxy2.By using the cited identity yx=qxy+[y,x]q on each summand of Q1,3(x,y) (other than xy3), we get(34)y3x=y2y,xq+qyy,xqy+q2y,xqy2+q3xy3y2xy=yy,xqy+qy,xqy2+q2xy3yxy2=y,xqy2+qxy3.Inserting these identities in the expansion of Q1,3(x,y) we get(35)4q!2q·Q1,3x,y=⋯=-q4+q3+q3-q4q3+q5-2q4+2q3-q2q2+q5-2q4+2q3-q2q·xy3+q3-q4y2y,xq+q3-q4q+q5-2q4+2q3-q2yy,xqy+q3-q4q2+q5-2q4+2q3-q2q+q5-2q4+2q3-q2y,xqy2=0·xy3-q3q-1y2y,xq-q2q-12yy,xqy+q2q-1y,xqy2.Obviously, the cancelation of the summand xy3 (see the curly braces) is not sheer chance, but it derives from the fact that Q1,3 belongs to I. We next apply the technique exemplified above, based on (9), thus obtaining the presentation(36)Q1,3x,y=-q24q!2qy,xq,yq,yq+y,y,xq,yqq.Analogously one gets(37)Q3,1x,y=-q24q!2qx,x,y,xqqq+x,y,xqq,xq.We explicitly remark that in order to get simplifications for Q1,3(x,y) and Q3,1(x,y) one also needs to take into account the fact that (38)y,y,xqq,yq=y,y,xq,yqq,which is a particular case of an identity which will take a crucial role in the sequel:(39)a,bq,aq=a,b,aqqforeverya,b.No more relations intervene among the q-mutators(40)y,y,y,xqqq,y,xq,yq,yq,y,y,xq,yqq,which are linearly independent; it is a striking fact, however, that Q1,3(x,y) can be written by means of the last two only. With the same techniques we obtained the fourth-degree expansion in (11).
The above methodology in attacking the study of the q-BCH Formula shows that it is of relevance to study the following issues:
to obtain an explicit expression of the q-BCH summands in (32) in terms of iterated q-mutators centered at [y,x]q;
to obtain identities among the iterated q-mutators centered at [y,x]q, allowing simplifying the presentation of Qi,j(x,y)’s and studying bases/dependence-relations in the spaces of the q-mutators centered at [y,x]q.
The answer to the first issue is given by the following theorem which we prove in Section 4; the second problem is investigated in the next section.
Theorem 1.
The q-BCH series has form (32), where any homogeneous summand Qi,j(x,y) is given by the following formula, as a linear combination of iterated [y,x]q-centered q-mutators:(41)Qi,jx,y=∑n=1i+jλn∑i1,j1,…,in,jn≠0,0i1+⋯+in=i,j1+⋯+jn=j1i1q!j1q!⋯inq!jnq!·11-q2i-1∑r=1n-1qi2j1+⋯+irj1+⋯+jr-11-q2jr+1+jr+2+⋯+jn∑k=0j1+⋯+jr-1qk1-q2kj1+⋯+jrk∑h=0ir+1-1qhj1+⋯+jradq∗y+qadqyjn⋯adq∗x+qadqxir+2adq∗y+qadqyjr+1adq∗x+qadqxir+1-h-1adq∗y+qadqykadqx+qadq∗xh+i1+⋯+iradqyj1+⋯+jr-kx.Here the numbers λn are the coefficients of the expansion of the q-logarithm in (27). Finally, any power of adqz+qadq∗z and adq∗z+qadqz (with z=x or y) can be further expanded by Newton’s binomial, since adqz and adq∗z commute for every z (as it derives from (39); see also (5) for the meaning of adq,adq∗).
3. The Identity-Generating Technique
In this section we provide one of our two main results: an algorithm for the generation of identities between iterated q-mutators. We fix the definitions used in the sequel (see also the notations for A,Ai,j,I,Ii,j introduced at the beginning of Section 2).
Definition 2.
One gives the following three definitions.
Fixing b∈A, one sets(42)Rb:A⟶A,Rba≔ab,(43)Lb:A⟶A,Lba≔ba.
In other words, Rb and Lb are, respectively, the right and left multiplications in the associative algebra A.
Let one use notation (5) for the right/left q-adjoint operators adq∗ and adq; given z∈A, any q-mutator of the form(44)adq∗xh1adq∗yk1adqxh1′adqyk1′⋯adq∗xhnadq∗yknadqxhn′adqykn′z
(with n∈N and h1,k1,h1′,k1′,…,hn,kn,hn′,kn′∈N∪0) will be called an iterated q-mutator (of x and y) centered at z (or z-centered). For one’s aims, one shall be interested in [y,x]q-centered q-mutators only.
With the above notation, for every i,j∈N one denotes by Si,j the subspace of Ii,j spanned by the [y,x]q-centered q-mutators (44) additionally satisfying (45)h1+h1′+⋯+hn+hn′=i-1,k1+k1′+⋯+kn+kn′=j-1.
If i=0 or j=0, one sets Si,j≔{0}. Furthermore one sets(46)S≔∑i,j∈N∪0Si,j,
to denote the formal power series in A with summands in the sets Si,j’s.
The letter “S” has been chosen to remind us of Schützenberger’s result (7). For example, x3y[y,x]qy belongs to I4,3, while [x,[x,[y,[[[y,x]q,x]q,y]q]q]q]q belongs to S4,3. A priori, whereas it is trivial that Si,j⊆Ii,j, it is not at all obvious that Ii,j=Si,j, which is stated in the next result.
Lemma 3.
With the notation in Definition 2, one has(47)Ii,j=Si,jfor every i,j≥0.
The proof of this result is contained in Proposition 14.
3.1. Some Dimensions
We next take into account the space S of the [y,x]q-centered polynomials of bidegree (i,j). For example, we have (all spans are understood over the field K(q))(48)S1,1=spany,xqS2,1=spanx,y,xqq,y,xq,xqS1,2=spany,y,xqq,y,xq,yqS3,1=spanx,x,y,xqqq,x,y,xqq,xq,x,y,xq,xqq,y,xq,xq,xqS2,2=spany,x,y,xqqq,x,y,xqq,yq,y,y,xq,xqq,y,xq,xq,yq,x,y,y,xqqq,y,y,xqq,xq,x,y,xq,yqq,y,xq,yq,xqS1,3=spany,y,y,xqqq,y,y,xqq,yq,y,y,xq,yqq,y,xq,yq,yq.The above spaces are expressed in terms of generators, not all of which may be linearly independent (nor different!). For example one has (due to (39))(49)x,y,xqq,xq=x,y,xq,xqq,y,y,xqq,yq=y,y,xq,yqq,and it can be proved that no other dependency relations hold among the generators of S3,1 or the generators of S1,3, so that dim(S3,1)=dim(S1,3)=3.
The problem of determining the dimension of Si,j is rather simple (see Proposition 4), whereas the problem of discovering the dependency relations among the generators of a given Si,j is much more difficult: here we determine the pertinent number of relations and we propose an algorithm for discovering all of them.
For example, we consider the case of total degree i+j=4: one can prove that dim(S2,2)=5 and that the dependency relations among the 8 generators of S2,2 are the following three: (50)x,y,xqq,yq-x,y,xq,yqq=y,y,xq,xqq-y,y,xqq,xq=y,x,y,xqqq-y,xq,yq,xq=q3qx,y,y,xqqq-y,xq,xq,yq.These identities may obviously produce infinitely many others; for example (as we shall see by a very general procedure for obtaining identities), hidden in the above identities one has(51)y,xq,xq,yq-y,xq,yq,xq+y,x,y,xqqq-x,y,y,xqqq=q+1q·y,y,xqq,xq-y,y,xq,xqq.
In the next result it is understood that the field underlying all vector space structures is K(q). Along with other dimensional facts, we aim to count the following set of generators of Si,j: these are the iterated q-mutators of the form(52)β1∘β2∘⋯∘βi+j-2y,xq,where β1,…,βi+j-2 all belong to the set of maps {adq∗x,adq∗y,adqx,adqy} in such a way that x appears exactly i-1 times and y appears exactly j-1 times (if i=1 or j=1 it is understood that these maps are not counted).
Proposition 4 (dimensions).
Let i,j∈N. Let the vector space Si,j⊆Ai,j be as in Definition 2. Then one has the following:
dim(Ai,j)=i+ji;
d(i,j)≔dim(Si,j)=i+ji-1;
the number of possible [y,x]q-centered q-mutators writable as in (52) defining Si,j is(53)ni,j≔i+j-2i-1·2i+j-2;
the number of the linearly independent dependency relations among the list of the q-mutators in part (iii) above is(54)ri,j≔ni,j-di,j=i+j-2i-1·2i+j-2-i+ji+1.
