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Advances in Mathematical Physics
Volume 2016 (2016), Article ID 9598409, 26 pages
Research Article

Generating -Commutator Identities and the -BCH Formula

1Dipartimento di Matematica, Università degli Studi di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, Italy
2Department of Chemistry, Technion-Israel Institute of Technology, 32000 Haifa, Israel

Received 21 June 2016; Accepted 1 August 2016

Academic Editor: Antonio Scarfone

Copyright © 2016 Andrea Bonfiglioli and Jacob Katriel. 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.


Motivated by the physical applications of -calculus and of -deformations, the aim of this paper is twofold. Firstly, we prove the -deformed analogue of the celebrated theorem by Baker, Campbell, and Hausdorff for the product of two exponentials. We deal with the -exponential function , where denotes, as usual, the th -integer. We prove that if and are any noncommuting indeterminates, then , where is a sum of iterated -commutators of and (on the right and on the left, possibly), where the -commutator has always the innermost position. When , this expansion is consistent with the known result by Schützenberger-Cigler: . Our result improves and clarifies some existing results in the literature. Secondly, we provide an algorithmic procedure for obtaining identities between iterated -commutators (of any length) of and . These results can be used to obtain simplified presentation for the summands of the -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 -deformed BCH Formula, and an algorithm for generating -commutator identities), we first review what is known so far as the -analogue (or -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 -deformation goes back to Euler in the mid eighteenth century and to Gauss in the early nineteenth century. If one defines the th -integer as and, accordingly, if the -factorial is defined as (where ), then the -exponential isIt is well known that Jackson’s -derivative, defined by the ratio satisfies (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 -calculus. The advent of quantum groups, some thirty years ago [6], gave rise to vigorous renewed interest in -deformations, although the symmetric -integers introduced in that context, namely, (hence the corresponding -exponential), differ from those defined above.

The earliest introduction of -deformations into physics is probably due to Arik and Coon [7], who studied the -deformed oscillator, whose creation and annihilation operators, and , satisfy . The conventional boson operators satisfy a special case of the BCH identity:which allows transformation from normal to symmetric ordering of the boson operators. The failure to obtain -analogue of this relation was referred to in [8] as an instance of the-Campbell-Baker-Hausdorff enigma.” In connection with this problem, in a very recent note [3] we addressed this “enigma,” and we announced the -deformed BCH Theorem (which we prove here), providing improvements of existing partial results previously given in [8, 9].

The -deformation of the BCH identity has also been considered, from very different points of view, in [1013]; as a related topic to -exponentiation, see [1417] for the -analogue of the Zassenhaus formula; as another related topic, see also the very recent works [1820] for the -analogue of the so-called pre-Lie Magnus expansion. -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 [2325].

One mathematical motivation for the study of -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 requires the specification of the eigenvalues of all the 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 rather than integers. An encyclopedic treatment of -deformed special functions is provided by Vilenkin and Klimyk [27].

In physical applications one can identify two types of motivations for the study of -deformations. On the one hand, -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 -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 -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 -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 -deformation and -exponentiation, we now focus on the -analogue of the BCH Theorem and on the identities between -commutators. Let us denote by the so-called -commutator (also abbreviated as -mutator) of and . If and are any symbols, by an iterated -commutator centered at (see also Definition 2) we mean any arbitrarily long polynomial in of the formwhere may be any of and and where denotes indifferently a left or a right -commutator operator, that is, any of the maps and defined byFor example, is an iterated -commutator centered at , whereas or are not centered at .

In a very recent note [3], we announced the following result: if is as in (1), thenwhere the formal power series can be represented by infinitely many summands, each of which is an iterated -commutator centered at . 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 thatWhereas from (7) it follows that the series satisfying (6) is of the form , where is an infinite sum of polynomials each containing the factor , it does not follow the fact that is a series of iterated -commutators centered at . About this question, which we now answer, we recall that(a)it was posed and (only) partially solved in the 1995 work [8] by the second-named author and Duchamp;(b) in 1983, Reiner [9] showed that is a series of right-nested -commutators of and : by the latter we mean any expression of the form (see also (5))where may be any of or ;(c) a crucial tool in our arguments is provided by the following identity:transforming products into -mutators.According to Reiner’s result recalled in (b), the innermost -commutator may be (and often will be) any of but this expansion does not imply Schützenberger’s result, where only 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 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 -commutators centered at .

