Generating 𝑞 -Commutator Identities and the 𝑞 -BCH Formula

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 exp 𝑞 (𝑥) = ∑ ∞𝑛=0 (𝑥 𝑛 /[𝑛] 𝑞 !) , where [𝑛] 𝑞 = 1 + 𝑞 + ⋅⋅ ⋅ + 𝑞 𝑛−1 denotes, as usual, the 𝑛 th 𝑞 -integer. We prove that if 𝑥 and 𝑦 are any noncommuting indeterminates, then exp 𝑞 (𝑥) exp 𝑞 (𝑦) = exp 𝑞 (𝑥 + 𝑦 + ∑ ∞𝑛=2 𝑄 𝑛 (𝑥,𝑦)) , where 𝑄 𝑛 (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 ﬂ 𝑦𝑥 − 𝑞𝑥𝑦 has always the innermost position. When [𝑦,𝑥] 𝑞 = 0 , this expansion is consistent with the known result by Sch¨utzenberger-Cigler: exp 𝑞 (𝑥) exp 𝑞 (𝑦) = exp 𝑞 (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.


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 theanalogue (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 [ ] = 1 + + 2 + ⋅ ⋅ ⋅ + −1 and, accordingly, if the -factorial is defined as (1) It is well known that Jackson's -derivative, defined by the ratio satisfies exp ( ) = exp ( ) (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 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 theanalogue of the BCH Theorem and on the identities between -commutators. Let us denote by [ , ] fl − the so-called -commutator (also abbreviated as -mutator) of and . If and are any symbols, by an iteratedcommutator centered at [ , ] (see also Definition 2) we mean any arbitrarily long polynomial in , of the form where 1 , . . . , may be any of and and where denotes indifferently a left or a right -commutator operator, that is, any of the maps ad and ad * defined by For 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 exp is as in (1), then exp ( ) exp ( ) = exp ( ( , )) , where 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).
Whereas 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 0 , 1 , . . . , may be any of or ; Advances in Mathematical Physics 3 (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 innermostcommutator may be (and often will be) any of [ , ] , [ , ] , [ , ] , [ , ] , 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 4 in terms of [ , ] -centered -mutators only: ( , ) = + − 1 [2] [ , ] − [3] 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 → 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 → 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 -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 | | < 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 -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 K( )( , ) (the algebra of the formal power series in , with coefficients in K( )) 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 > 0, 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 K( )( , ), 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.

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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: (see the notation in (5)), we obtain new identities betweencommutators. 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 [ , ] -centeredmutators of bidegree (2,3) in , (i.e., of degree 2 in and degree 3 in ): denoting the generators by we have the following 15 independent relations among them:

Method: The -Deformed BCH Formula
Notation. We fix the algebraic setting we work in: = K( )( , ) will denote the associative algebra of the formal power series in two noncommuting indeterminates and , with coefficients in K( ), which is the field of the rational functions in the symbol over a field K of characteristic 0. (We recall that whereas K[ ] usually denotes the ring of the polynomials in the indeterminate , by K( ) one identifies the field of the quotients of the ring K[ ].) The associative multiplication in is the usual Cauchy product of formal power series. The notation K( )⟨ , ⟩ will stand for the associative algebra of the polynomials in and over K( ). From now on, we introduce on the bilinear map We say that [ , ] is the -mutator (shortcut of "commutator") of and . Given , ∈ N ∪ {0}, , 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 ∈ N ∪ {0}, Thus, for example, Obviously, we have the direct sum/product decompositions The typical element of is therefore Finally, we denote by the two-sided ideal in generated by [ , ] and we set , fl ∩ , .
We 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 (1, 1), then = ∏ , ≥0 , . Notice that, obviously, Since the lowest order term in the -exponential series (1) is 1, there exists the inverse map of exp , say log (called the -logarithm), defined on the set 1 + + , where + is the set of the formal power series in , whose zero-degree term is null. We use the notation where the coefficients ∈ K( ) are given by the recurrence formula: Thus, the unique series ( , ) closing the identity (6) is referred to as the -Baker-Campbell-Hausdorff series, shortly, -BCH series. Therefore, an explicit expression for ( , ) in terms of polynomials is where 's are as in (26).

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Starting from the (tautological) identity = + [ , ] , any monomial 1 1 ⋅ ⋅ ⋅ in (29) can be rewritten, modulo , by moving any on the left and any on the right. Namely, one has Since, 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 3 = 2 4 + {element of }; explicitly (by applying four times (9) in the last four equalities) Advances in Mathematical Physics 7 This methodology can be applied to any summand in (29).
For example, if we use the bidegree notation where , ( , ) has degree with respect to and degree with respect to , we can readily obtain the associative presentation of 1,3 ( , ): By using the cited identity = + [ , ] on each summand of 1,3 ( , ) (other than 3 ), we get Inserting these identities in the expansion of 1,3 ( , ) we get Obviously, the cancelation of the summand 3 (see the curly braces) is not sheer chance, but it derives from the fact that 1,3 belongs to . We next apply the technique exemplified above, based on (9), thus obtaining the presentation We explicitly remark that in order to get simplifications for 1,3 ( , ) and 3,1 ( , ) 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: for every , .
No more relations intervene among the -mutators which are linearly independent; it is a striking fact, however, that 1,3 ( , ) 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/dependencerelations 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 ad + ad * and ad * + ad (with = or ) can be further expanded by Newton's binomial, since ad and ad * commute for every (as it derives from (39); see also (5) for the meaning of ad , ad * ).