A clarification of point (iii) above is needed: here we are counting separately any of the formal objects in (52) even if, a posteriori, some of these q-mutators may be equal. In other words, we count the (i+j-2)-tuples (β1,β2,…,βi+j-2), where β1,…,βi+j-2 belong to {adq∗x,adq∗y,adqx,adqy} in such a way that x appears exactly i-1 times and y appears exactly j-1 times (if i=1 or j=1 it is understood that these maps are not counted).
Proof.
We split the proof into four steps.
dim(Ai,j) follows from the cardinality of the set (55)xi1yj1⋯xinyjn:i1+⋯+in=i,j1+⋯+jn=j.
We claim that dim(Si,j) is one unit less than dim(Ai,j). To this aim, we recall that in Section 2 we introduced on A=∏i,jAi,j the quotient modulo I, the two-sided ideal generated by [y,x]q. Due to homogeneity and degree reasons, we can also consider this quotient on each Ai,j separately and we can infer that Ai,j/I is isomorphic to Ai,j/Ii,j. We therefore get dim(Ai,j/Ii,j)=dim(Ai,j/I)=1. This last identity follows by (30). From Lemma 3 we know that Ii,j=Si,j, so that dim(Ai,j/Si,j)=1. Since Si,j is a vector subspace of Ai,j, this gives the claimed dim(Si,j)=dim(Ai,j)-1.
If i=j=1 the only object in (52) is [y,x]q and (53) is correct, providing n(1,1)=1. If i≠1 or j≠1, we need to count the (i+j-2)-tuples (β1,β2,…,βi+j-2), where β1,…,βi+j-2∈{adq∗x,adq∗y,adqx,adqy}, and x, y, respectively, appear i-1 and j-1 times. These (i+j-2)-tuples are in bijective relation with the formal monomials(56)Xa1Yb1⋯XanYbnwith a1+⋯+an=i-1b1+⋯+bn=j-1,
where X∈{adq∗x,adqx} and Y∈{adq∗y,adqy}, via the identification (57)β1,…,βi+j-2⟷X,…,X︸a1 times,Y,…,Y︸b1 times,…,X,…,X︸an times,Y,…,Y︸bn times.
Now, the number of the monomials in (56) is precisely i+j-2i-1, and for any such a monomial we can choose X’s in 2a1+⋯+an=2i-1 different ways and Y’s in 2b1+⋯+bn=2j-1 different ways; this proves (53).
This follows from (ii) and (iii).
Remark 5.
Proposition 4 provides us with a very simple basis for Si,j (which is not, however, constituted of iterated q-mutators of x and y as in (52)). Indeed, from (30) we know that(58)xi1yj1⋯xinyjn-qi2j1+⋯+inj1+j2+⋯+jn-1xiyj∈Si,j,whenever i1+⋯+in=i and j1+⋯+jn=j. Taking into account that dim(Si,j)=dim(Ai,j)-1, it is then very easy to construct a basis for Si,j by means of this procedure.
An example will clarify this. A2,4 is spanned by the 15 monomials(59)x2y4,xyxy3,xy2xy2,xy3xy,xy4xy0,x0yx2y3,x0yxyxy2,x0yxy2xy,x0yxy3xy0,x0y2x2y2,x0y2xyxy,x0y2xy2xy0,x0y3x2y,x0y3xyxy0,x0y4x2y0.We apply the procedure in (58) to all of these monomials except for the first:(60)xyxy3-qx2y4,xy2xy2-q2x2y4,xy3xy-q3x2y4,xy4xy0-q4x2y4,yx2y3-q2x2y4,yxyxy2-q3x2y4,yxy2xy-q4x2y4,yxy3x-q5x2y4,y2x2y2-q4x2y4,y2xyxy-q5x2y4,y2xy2x-q6x2y4,y3x2y-q6x2y4,y3xyx-q7x2y4,y4x2-q8x2y4.Due to (58) these 14 polynomials all belong to S2,4 and they are (clearly) linearly independent (as they are obtained from linearly independent vectors by subtracting multiples of a given vector); since dim(S2,4)=14 by Proposition 4-(ii), they form a basis for S2,4. This also means that each of them can be written as a sum of [y,x]q-centered q-mutators: it is not difficult to obtain such a representation for each of them by using the technique described in Section 2.
3.2. Producing General Identities: The Basic Maps
We are ready to provide a general technique which produces q-mutator identities. For later reference, we give for each formula/procedure a one-letter name. In the sequel,(61)B,C∘=B∘C-C∘Bdenotes the commutator of two operators B, C with respect to the composition ∘ of maps (whenever this makes sense).
Here is the list of our procedures for obtaining q-mutator identities:
We say that identity (9) is the Transformation Rule; it allows transforming polynomials (under their associative presentation) into a linear combination of iterated q-mutators. With the notation in Definition 2, (9) can be rewritten as(62)Rb=11-q2adq∗b+qadqb,La=11-q2adqa+qadq∗a,
holding true for every a and b.
The following identity (see also (39)) is implicitly contained in the work [9] by Reiner; we call it Reiner’s identity:(63)a,bq,aq=a,b,aqqfor every a,b.
With the formalism in Definition 2, it can be rewritten as the commuting relation(64)adq∗a∘adqa≡adqa∘adq∗a,for every a,
which is also equivalent to(65)adq∗a,adqa∘≡0.
We introduce the following identity involving three letters a, b, c:(66)a,c,bqq+b,c,aqq=b,cq,aq+a,cq,bq.
It can be written as(67)adqa∘adq∗b+adqb∘adq∗a≡adq∗a∘adqb+adq∗b∘adqa,
for every a,b, or alternatively as a relation involving the ∘-commutators of left and right q-mutator operators:(68)adqa,adq∗b∘≡adq∗a,adqb∘.
Identity (67) can be proved starting from identity (64) by the substitution of a with a+b (and then by two cancelations, using (64)). We note that (66) is symmetric with respect to an interchange of a with b. Finally, when a=b, (67) gives at once (64). Therefore (R) and (A) are equivalent, but, for our purposes, we shall use them in different ways, so it is more convenient to keep them separated.
We introduce another identity for three letters a, b, c; namely,(69)a,c,bqq-a,cq,bq=qq2+1c,aq,bq-c,bq,aq+b,a,cqq-a,b,cqq.
It can be written as(70)adqa∘adq∗b-adq∗b∘adqa≡qq2+1adq∗b∘adq∗a-adq∗a∘adq∗b+adqb∘adqa-adqa∘adqb.
This gives an alternative way of writing the ∘-commutator of left and right q-mutator operators by means of the ∘-commutators of two right and two left q-mutator operators: (71)adqa,adq∗b∘=qq2+1adq∗b,adq∗a∘+adqb,adqa∘.
The proof of (69) follows by applying twice the Transformation Rule (T) to acb, by writing the latter alternatively as (a·c)·b and a·(c·b) and then using identity (A). If we interchange a and b in (70), the right-hand side changes sign; it then easily follows that (70) implies (67), whence (B) implies (A). Furthermore, if a=b, identity (69) reduces to (63).
Let p be any monomial of the form xi1yj1⋯xinyjn. We consider the tautological identity (72)Y·p·yx-qxy=yx-qxy·p·Y,where Y=y,xq.
We repeatedly apply the Transformation Rule (T) to both of its sides, in the following way: we write the left-hand side as (73)Y·x⋯x︸i1 timesy⋯y︸j1 times⋯x⋯x︸in timesy⋯y︸jn times·y·x-q…analogous…,
and we apply (T) from left to right (to both summands), without breaking Y into its summands yx-qxy, so that Y will always appear in the innermost position of a sum of iterated q-mutators of x and y (ultimately producing a linear combination of [y,x]q-centered polynomials); we do the same on the right-hand side, starting from right to left, in order to preserve again Y in innermost positions. See Example 6 for an example of this technique.
Under the name inherited relations, we call any identity which can be directly obtained from lower order identities in (R), (A), (B), and (C) by applying either(74)adq∗x,adq∗y,adqxor adqy
to both sides. Furthermore, starting from total degree 6 (see Table 1) this procedure will also apply on lower order identities previously obtained by (I) itself.
A few numbers relative to the analysis of the identities obtained by our procedure, for [y,x]q-centered q-mutators of total degree ≤ 10.