An expansion of the series up to fourth order in terms of nested -commutators that depend both on and was obtained by the second-named author and Solomon [34], and it was claimed that the dependence on 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 up to degree in terms of -centered -mutators only:Incidentally, we observe a striking novelty of the -BCH series compared to the classical undeformed BCH series: the latter does not contain summands with three and one or three and one (this is due to the properties of the Bernoulli numbers; see [1]), whereas the -BCH series does. In the formal limit as 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 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 -commutators. Finally, we hope that an understanding of the -BCH Formula will shed light on -Zassenhaus and -Magnus expansions as well.

As an application (see the Appendix), we show that our explicit -mutator expansion is convergent in any Banach algebra (when ); 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 -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 (the algebra of the formal power series in with coefficients in ) to obtain a closed formula for the -BCH series, the proof of the convergence of the latter is obtained only by using the estimate for some constant , and this suggests that our presentation of the -BCH series in terms of -mutators may be of relevance for the study of other contexts, with nontrivial commutation identities. Despite the lack of nontrivial relations in , 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 -commutator-form of the -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 in Theorem 1, one has to take into account the linear dependency relations among the -mutators of the same bidegree in . In this paper we furnish an algorithm to obtain these identities for any bidegree.

Our procedure for generating -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 -commutations), we shall combine the following identities:holding true for any . Clearly, if we repeatedly apply to these identities any choice of (see the notation in (5)), we obtain new identities between -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 -centered -mutators of a fixed bidegree in and .

In order to show the efficiency of our procedure for generating -commutator identities (and the nontriviality of the dependency relations among -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 -centered -mutators of bidegree in (i.e., of degree in and degree in ): denoting the generators by we have the following 15 independent relations among them:

2. Method: The -Deformed BCH Formula

Notation. We fix the algebraic setting we work in: will denote the associative algebra of the formal power series in two noncommuting indeterminates and , with coefficients in , which is the field of the rational functions in the symbol over a field of characteristic . (We recall that whereas usually denotes the ring of the polynomials in the indeterminate , by one identifies the field of the quotients of the ring .) The associative multiplication in is the usual Cauchy product of formal power series. The notation will stand for the associative algebra of the polynomials in and over . From now on, we introduce on the bilinear mapWe say that is the -mutator (shortcut of “-commutator”) of and .

Given , denotes the set of the homogeneous polynomials in with degree with respect to and degree with respect to . We say that any element of has bidegree (with respect to , resp.). We also set, for any , Thus, for example,Obviously, we have the direct sum/product decompositionsThe typical element of is thereforeFinally, we denote by the two-sided ideal in generated by and we setWe can consider the quotient of modulo , denoted as usual by We also use the standard notation for the equivalence modulo : is an associative algebra with the obvious operations. As is a two-sided ideal generated by a homogeneous polynomial of bidegree , then . Notice that, obviously,

Since the lowest order term in the -exponential series (1) is , there exists the inverse map of , say (called the -logarithm), defined on the set , where is the set of the formal power series in whose zero-degree term is null. We use the notationwhere the coefficients are given by the recurrence formula:Thus, the unique series closing the identity (6) isreferred to as the -Baker-Campbell-Hausdorff series, shortly, -BCH series. Therefore, an explicit expression for in terms of polynomials iswhere ’s are as in (26).

Starting from the (tautological) identity , any monomial in (29) can be rewritten, modulo , by moving any on the left and any on the right. Namely, one hasSince, as we prove in Section 4 (starting from (7)), we have with , by means of (30) one can write as a series whose summands (other than ) are polynomials in ; that is, they contain the factor .

Now, by means of the rearranging identity (9), we can write any element of in terms of iterated -mutators of and centered at . An explicit example will clarify this: from (30) we have ; explicitly (by applying four times (9) in the last four equalities) This methodology can be applied to any summand in (29). For example, if we use the bidegree notationwhere has degree with respect to and degree with respect to , we can readily obtain the associative presentation of :By using the cited identity on each summand of (other than ), we getInserting these identities in the expansion of we getObviously, the cancelation of the summand (see the curly braces) is not sheer chance, but it derives from the fact that belongs to . We next apply the technique exemplified above, based on (9), thus obtaining the presentationAnalogously one getsWe explicitly remark that in order to get simplifications for and one also needs to take into account the fact that which is a particular case of an identity which will take a crucial role in the sequel:No more relations intervene among the -mutatorswhich are linearly independent; it is a striking fact, however, that 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 -BCH Formula shows that it is of relevance to study the following issues:(1)to obtain an explicit expression of the -BCH summands in (32) in terms of iterated -mutators centered at ;(2)to obtain identities among the iterated -mutators centered at , allowing simplifying the presentation of ’s and studying bases/dependence-relations in the spaces of the -mutators centered at .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 -BCH series has form (32), where any homogeneous summand is given by the following formula, as a linear combination of iterated -centered -mutators:Here the numbers are the coefficients of the expansion of the -logarithm in (27). Finally, any power of and (with or ) can be further expanded by Newton’s binomial, since and commute for every (as it derives from (39); see also (5) for the meaning of ).