Bidegree
n(i,j)
d(i,j)
r(i,j)
I(i,j)
R(i,j)
A(i,j)
B(i,j)
C(i,j)
(1,1)
1
1
0
0
0
0
0
0
(1,2)
2
2
0
0
0
0
0
0
(1,3)
4
3
1
0
1
0
0
0
(2,2)
8
5
3
0
0
1
1
1
(1,4)
8
4
4
2
2
0
0
0
(2,3)
24
9
15
8
2
2
2
1
(1,5)
16
5
11
8
3
0
0
0
(2,4)
64
14
50
38
5
3
3
1
(3,3)
96
19
77
60
6
5
5
1
(1,6)
32
6
26
22
4
0
0
0
(2,5)
160
20
140
122
9
4
4
1
(3,4)
320
34
286
254
13
9
9
1
(1,7)
64
7
57
52
5
0
0
0
(2,6)
384
27
357
332
14
5
5
1
(3,5)
960
55
905
852
24
14
14
1
(4,4)
1280
69
1211
1144
28
19
19
1
(1,8)
128
8
120
114
6
0
0
0
(2,7)
896
35
861
828
20
6
6
1
(3,6)
2688
83
2605
2524
40
20
20
1
(4,5)
4480
125
4355
4232
54
34
34
1
(1,9)
256
9
247
240
7
0
0
0
(2,8)
2048
44
2004
1962
27
7
7
1
(3,7)
7168
119
7049
6932
62
27
27
1
(4,6)
14336
209
14127
13920
96
55
55
1
(5,5)
17920
251
17669
17420
110
69
69
1
For every bidegree (i,j) we have the following.
n (i,j): number of symbolic [y,x]q-centered q-mutators of bidegree (i,j).
d (i,j): dimension of the vector space Si,j.
r (i,j): number of relations among the symbolic q-mutators.
I(i,j): number of inherited relations (I).
R(i,j): number of relations according to Reiner’s identity (R).
A(i,j): number of relations according to identity (A).
B(i,j): number of relations according to identity (B).
C(i,j): number of relations according to procedure (C).
Example 6.
We give an example for the procedure (C), when p=x. We have(75)Y·x·yx-qxy=yx-qxy·x·Y,where Y=y,xq.The left-hand side is Y·x·y·x-qY·x·x·y. For the first summand we have(76)Y·x·y·x=Y·x·y·x=T11-q2Y,xq+qx,Yq·y·x=T11-q22Y,xq+qx,Yq,yq+qy,Y,xq+qx,Yqq·x=T11-q23Y,xq+qx,Yq,yq+qy,Y,xq+qx,Yqq,xq+qx,Y,xq+qx,Yq,yq+qy,Y,xq+qx,Yqqq. Analogously, the second summand is(77)-qY·x·x·y=⋯=-q1-q23Y,xq+qx,Yq,xq+qx,Y,xq+qx,Yqq,yq+qy,Y,xq+qx,Yq,xq+qx,Y,xq+qx,Yqqq. The right-hand side is y·x·x·Y-qx·y·x·Y. For the first summand we have(78)y·x·x·Y=y·x·x·Y=T11-q2y·x·x,Yq+qY,xq=T11-q22y·x,x,Yq+qY,xqq+qx,Yq+qY,xq,xq=T11-q23y,x,x,Yq+qY,xqq+qx,Yq+qY,xq,xqq+qx,x,Yq+qY,xqq+qx,Yq+qY,xq,xq,yq. Analogously, the second summand is (79)-qx·y·x·Y=⋯=-q1-q23x,y,x,Yq+qY,xqq+qx,Yq+qY,xq,yqq+qy,x,Yq+qY,xqq+qx,Yq+qY,xq,yq,xq. Putting the pieces together, we obtain an identity for nested q-mutators in S3,2.
3.3. Producing General Identities: Counting the Identities
Finally, we describe how to obtain identities in each space Si,j (see Definition 2). Let i,j∈N be fixed. According to Proposition 4, we know that the dimension of Si,j is d(i,j)=i+ji-1, while the total number of the formal [y,x]q-centered q-mutators spanning Si,j is n(i,j)=i+j-2i-1·2i+j-2.
We show, inductively, how to construct n(i,j)-d(i,j) identities among the generators of Si,j by using the procedures described above. We here conjecture that the identities that we are able to obtain are linearly independent, and we shall deal with the proof of this conjecture in a future investigation.
If (i,j)=(1,1) there is nothing to prove since d(1,1)=n(1,1)=1. The same is true for the bidegrees (1,2) and (2,1) since d(1,2)=d(2,1)=n(1,2)=n(2,1)=2. For the bidegrees (i,j) with total degree i+j=4 we have the following scenario.
For bidegree (1,3) we have d(1,3)=3 and n(1,3)=4; there is one relation from (R); namely (see (63) with b=[y,x]q and a=y),(80)y,y,xqq,yq=y,y,xq,yqq.
Nothing can be obtained from procedure (I) since in previous degree 3 there are no relations; nothing can be obtained as well from identities (A) and (B) since, with the notations in (66) and (69), one has to choose c=[y,x]q and this forces taking a=b=y, but we already know that (A) and (B) reduce to (R) when a and b are equal. Analogous facts hold for bidegree (3,1).
For bidegree (2,2) we have d(2,2)=5 and n(2,2)=8; one has to find 3 independent relations. Nothing can be obtained from procedure (I) since in cases (1,2) and (2,1) there are no relations; (R) is not useful either, since one is forced to choose b=[y,x]q in (63), but then a cannot be either x or y, due to bidegree (2,2). The three needed relations are instead provided by (A), (B), and (C); indeed
from (A) we get (see (66) with c=[y,x]q, a=x, and b=y) (81)x,y,xq,yqq+y,y,xq,xqq=y,y,xqq,xq+x,y,xqq,yq;
from (B) we get (see (69) with c=[y,x]q, a=x, and b=y)(82)x,y,xq,yqq-x,y,xqq,yq=qq2+1y,xq,xq,yq-y,xq,yq,xq+y,x,y,xqqq-x,y,y,xqqq;
from (C) we get (use the procedure with p=1)(83)y,xq,yq+qy,y,xqq,xq+qx,y,xq,yq+qy,y,xqqq-qy,xq,xq+qx,y,xqq,yq-q2y,y,xq,xq+qx,y,xqqq=y,x,y,xqq+qy,xq,xqq+qx,y,xqq+qy,xq,xq,yq-qx,y,y,xqq+qy,xq,yqq-q2y,y,xqq+qy,xq,yq,xq.
It can be proved with some tedious linear algebra computations that these three identities are independent of each other.
After warming up with low degrees (which also serve for starting the induction), we are ready to take into account the general bidegree (i,j). In the sequel we can suppose that i+j>4 and we count the number of expected relations deriving from (I), (R), (A), (B), and (C), provided that we know these numbers for degrees strictly less than i+j.
Number of Relations from Procedure (I). Procedure (I) requires application of the following.
adq∗x or adqx: if i=1 there is nothing to do; otherwise we apply any of these maps to the relations of bidegree (i-1,j); there are precisely r(i-1,j) of these relations.
adq∗y or adqy: if j=1 there is nothing to do; otherwise we apply any of these maps to the relations of bidegree (i,j-1); there are precisely r(i,j-1) of these relations.
Summing up, after some simple computation on binomials, the total number of expected identities from procedure (I) is (see also (54))(84)Ii,j≔2ri-1,j+2ri,j-1=0if i=j=1i+j-2i-12i+j-2-2i+ji+4if i≥2 or j≥2.
Number of Relations from Identity (R). We apply identity (63) with a=x or with a=y. In the first case we choose as b any of the members of a basis of the set Si-2,j; in the second case we choose as b any of the members of a basis of Si,j-2. Since the iterated q-mutator b must contain [y,x]q in the innermost position, the first case occurs only if i≥3 and the second case only if j≥3. Summing up, the expected relevant number of relations from identity (R) is (see also Proposition 4-(ii)):(85)Ri,j≔0if i≤2,j≤2i+j-2i-1if i≤2,j≥3i+j-2j-1if i≥3,j≤2i+j-2i+i+j-2j-2if i≥3,j≥3.
Number of Relations from Identity (A). As we already remarked, identity (66) boils down to (R) when a=b; hence we can take a≠b. Also, we do not get new information if a and b are interchanged; thus we can always choose a=x and b=y, so that (A) can be applied only when i≥2 and j≥2 (since c must contain [y,x]q). In place of c we can then take any member of a basis of Si-1,j-1. Summing up, taking into account the formula for d(i-1,j-1), the expected number of relations from identity (A) is(86)Ai,j≔0if i=1 or j=1i+j-2i-1-1if i≥2,j≥2.
Number of Relations from Identity (B). An argument similar to the above one applies for (B): we take a=x and b=y and c is any member of a basis of Si-1,j-1. Thus, the expected number of relations from identity (B) is(87)Bi,j≔0if i=1 or j=1i+j-2i-1-1if i≥2,j≥2.
Number of Relations from Identity (C). With the notation in the description of procedure (C), since x and y appear at least twice (remember that Y=[y,x]q), procedure (C) is inapplicable if i=1 or j=1. Our conjecture states that, denoting by C(i,j) the number of relations deriving from (C), there suffices one and only one such relation, when i≥2 and j≥2; thus we define(88)Ci,j≔0if i=1 or j=11if i≥2,j≥2.
(It is expected that this can be obtained by taking p=xi-2yj-2.)
Remark 7 (consistency of the number of identities).
In order to support our conjecture on the linear independence of the identities obtained via (I), (R), (A), (B), and (C) (and the minimal application of the latter), we verify that the sum of the numbers of the relations obtained above fill the number of the needed independent relations; in other words, for every i,j∈N, (89)Ii,j+Ri,j+Ai,j+Bi,j+Ci,j=ni,j-di,j.This is a simple verification, which we omit, based on the Pascal rule for binomials and on formulas (84) to (88).
See Table 1 for the computation of the above numbers in (84) to (88), up to degree 10. Up to degree 10 it has been verified, with the help of the computer algebra system Reduce [35], that our conjecture is true.
4. The q-Deformed CBH Theorem
It is understood for the rest of the paper that the notations of Section 2 are fixed. To make our study of the q-BCH Formula precise, we need to endow A with a metric structure. Indeed, as in [1, Theorem 2.58, p. 94], A can be equipped with a metric space structure by the distance(90)da,b≔exp-mda-b,a,b∈A,where md(0)≔∞ and if p=∑npn≠0 (see the notation in (21)), we set(91)mdp≔minn≥0:pn≠0,with the convention exp(-∞)=0.
The metric space (A,d) is complete and it is an isometric completion of Kqx,y (as a metric subspace); moreover it is ultrametric; that is,(92)da,b≤maxda,c,dc,b,a,b,c∈A.
Remark 8.
As a consequence of these facts and of the invariant property d(a,b)=d(a+c,b+c), any series ∑nan in A (where an∈A for any n) is convergent if and only if limn→∞an=0 in A, that is, if and only if limn→∞md(an)=∞. (See [1, Section 2.3.1] for all the details.) In the sequel we shall tacitly use the well-behaved properties of the topology of (A,d) allowing us to easily perform any passage to the limit or limit/series interchange.
With this topology, the series in (21) not only is a formal expression but also becomes a genuine convergent series in (A,d), since md(pn)≥n→∞ as n→∞ (because pn∈An).
Since, for any p∈A+, one has md(pn)→∞ as n→∞, Remark 8 ensures that the following maps are well posed, as convergent series in the metric space A:(93)exp:A+⟶1+A+,expp=∑n=0∞pnn!,expq:A+⟶1+A+,expqp=∑n=0∞pnnq!.We have the following results, whose simple proofs are omitted.
(i) Each of the maps exp and expq in (93) admits an inverse function, which we, respectively, denote by log and logq, from 1+A+ to A+. We say that logq is the q-logarithm.
(ii) There exists a map ϕq:A+→A+ such that(94)expϕqp=expqp,expqϕq-1p=exppfor every p∈A+.It is known that ϕq has the explicit expansion (see [16, 37])(95)ϕqp=∑n=1∞1-qn-1nnqpn,p∈A+.Then we infer that in the associative algebra A there exists one and only one formal power series zq(x,y) such that (6) is valid, and this is defined as in (28). The series zq(x,y) is referred to as the q-Baker-Campbell-Hausdorff series (shortly, the q-BCH series), and it will be also denoted by x⋄qy. Its associative presentation is (29). Grouping together the summands of the same degree, we use the notation(96)x⋄qy=∑n=1∞Qnx,y,with Qnx,y∈An for every n≥1.The operation ⋄q gives the q-deformation of the classical BCH Formula(97)x⋄y=x+y+12x,y+112x,x,y+y,y,x-124x,y,x,y+⋯.
Remark 9 (intertwining of the deformed/undeformed BCH series).
By using the map ϕq in (95) we can obtain a representation for x⋄qy deriving from the following argument:(98)expqx⋄qy=6expqxexpqy=94expϕqxexpϕqy=97expϕqx⋄ϕqy=94expqϕq-1ϕqx⋄ϕqy.Since expq is injective we get the identity(99)x⋄qy=ϕq-1ϕqx⋄ϕqy,intertwining the undeformed and the q-deformed BCH series. Due to explicit form (95) of ϕq, identity (99) can be used for computational issues to derive explicit summands of ⋄q starting from those of ⋄; unfortunately, computations become cumbersome very rapidly.
As it happens for the undeformed case of the classical Campbell-Baker-Hausdorff-Dynkin series, the most natural problem is the study of the distinguished algebraic properties of the polynomials Qn(x,y) in (96). We next claim that, apart from Q1(x,y)=x+y, any Qn(x,y) (with n≥2) is a sum of polynomials containing yx-qxy as a factor. This fact can be seen as a consequence of Schützenberger’s result (7) recalled in the Introduction.
In order to prove the above claim, we first observe that, from yx-qxy∈I, one gets(100)yIxI=qxIyI.By arguing inductively one obtains(101)yIjxIi=qijxIiyIj,for every i,j≥1.By the aid of this identity, we can provide a basis for Ai,j/I made of one single element; for instance,(102)Ai,jI=spanKqxIiyIj,for everyi,j≥0.Clearly, [x]Ii[y]Ij is nonvanishing, since xiyj∉I for any i,j≥0, whence the dimension of Ai,j/I is 1 precisely. Identity (101) has another remarkable consequence, resemblant to Newton’s binomial formula, namely (see Schützenberger [32]; see also Cigler [33]),(103)xI+yIn=∑k=0nnkqxIkyIn-k,for every n≥1,where(104)nkq≔nq!kq!n-kq!,0≤k≤n.Indeed, one can easily prove (103) by induction on n, using (101) and the well-known q-Pascal rule (see, e.g., [4]) (105)n+1kq=nk-1q+qknkq,1≤k≤n.By the aid of identity (103) solely, one can prove the next result. We name it after Schützenberger, even if its original formulation was in terms of q-commuting variables (see (7)).
Lemma 10 (Schützenberger [32]).
The q-BCH series has the decomposition(106)x⋄qy=∑i,j≥0Qi,jx,y,with Qi,jx,y∈Ai,jwith the following properties:
Q0,0(x,y)=0, Q(1,0)(x,y)=x, Q(0,1)(x,y)=y;
Q(i,0)(x,y)=Q(0,j)(x,y)=0 for every i,j≥2;
for any i,j with i,j≥1, one has Qi,j(x,y)∈Ii,j.
For example (see also [38]), one has(107)Q1,1x,y=-12qy,xq,Q2,1x,y=-q2q3q!x,y,xqq,Q1,2x,y=-q2q3q!y,xq,yq.
Proof.
We first equip A/I with the structures of a topological algebra and of a complete ultrametric space, by imitating the corresponding structures on A. Then one can define a q-exponential on A/I as well, denoted by exp~q. Since the projection [·]I is continuous morphism of the underlying algebras one obtains(108)expqxexpqyI=exp~qx⋄qyI.On the other hand, (103) gives(109)exp~qxI+yI=∑n=0∞1nq!∑k=0nnkqxIkyIn-k=∑k,j≥0xkyjkq!jq!I=expqxexpqyI.The injectivity of exp~q implies that [x⋄qy]I=[x]I+[y]I or equivalently x⋄qy-(x+y)∈I, which immediately proves the theorem.
Next we describe another feature of the q-BCH series.
Definition 11 (nested q-mutators).
One says that any element of the form(110)zk,⋯z3,z2,z1qq⋯q,with k≥2 and z1,z2,…,zk∈{x,y}, is a right-nested q-mutator of x and y of length k. Analogously, one says that any element of the form (111)⋯z1,z2q,z3q⋯,zkqis a left-nested q-mutator of x and y of length k. For k=1, one qualifies x and y as the right-nested (and the left-nested) q-mutators of length 1.
We use the following result.
Theorem 12 (Reiner [9]).
Let i,j∈N∪{0}. Let one construct the set Ri,j of the polynomials in Ai,j consisting of the left-nested q-mutators(112)x,xq⋯x︸i1 timesqyq⋯y︸j1 timesq⋯xq⋯x︸in timesqyq⋯y︸jn timesq,with i1+⋯+in=i and j1+⋯+jn=j.