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 -mutators. We fix the definitions used in the sequel (see also the notations for introduced at the beginning of Section 2).

Definition 2. One gives the following three definitions.(i) Fixing , one setsIn other words, and are, respectively, the right and left multiplications in the associative algebra .(ii) Let one use notation (5) for the right/left -adjoint operators and ; given , any -mutator of the form(with and ) will be called an iterated -mutator (of and ) centered at (or -centered). For one’s aims, one shall be interested in -centered -mutators only.(iii) With the above notation, for every one denotes by the subspace of spanned by the -centered -mutators (44) additionally satisfying If or , one sets . Furthermore one setsto denote the formal power series in with summands in the sets ’s.

The letter “” has been chosen to remind us of Schützenberger’s result (7). For example, belongs to , while belongs to . A priori, whereas it is trivial that , it is not at all obvious that , which is stated in the next result.

Lemma 3. With the notation in Definition 2, one has

The proof of this result is contained in Proposition 14.

3.1. Some Dimensions

We next take into account the space of the -centered polynomials of bidegree . For example, we have (all spans are understood over the field )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))and it can be proved that no other dependency relations hold among the generators of or the generators of , so that .

The problem of determining the dimension of is rather simple (see Proposition 4), whereas the problem of discovering the dependency relations among the generators of a given 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 : one can prove that and that the dependency relations among the generators of are the following three: 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

In the next result it is understood that the field underlying all vector space structures is . Along with other dimensional facts, we aim to count the following set of generators of : these are the iterated -mutators of the formwhere all belong to the set of maps in such a way that appears exactly times and appears exactly times (if or it is understood that these maps are not counted).

Proposition 4 (dimensions). Let . Let the vector space be as in Definition 2. Then one has the following:(i) ;(ii) ;(iii) the number of possible -centered -mutators writable as in (52) defining is(iv) the number of the linearly independent dependency relations among the list of the -mutators in part (iii) above is

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 -mutators may be equal. In other words, we count the -tuples , where belong to in such a way that appears exactly times and appears exactly times (if or it is understood that these maps are not counted).

Proof. We split the proof into four steps.(i) follows from the cardinality of the set (ii)We claim that is one unit less than . To this aim, we recall that in Section 2 we introduced on the quotient modulo , the two-sided ideal generated by . Due to homogeneity and degree reasons, we can also consider this quotient on each separately and we can infer that is isomorphic to . We therefore get This last identity follows by (30). From Lemma 3 we know that , so that . Since is a vector subspace of , this gives the claimed .(iii)If the only object in (52) is and (53) is correct, providing . If or , we need to count the -tuples , where , and , , respectively, appear and times. These -tuples are in bijective relation with the formal monomialswhere and , via the identification Now, the number of the monomials in (56) is precisely , and for any such a monomial we can choose ’s in different ways and ’s in different ways; this proves (53).(iv)This follows from (ii) and (iii).

Remark 5. Proposition 4 provides us with a very simple basis for (which is not, however, constituted of iterated -mutators of and as in (52)). Indeed, from (30) we know thatwhenever and . Taking into account that , it is then very easy to construct a basis for by means of this procedure.

An example will clarify this. is spanned by the monomialsWe apply the procedure in (58) to all of these monomials except for the first:Due to (58) these polynomials all belong to and they are (clearly) linearly independent (as they are obtained from linearly independent vectors by subtracting multiples of a given vector); since by Proposition 4-(ii), they form a basis for . This also means that each of them can be written as a sum of -centered -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 -mutator identities. For later reference, we give for each formula/procedure a one-letter name. In the sequel,denotes the commutator of two operators , with respect to the composition of maps (whenever this makes sense).

Here is the list of our procedures for obtaining -mutator identities:(T) We say that identity (9) is the Transformation Rule; it allows transforming polynomials (under their associative presentation) into a linear combination of iterated -mutators. With the notation in Definition 2, (9) can be rewritten asholding true for every and .(R) The following identity (see also (39)) is implicitly contained in the work [9] by Reiner; we call it Reiner’s identity:With the formalism in Definition 2, it can be rewritten as the commuting relationwhich is also equivalent to(A) We introduce the following identity involving three letters , , :