Then Ri,j is a basis of Ai,j as well.
Clearly one can obtain an analogous result with right-nested q-mutators. For brevity, we shall refer to Ri,j as the left-nested Reiner basis of Ai,j (the choice of the notation “R” refers to “Reiner”). Since we have an associative presentation of x⋄qy, we immediately get, from Reiner’s Theorem 12, the following result.
Corollary 13.
With the notation in Lemma 10, any Qi,j(x,y) can be expressed in a unique way as a linear combination of elements of the left-nested Reiner basis Ri,j of Ai,j. Therefore, the q-BCH series x⋄qy admits a presentation as a series of left-nested q-mutators of x and y as in (112) (with coefficients in K(q)).
An analogous result holds for right-nested q-mutators.
The disadvantage of the above nested presentation of the q-BCH series, based on Reiner’s Theorem 12, is that it (necessarily) allows for summands of the form(113)x,xq,⋯q,x,yq,⋯q,y,xq,⋯q,y,yq,⋯q,which are not manifestly consistent with what is known from Schützenberger’s result (7) (encoded in Lemma 10); for example, already in degree three the nested presentation differs from (107) in that(114)Q2,1x,y=-q2q3q!x,yq,xq.Compared to our expression (107) for Q2,1, we get general identity (39) (implicitly contained in [9]). Our main task is to compound Lemma 10 and Corollary 13 and prove that x⋄qy admits a presentation with q-mutators (not necessarily nested) where the innermost q-mutator is [y,x]q, consistently with the mentioned Schützenberger’s result. This is not obvious since, for example, [[x,x]q,y]q (which is a priori legitimate in Reiner’s presentation) is not a linear combination of [[y,x]q,x]q and [x,[y,x]q]q.
Proposition 14.
For any i,j≥1, the summand Qi,j(x,y) in q-BCH series (106) belongs to Si,j (see Definition 2); that is, Qi,j(x,y) can be expressed as a linear combination (with coefficients in K(q)) of [y,x]q-centered q-mutators.
More generally, for every i,j∈N∪{0} we have Ii,j=Si,j and I=S.
Proposition 14 improves the result provided in [34], where Q1,…,Q4 are given in terms of nested q-mutators centered at [y,x]q or centered at [x,y]q.
Proof.
We prove that Ii,j=Si,j for every i,j∈N∪{0}. Since Si,j⊆Ii,j, we are left to prove the reverse inclusion. From (25) (and the definition of Si,j when i or j vanishes) we have Ii,j={0}=Si,j whenever i or j is 0. We can therefore suppose that i,j≥1. If i=j=1 we trivially have I1,1=span{[y,x]q}=S1,1.
We can thus suppose that i+j≥3. Any element of Ii,j is, by definition, a linear combination (with coefficients in K(q)) of(115)B1∘⋯∘Bi+j-2y,xq,where B1,B2,…,Bi+j-2 all belong to the set of maps {Rx,Ry,Lx,Ly} (see Definition 2), in such a way that x appears exactly i-1 times and y appears exactly j-1 times. By (9), any of the maps B1,B2,…,Bi+j-2 in (115) is a linear combination of suitable maps belonging to {adq∗x,adq∗y,adqx,adqy} (preserving the total number of x’s and y’s).
Therefore (115) is a [y,x]q-centered polynomial in Ai,j. This shows that Ii,j⊆Si,j. In particular, since Qi,j(x,y)∈Ii,j (see Lemma 10-(3)), we get Qi,j(x,y)∈Si,j.
5. A Criterion for [y,x]q-Centered q-Mutators
Due to our interest in [y,x]q-centered q-mutators, we introduce a criterion for characterizing the elements of Ii,j (or equivalently, of Si,j).
Any nonvanishing monomial in A can be written in a unique way as a scalar multiple of the following basis1 monomials (we agree that x0=y0=1K, the identity of K):(116a)xi1yj1with i1,j1≥0,(116b)xi1yj1⋯xinyjnwith n≥2,i1,jn≥0,j1,i2,…,jn-1,in≥1.We denote by m any of the above monomials. In the sequel, we also agree that any monomial in x,y has been written in the above unique way.
Definition 15.
Let M={m} denote the collection of the monomials in (116a)-(116b). We set(117)H:M⟶N∪0Hm≔0if m is as in116ai2j1+⋯+inj1+j2+⋯+jn-1if m is as in116b.By an abuse of notation, we agree that the map H is also defined on the multi-indices appearing in (116a)-(116b), so that we also write H(i1,j1)=0 if the indexes are as in (116a), and(118)Hi1,j1,…,in,jn=i2j1+⋯+inj1+⋯+jn-1,if the indexes are as in (116b).
Starting from (101), which can be rewritten as ynxm≡Iqnmxmyn, by an inductive argument one gets (30); namely,(119)xi1yj1⋯xinyjn≡Iqi2j1+⋯+inj1+j2+⋯+jn-1xi1+⋯+inyj1+⋯+jn.The following map plays, in a certain sense, the same role played by the Dynkin-Specht-Wever map (see, e.g., [1, Lemma 3.26]) in detecting the Lie-polynomials.
Lemma 16 (criterion for [y,x]q-centrality).
With the notation in Definition 15, we consider the unique (continuous) K(q)-linear map φ:A→A defined on monomials as follows:(120)φm≔m-qHmxi1yj1=0if m is as in116am-qHmxi1+⋯+inyj1+⋯+jnif m is as in116b.Then, given p∈A, one has p∈I (or, equivalently, p∈S) if and only if φ(p)=p. Moreover, φ is valued in I=S so that φ is a projection onto S.
By using the abused notation following Definition 15, a homogeneous polynomial p belonging to Ai,j, say (121)p=∑i1+⋯+in=ij1+⋯+jn=jci1,j1,…,in,jnxi1yj1⋯xinyjn,belongs to S if and only if(122)∑i1+⋯+in=ij1+⋯+jn=jci1,j1,…,in,jnqHi1,j1,…,in,jn=0.Note that the latter is simply an identity in K(q).
Example 17.
We consider the polynomial p in A2,2 defined by(123)p=1+q2xyxy-qx2y2+q2-qyx2y-q3+qyxyx+q2y2x2.The associated scalar as in (83) is(124)1+q2q-qq0+q2-qq2-q3+qq3+q2q4.Since this is evidently null (as one can check upon expansion), we can infer that p∈S. Actually one can verify that p is equal to [[x,[y,x]q]q,y]q∈S2,2.
Proof of Lemma 16.
We split the proof into five steps.
(I) First we have φ(A)⊆I thanks to (119): indeed, for any m∈M we have either φ(m)=m-qH(m)xi1yj1=0∈I (in case (116a)) or(125)φm=m-qHmxi1+⋯+inyj1+⋯+jn≡I0in case (116b) (i.e., φ(m)∈I). By linearity (and continuity) this gives φ(a)∈I for any a∈A.
(II) If p∈A is such that φ(p)=p, then, by part (I), we infer that p=φ(p)∈φ(A)⊆I, whence p∈I.
(III) Conversely, suppose that p∈I; we need to show that φ(p)=p, or equivalently we have to prove that φI is the identity on I. To this aim, it suffices to show that φ is the identity on any Ii,j. To this end, let p∈Ii,j; like any polynomial, p can be uniquely written in the basis (116a)-(116b) as(126)p=∑m∈Fpcpmm,where Fp is a finite family of basis monomials in M and cp(m)∈K(q) for any m∈Fp. Then, by recalling that the (nonzero) monomials which span Ii,j have bidegree (i,j), we infer(127)φp=∑m∈Fpcpmm-qHmxiyj=∑m∈Fpcpmm-∑m∈FpcpmqHmxiyj=p-∑m∈FpcpmqHmxiyj.Moving terms around we get (∑m∈Fpcp(m)qH(m))xiyj=φ(p)-p. Now, the right-hand term belongs to I since φ is I-valued (see part (I) of the proof) and since p∈I by assumption; so the same is true of the left-hand side, but a scalar multiple of xiyj can belong to I iff the scalar factor is null. Hence, from (127) we get φ(p)=p.
(IV) The surjectivity of φ is a trivial consequence of part (III).
(V) We have to prove the last assertion of the theorem. On the one hand, let p∈S (this part of the proof does not require p∈Ai,j). By part (III) of the proof we know that p=φ(p); hence, if we write p in the basis monomials as(128)p=∑ci1,j1,…,in,jnxi1yj1⋯xinyjn,we get(129)p=φp=∑ci1,j1,…,in,jnφxi1yj1⋯xinyjn=∑ci1,j1,…,in,jnxi1yj1⋯xinyjn-qHi1,j1,…,in,jnxi1+⋯+inyj1+⋯+jn=p-∑ci1,j1,…,in,jnqHi1,j1,…,in,jnxi1+⋯+inyj1+⋯+jn.After canceling p and grouping terms of the same bidegree we get(130)0=∑i1+⋯+in=ij1+⋯+jn=jci1,j1,…,in,jnqHi1,j1,…,in,jn,for every i,j≥0,as needed (this gives precisely (83) when p∈Ai,j).
Conversely, let p∈Ai,j and suppose that after we have written p as(131)p=∑ci1,j1,…,in,jnxi1yj1⋯xinyjn,it is known that (83) holds true. In the preceding computations we proved that(132)φp=p-∑ci1,j1,…,in,jnqHi1,j1,…,in,jnxi1+⋯+inyj1+⋯+jn.If p∈Ai,j this is just(133)φp=p-∑i1+⋯+in=ij1+⋯+jn=jci1,j1,…,in,jnqHi1,j1,…,in,jnxiyj.By assumed (83), the term in the parenthesis is null, whence φ(p)=p. By part (II) of the proof we therefore get p∈S.
Remark 18.
Without any specification on the exponents i1,j1,…,in,jn, different-looking monomials p=xi1yj1⋯xinyjn can produce the same monomial m as in (116a)-(116b): for example,(134)x1y0x2y0x1y1x0y0x0y2xy2=x4y3xy2=x4y3x1y0x0y1x0y1.However, it can be easily checked that the definition of H is unambiguous for any monomial p and it leads to the same result; that is,(135)Hxi1yj1⋯xinyjn≔i2j1+⋯+inj1+j2+⋯+jn-1,with the convention (which we tacitly assume in the sequel) that the sum is 0 if n=1. Accordingly, the map φ is well posed for every monomial:(136)φxi1yj1⋯xinyjn=xi1yj1⋯xinyjn-qi2j1+⋯+inj1+j2+⋯+jn-1xi1+⋯+inyj1+⋯+jn.
The next section provides a closed formula for the terms Qi,j in the q-BCH series, only depending on the coefficients of the q-logarithm. The main tool is Lemma 16.
6. An Explicit Formula for the q-BCH Series
We already showed the basic associative presentation of x⋄qy in (29), where the coefficients λn (from the expansion of logq) are used: they can be derived, for example, by the recursion formula (27). In this section we provide an explicit formula for Qi,j in terms of iterated q-mutators. The procedure is quite technical, so that the reader may first want to consult an example, describing the idea behind our formula for the q-BCH series with an example: this is given in Section 7. For obtaining an explicit formula for Qi,j in terms of iterated q-mutators, we first need some lemmas whose proofs (mainly, some inductive arguments) are omitted.
Lemma 19.
For any j≥1 one has (137)yjx=qjxyj+∑k=0j-1qkjkadqyj-kxyk,where jk denotes the classical (undeformed) binomial coefficient, and the notation adq denotes the left q-commutation map in (42).
Note that the sum over k in the right-hand side of (137) is an element of I, the bilateral ideal generated by [y,x]q. A direct application of formula (62) proves the following result, starting from (137).
Lemma 20.
For any j≥1 one has(138)yjx=qjxyj+∑k=0j-1q1-q2kjkadq∗y+qadqykadqyj-kx.
Remark 21.
Formula (138) could be written in an even more explicit form: indeed, the operators adq∗a and qadqa commute, for every a∈A, as identity (39) proves. Hence one can apply Newton’s binomial formula to obtain(139)adq∗a+qadqak=∑h=0kqhkhadq∗ak-hadqah,∀a∈A.Moreover, since any right multiplication Rb commutes with any left multiplication La (by associativity), we infer that(140)anymapadq∗b+qadqbcommuteswithanymapadqa+qadq∗a.
We now obtain a formula, generalizing the above lemma, which also expresses in a “quantitative” way the congruence yjxi≡qjixiyj(modI).
Lemma 22.
For any i,j≥1 one has(141)yjxi-qjixiyj=11-q2i-1∑k=0j-1∑h=0i-1qk+hj1-q2kjkadq∗x+qadqxi-1-hadqx+qadq∗xhadq∗y+qadqykadqyj-kx.
As stated in Remark 21, this formula can be made even more explicit by unraveling the powers of adq∗x+qadqx, adqx+qadq∗x, and adq∗y+qadqy by means of (139).
Note that the right-hand side of identity (141) is a linear combination of iterated q-mutators, centered at [y,x]q, since j-k>0. In other words it is an element of S=I. Our final prerequisite is to find an explicit form (in terms of iterated [y,x]q-centered q-mutators) for the projection φ defined in Lemma 16, when it acts on a generic monomial xi1yj1⋯xinyjm. This is given in the next result.
Lemma 23.
For any n≥2 and any i1,j1,…,in,jn≥0 one has (142)xi1yj1⋯xinyjn-qi2j1+i3j1+j2+⋯+inj1+⋯+jn-1xi1+⋯+inyj1+⋯+jn=11-q2i1+⋯+in-1∑r=1n-1qi2j1+⋯+irj1+⋯+jr-11-q2jr+1+jr+2+⋯+jn∑k=0j1+⋯+jr-1qk1-q2kj1+⋯+jrk∑h=0ir+1-1qhj1+⋯+jradq∗y+qadqyjn⋯adq∗x+qadqxir+2adq∗y+qadqyjr+1adq∗x+qadqxir+1-h-1adq∗y+qadqykadqx+qadq∗xh+i1+⋯+iradqyj1+⋯+jr-kx.We agree that, when r=1, the exponent i2j1+⋯+ir(j1+⋯+jr-1) has to be considered 0; furthermore we agree that summations over an empty set of indices are to be omitted.
Again (see Remark 21) the above formula can be made more explicit (although more cumbersome) by unraveling the powers of adq∗z+qadqz and of adqz+qadq∗z (with z=x,y), by means of (139). We are ready for the proof of Theorem 1.
Proof of Theorem 1.
Let i,j≥1 be fixed. From Proposition 14 we know that Qi,j(x,y) belongs to Si,j⊂S. From Lemma 16 we derive that φ(Qi,j(x,y))=Qi,j(x,y), since φ is a projection onto S. From the associative presentation (29) of x⋄qy we infer the explicit formula(143)Qi,jx,y=∑n=1i+jλn∑i1,j1,…,in,jn≠0,0i1+⋯+in=i,j1+⋯+jn=jxi1yj1⋯xinyjni1q!j1q!⋯inq!jnq!,where the λn are as in (27). We observe that, from (29), we have isolated the summands with 1≤n≤i+j since (i1,j1),…,(in,jn)≠(0,0), so that(144)i+j=i1+j1+⋯+in+jn≥n.Taking into account the above facts (and the definition of φ), if we apply φ to both sides of (129) we get (see also Remark 18)(145)Qi,jx,y=∑n=1i+jλn∑i1,j1,…,in,jn≠0,0i1+⋯+in=i,j1+⋯+jn=j1i1q!j1q!⋯inq!jnq!xi1yj1⋯xinyjn-qi2j1+⋯+inj1+j2+⋯+jn-1xi1+⋯+inyj1+⋯+jn.Finally, the polynomial in the above parentheses has been explicitly written as a [y,x]q-centered q-mutator in Lemma 23: this proves the theorem.
7. An Example of the Rearrangement Technique
We know from (101) that xyxy3 is congruent to qx2y4 modulo I. By means of a repeated application of the trivial identity(146)yx=y,xq+qxy,we can write xyxy3-qx2y4 as an element of I (i.e., as a sum of polynomials factorizing [y,x]q); subsequently, we can apply basic procedure (9) to write it as an element of S (i.e., as a linear combination of [y,x]q-centered q-mutators). This is done in the next computation: (147)xyxy3-qx2y4=xy,xq+qxyy3-qx2y4=xy,xqy3=911-q2x,y,xqq+qy,xq,xq·y3=911-q22x,y,xqq,yq+qy,x,y,xqqq+qy,xq,xq,yq+q2y,y,xq,xqq·y2=911-q23x,y,xqq,yq,yq+qy,x,y,xqq,yqq+qy,x,y,xqqq,yq+q2y,y,x,y,xqqqq+qy,xq,xq,yq,yq+q2y,y,xq,xq,yqq+q2y,y,xq,xqq,yq+q3y,y,y,xq,xqqq·y=911-q24x,y,xqq,yq,yq,yq+qy,x,y,xqq,yq,yqq+qy,x,y,xqq,yqq,yq+q2y,y,x,y,xqq,yqqq+qy,x,y,xqqq,yq,yq+q2y,y,x,y,xqqq,yqq+q2y,y,x,y,xqqqq,yq+q3y,y,y,x,y,xqqqqq+qy,xq,xq,yq,yq,yq+q2y,y,xq,xq,yq,yqq+q2y,y,xq,xq,yqq,yq+q3y,y,y,xq,xq,yqqq+q2y,y,xq,xqq,yq,yq+q3y,y,y,xq,xqq,yqq+q3y,y,y,xq,xqqq,yq+q4y,y,y,y,xq,xqqqq.In Lemma 23 we provided a formula for the above procedure for any monomial(148)xi1yj1⋯xinyjn.
This procedure allows us to write any Qi,j as a [y,x]q-centered q-mutator. For example, a direct computation based on (129) and (27) gives out the associative presentation of Q1,3:(149)Q1,3=14q!2q·q3-q+1·xy3+q3-q+1·y3x+q2q3-2q2+2q-1·y2xy+q2q3-2q2+2q-1·yxy2.We then use the identity yx=[y,x]q+qxy on each summand of Q1,3 (other than xy3) to eventually produce, modulo I, the monomial xy3. For instance,(150)y3x=y2y,xq+qyy,xqy+q2y,xqy2+q3xy3y2xy=yy,xqy+qy,xqy2+q2xy3yxy2=y,xqy2+qxy3.(These formulas can be improved by shifting all the factors y on the right as in Lemma 19.) Inserting these identities in the expansion of Q1,3 we get(151)4q!2q·Q1,3=⋯=-q4+q3+q3-q4q3+q5-2q4+2q3-q2q2+q5-2q4+2q3-q2q·xy3+q3-q4y2y,xq+q3-q4q+q5-2q4+2q3-q2yy,xqy+q3-q4q2+q5-2q4+2q3-q2q+q5-2q4+2q3-q2y,xqy2=0·xy3-q3q-1y2y,xq-q2q-12yy,xqy+q2q-1y,xqy2.Obviously, the cancelation of the summand xy3 is not sheer chance, but it derives from the fact that Q1,3 belongs to I. We next apply the technique exemplified above, based on (9), thus obtaining the presentation2(152)Q1,3=-q24q!2qy,xq,yq,yq+y,y,xq,yqq.Analogously one gets(153)Q3,1=-q24q!2qx,x,y,xqqq+x,y,xqq,xq.
AppendixThe Convergence of the q-BCH Series Near Zero
To test the applicability of our formula for the q-BCH series, we prove the following result concerning the convergence of the series for Banach algebras; we observe that our arguments can also be applied in wider context than associative algebras, as the proof will show (see also Remark A.2).
We recall that a Banach algebra is triple (A,∗,·) where (A,∗) is an associative algebra over R (or C), (A,·) is a normed Banach space, and (a,b)↦a∗b is a continuous map (A×A is equipped with the product topology of the space A).
Theorem A.1.
Let (A,∗,·) be a Banach algebra over R. Let q∈R be such that |q|<1. Then there exists an open neighborhood Uq of 0∈A such that the series ∑i,j≥1Qi,j(x,y) converges normally for x,y∈Uq.
Here Qi,j(x,y) is given by formula (41) and [a,b]q=a∗b-qb∗a for any a,b∈A. Finally the numbers λn are defined as in (27) relative to R.3
In particular, the q-BCH series is convergent in any (real or complex) matrix algebra and—more generally—in any finite dimensional (real or complex) associative algebra.
Remark A.2.
Normal convergence on Uq means, as usual,(A.1)∑i,j≥1supx,y∈UqQi,jx,y<∞.Since A is a Banach space, normal convergence implies uniform and pointwise convergence. Theorem A.1 also holds true if the underlying field is C, as the proof will show. Finally, we do not use explicitly the associative structure of A, but only the basic inequality(A.2)a,bq≤Mqa·b,∀a,b∈A,for some constant Mq>0. Hence the same proof works for more general settings than Banach (associative) algebras.
Proof.
Since A is a Banach algebra, there exists a constant c≥1 such that(A.3)a∗b≤ca·b,∀a,b∈A.To simplify the notation, by replacing the norm · with the equivalent norm c·, we can assume that the above inequality holds true with c=1. As a consequence (A.2) holds true with Mq≔(1+|q|)≥1. From (A.2) one easily obtains the estimates4(A.4)adqajb≤Mqjaj·b,(A.5)adqa+qadq∗ajb,adq∗a+qadqajb≤Mq2jaj·b,for any a,b∈A and any j∈N, where adq and adq∗ have the obvious meanings.
We let ε>0 be small (it will be conveniently chosen in due course) and we take any x,y∈A such that x,y<ε. Then, by the triangle inequality and a repeated application of (A.4)-(A.5), we have(A.6)Qi,jx,y≤∑n=1i+jλn∑i1,j1,…,in,jn≠0,0i1+⋯+in=i,j1+⋯+jn=j1i1q!j1q!⋯inq!jnq!11-q2i-1∑r=1n-1qi2j1+⋯+irj1+⋯+jr-11-q2jr+1+jr+2+⋯+jn∑k=0j1+⋯+jr-1qk1-q2kj1+⋯+jrk∑h=0ir+1-1qhj1+⋯+jrMq2jn+in+⋯+ir+2+jr+1+ir+1-h-1+k+h+i1+⋯+ir+j1+⋯+jr-kyjn+⋯+jr+1+k+j1+⋯+jr-kxin+⋯+ir+2+ir+1-h-1+h+i1+⋯+ir+1≤Mq2i+jxiyj∑n=1i+jλn∑i1,j1,…,in,jn≠0,0i1+⋯+in=i,j1+⋯+jn=j1i1q!⋯jnq!11-q2i-1∑r=1n-111-q2jr+1+jr+2+⋯+jn∑k=0j1+⋯+jr-1Mqkqk1-q2kj1+⋯+jrk∑h=0ir+1-11≕⋆.In the last inequality we also used |q|<1 and Mq≥1.
In order to estimate (⋆) we observe the following facts:
the inner sum on h equals ir+1 and is therefore bounded above by i≤Mqi;
setting A=Mqq/1-q2, the sum over k is majorized by(A.7)∑k=0j1+⋯+jrAkj1+⋯+jrk=1+Aj1+⋯+jr.
As a consequence we have(A.8)⋆≤Mq3i+2j1-q2i-1xiyj∑n=1i+jλn∑i1,j1,…,in,jn≠0,0i1+⋯+in=i,j1+⋯+jn=j1i1q!⋯jnq!∑r=1n-11+Mqq1-q2j1+⋯+jr·11-q2jr+1+⋯+jn≕2⋆.In order to estimate (2⋆) we observe that |1-q2|<1 (since |q|<1 by assumption), so that the inner sum over r is bounded above by(A.9)∑r=1n1+Mqq1-q2j1+⋯+jn·11-q2j1+⋯+jn=n1-q2+Mqq1-q22j.Here we used the fact that the second sum in (2⋆) specifies that j1+⋯+jn=j. Furthermore, from the first sum in 2⋆ we know that n≤i+j, and the latter is obviously ≤Mqi+j (since Mq≥1). Therefore we get(A.10)2⋆≤1-q2+Mqq1-q22j·Mq4i+3j1-q2ixiyj∑n=1i+jλn∑i1,j1,…,in,jn≠0,0i1+⋯+in=i,j1+⋯+jn=j1i1q!j1q!⋯inq!jnq!≕3⋆.(Again we used |1-q2|<1.) We set(A.11)X=Mq41-q2x,Y=1-q2+Mqq1-q22·Mq3y, so that (3⋆) is equal to(A.12)∑n=1i+jλn∑i1,j1,…,in,jn≠0,0i1+⋯+in=i,j1+⋯+jn=jXi1⋯Xini1q!⋯inq!·Yj1⋯Yjnj1q!⋯jnq!.Now we get to the crucial part of the estimate (see also the analogy to the classical case [1, Section 5.2]). Summing over all the indices i,j≥1 we get(A.13)∑i,j≥1Qi,jx,y≤∑i,j≥1∑n=1i+jλn∑i1,j1,…,in,jn≠0,0i1+⋯+in=i,j1+⋯+jn=jXi1⋯Xini1q!⋯inq!·Yj1⋯Yjnj1q!⋯jnq!≤∑n=1∞λn∑i1,j1,…,in,jn≠0,0Xi1⋯Xini1q!⋯inq!·Yj1⋯Yjnj1q!⋯jnq!=∑n=1∞λn∑i,j≠0,0Xiiq!·Yjjq!n=∑n=1∞λn∑i=0∞Xjiq!·∑j=0∞Yjjq!-1n≕4⋆.To end the proof, we recall that (as is well known, see, e.g., [39]), the complex series expq(z)=∑i=0∞(zi/[i]q!) has a positive radius of convergence (depending on q), say αq∈]0,∞]; for |z|<αq one obviously also has(A.14)Ez≔∑i=0∞ziiq!<∞.The inverse function logq(z) of expq(z) is convergent for z in some neighborhood of 1; as a consequence the series logq(1+z)=∑n=1∞λnzn has a positive radius of convergence, say βq>0. Analogously, for z<βq one has(A.15)Lz≔∑n=1∞λnzn<∞.Summing up, from(A.16)4⋆=LEXEY-1,we infer that ∑i,j≥1Qi,j(x,y)is convergent if the right-hand side of (A.16) is finite, and—in its turn—the latter happens if(A.17)EXEY-1<βq.By continuity (since E(0)E(0)=1), there exists a small εq>0 (with εq<αq) such that EXEY-1<βq whenever X,Y≤εq.
For this choice of εq and by (A.11), we deduce that the series ∑i,j≥1Qi,j(x,y) is finite whenever (A.18)x≤εq1-q21+q4,y≤εq11+q3·1-q221-q2+1+qq.This completes the proof of the absolute convergence of ∑i,jQi,j(x,y). The proof of the normal convergence is completely analogous, by refining (A.17): for instance, it suffices to require that εq satisfy(A.19)EXEY-1<βq2if X,Y≤εq.This ends the proof.
Competing Interests
The authors declare that they have no competing interests.
Acknowledgments
The authors express their gratitude to Rüdiger Achilles for his contribution to some of the topics presented in this paper, published in the recent note [3]. Part of the paper was prepared during the “Senior Fellowship” period of the second-named author at the Institute of Advanced Studies (ISA) of Bologna University (Spring 2014). The second-named author wishes to thank ISA for its hospitality. Part of the paper was prepared during the visit period of the first-named author at the Henri Poincaré Institute (IHP) of Paris, during the Trimester “Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds” (2014). The first-named author wishes to thank IHP for its hospitality.
Endnotes
Here the word basis refers to the space Kqx,y.
In order to get simplifications one also needs to take into account the fact that∗y,y,xqq,yq=y,y,xq,yqq.The q-mutators [y,[y,[y,x]q]q]q, [[[y,x]q,y]q,y]q, and [y,[[y,x]q,y]q]q are linearly independent; it is a striking fact that Q1,3 can be written by means of the last two only.
λn are well posed in R for any q since [n]q=1+q+⋯+qn-1≠0 for any q∈R∖{-1} (and for any q∈C with q<1).
For obtaining (A.5), one can also use (139).
BonfiglioliA.FulciR.2012Heidelberg, GermanySpringerWilcoxR. M.Exponential operators and parameter differentiation in quantum physics1967896298210.1063/1.1705306MR02346892-s2.0-0042638530AchillesR.BonfiglioliA.KatrielJ.The q-deformed Campbell-Baker-Hausdorff-Dynkin theorem201522324510.3934/era.2015.22.32MR3370942KacV.CheungP.2002New York, NY, USASpringerUniversitext10.1007/978-1-4613-0071-7MR1865777ErnstT.2012Basel, SwitzerlandBirkhäuser-SpringerDrinfel'dV. G.Quantum groups198841289891510.1007/bf01247086ArikM.CoonD. D.Hilbert spaces of analytic functions and generalized coherent states197617452452710.1063/1.522937MR04959712-s2.0-36749115321KatrielJ.DuchampG.Ordering relations for q-boson operators, continued fraction techniques and the q-CBH enigma199528247209722510.1088/0305-4470/28/24/018MR13814142-s2.0-36149033832ReinerD. L.A q-analog of the Campbell-Baker-Hausdorff formula198343112512910.1016/0012-365x(83)90030-4MR6803142-s2.0-49049128915BonechiF.CeleghiniE.GiachettiR.PereñaC. M.SoraceE.TarliniM.Exponential mapping for nonsemisimple quantum groups19942741307131510.1088/0305-4470/27/4/023MR12690512-s2.0-21344489322Drinfel'dV. G.On some unsolved problems in quantum group theory19921510Berlin, GermanySpringer18Lecture Notes in MathematicsJørgensenP. E.WernerR. F.Coherent states of the q-canonical commutation relations1994164345547110.1007/bf02101486MR12912402-s2.0-21844521167ReshetikhinN.Quantization of Lie bialgebras1992714315110.1155/S1073792892000163MR1174619ChungW.-S.q-deformed Baker-Campbell-Hausdorff formula19938272569257110.1142/s0217732393002932MR1236185KatrielJ.RasettiM.SolomonA. I.The q-Zassenhaus formula1996371111310.1007/bf00400134MR1392142QuesneC.Disentangling q-exponentials: a general approach200443254555910.1023/b:ijtp.0000028885.42890.f5MR20805482-s2.0-3542996917SridharR.JagannathanR.On the q-analogues of the Zassenhaus formula for disentangling exponential operators20031601-229730510.1016/s0377-0427(03)00633-2Ebrahimi-FardK.ManchonD.A magnus- and Fer-type formula in dendriform algebras20099329531610.1007/s10208-008-9023-3MR24965532-s2.0-64749096348Ebrahimi-FardK.ManchonD.Dendriform equations2009322114053407910.1016/j.jalgebra.2009.06.002MR25561382-s2.0-70349913657Ebrahimi-FardK.ManchonD.Twisted dendriform algebras and the pre-Lie Magnus expansion2011215112615262710.1016/j.jpaa.2011.03.004MR28021512-s2.0-79955975930BerensteinA.Group-like elements in quantum groups, and Feigin's conjecturehttps://arxiv.org/abs/q-alg/9605016BerensteinA.GreensteinJ.Quantum Chevalley groupshttps://arxiv.org/abs/1205.7020v2BridgelandT.Quantum groups via Hall algebras of complexes2013177273975910.4007/annals.2013.177.2.9MR30108112-s2.0-84874807410GreenJ. A.Hall algebras, hereditary algebras and quantum groups1995120236137710.1007/BF01241133MR13290462-s2.0-0001322565RingelC. M.Hall algebras and quantum groups1990101358359110.1007/BF01231516MR10627962-s2.0-0000827429KatrielJ.AbdesselamB.ChakrabartiA.The fundamental invariant of the Hecke algebra Hn(q) characterizes the representations of Hn(q),Sn,SUq(N), and SU(N)19953651395158VilenkinN. Ja.KlimykA. U.1992Dordrecht, The NetherlandsKluwer AcademicWachterH.q-exponentials on quantum spaces200437337938910.1140/epjc/s2004-01999-5MR20997602-s2.0-8744265067BonatsosD.DaskaloyannisC.Model of n coupled generalized deformed oscillators for vibrations of polyatomic molecules19934853611361610.1103/physreva.48.36112-s2.0-0000703393GavrilikA. M.MishchenkoY. A.Deformed bose gas models aimed at taking into account both compositeness of particles and their interaction201358121171117710.15407/ujpe58.12.11712-s2.0-84891295523InomataA.KirchnerS.Bose-Einstein condensation of a quon gas19972315-631131410.1016/s0375-9601(97)00345-9MR1469158SchützenbergerM.-P.Une interprétation de certains solutions de l'equation fonctionnelle: F(x+y)=F(x)F(y)1953236352353CiglerJ.Operatormethoden für q-identitäten19798828710510.1007/bf01319097MR551934KatrielJ.SolomonA. I.A q-analogue of the Campbell-Baker-Hausdorff expansion19912419L1139L114210.1088/0305-4470/24/19/003MR11323202-s2.0-0142161616HearnA. C.Reduce, a portable general-purpose computer algebra systemhttp://reduce-algebra.sourceforge.net/DynkinE. B.Normed Lie algebras and analytic groups195051135186MR0035285QuesneC.Jackson's q-exponential as the exponential of a serieshttp://arxiv.org/abs/math-ph/0305003KatrielJ.SolomonA. I.A no‐go theorem for a Lie‐consistent q‐Campbell-Baker-Hausdorff expansion199435116172617810.1063/1.530736MR12999452-s2.0-36449006714ExtonH.1983Chichester, UKEllis HorwoodMR708496