The Identity-Generating Technique
In this section we provide one of our two main results: an algorithm for the generation of identities between iteratedmutators. 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 sets In other words, and are, respectively, the right and left multiplications in the associative algebra .
(iii) With the above notation, for every , ∈ N one denotes by , the subspace of , spanned by the [ , ] -centered -mutators (44) additionally satisfying to 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, 3  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. 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 3,1 or the generators of 1,3 , so that dim 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 + = 4: one can prove that dim( 2,2 ) = 5 and that the dependency relations among the 8 generators of 2,2 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 K( ). Along with other dimensional facts, we aim to count the following set of generators of , : these are the iterated -mutators of the form where 1 , . . . , + −2 all belong to the set of maps {ad * , ad * , ad , ad } in such a way that appears exactly − 1 times and appears exactly − 1 times (if = 1 or = 1 it is understood that these maps are not counted).

Proposition 4 (dimensions).
Let , ∈ N. Let the vector space , ⊆ , be as in Definition 2. Then one has the following: (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 ( + −2)-tuples ( 1 , 2 , . . . , + −2 ), where 1 , . . . , + −2 belong to {ad * , ad * , ad , ad } in such a way that appears exactly − 1 times and appears exactly − 1 times (if = 1 or = 1 it is understood that these maps are not counted).
Proof. We split the proof into four steps.
Remark 5. Proposition 4 provides us with a very simple basis for , (which is not, however, constituted of iteratedmutators of and as in (52)). Indeed, from (30) we know that whenever 1 + ⋅ ⋅ ⋅ + = and 1 + ⋅ ⋅ ⋅ + = . Taking into account that dim( , ) = dim( , ) − 1, it is then very easy to construct a basis for , by means of this procedure.
An example will clarify this. 2,4 is spanned by the 15 monomials We apply the procedure in (58) to all of these monomials except for the first: Due to (58) these 14 polynomials all belong to 2,4 and they are (clearly) linearly independent (as they are obtained from linearly independent vectors by subtracting multiples of a given vector); since dim( 2,4 ) = 14 by Proposition 4-(ii), they form a basis for 2,4 . 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.

Producing General Identities:
The Basic Maps. We are ready to provide a general technique which producesmutator 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 as holding 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 relation which is also equivalent to It can be written as for every , , or alternatively as a relation involving the ∘-commutators of left and right -mutator operators: Identity (67) can be proved starting from identity (64) by the substitution of with + (and then by two cancelations, using (64)). We note that (66) is symmetric with respect to an interchange of with . Finally, when = , (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.
It can be written as This gives an alternative way of writing the ∘commutator of left and right -mutator operators by means of the ∘-commutators of two right and two left -mutator operators: The proof of (69) follows by applying twice the Transformation Rule (T) to , by writing the latter alternatively as ( ⋅ ) ⋅ and ⋅ ( ⋅ ) and then using identity (A). If we interchange and in (70), the right-hand side changes sign; it then easily follows that (70) implies (67), whence (B) implies (A). Furthermore, if = , identity (69) reduces to (63).
(C) Let be any monomial of the form 1 1 ⋅ ⋅ ⋅ . We consider the tautological identity We repeatedly apply the Transformation Rule (T) to both of its sides, in the following way: we write the left-hand side as and we apply (T) from left to right (to both summands), without breaking into its summands − , so that will always appear in the innermost position of a sum of iterated -mutators of and (ultimately producing a linear combination of [ , ]centered polynomials); we do the same on the righthand side, starting from right to left, in order to preserve again in innermost positions. See Example 6 for an example of this technique.
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.
Example 6. We give an example for the procedure (C), when = . We have The left-hand side is ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ . For the first summand we have Analogously, the second summand is The right-hand side is ⋅ ⋅ ⋅ − ⋅ ⋅ ⋅ . For the first summand we have Analogously, the second summand is Putting the pieces together, we obtain an identity for nested -mutators in 3,2 .

Producing General Identities: Counting the Identities.
Finally, we describe how to obtain identities in each space , (see Definition 2). Let , ∈ N be fixed. According to Proposition 4, we know that the dimension of , is ( , ) = ( + ) − 1, while the total number of the formal [ , ]centered -mutators spanning , is ( , ) = ( + −2 −1 ) ⋅ 2 + −2 . We show, inductively, how to construct ( , ) − ( , ) identities among the generators of , 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.
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 ( , ). In the sequel we can suppose that + > 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 + .
(iii) Number of Relations from Identity (A). As we already remarked, identity (66) boils down to (R) when = ; hence we can take ̸ = . Also, we do not get new information if and are interchanged; thus we can always choose = and = , so that (A) can be applied only when ≥ 2 and ≥ 2 (since must contain [ , ] ). In place of we can then take any member of a basis of −1, −1 . Summing up, taking into account the formula for ( − 1, − 1), the expected number of relations from identity (A) is Advances in Mathematical Physics

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(v) Number of Relations from Identity (C). With the notation in the description of procedure (C), since and appear at least twice (remember that = [ , ] ), procedure (C) is inapplicable if = 1 or = 1. Our conjecture states that, denoting by C( , ) the number of relations deriving from (C), there suffices one and only one such relation, when ≥ 2 and ≥ 2; thus we define 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.

The -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 -BCH Formula precise, we need to endow with a metric structure. Indeed, as in [1, Theorem 2.58, p. 94], can be equipped with a metric space structure by the distance where md(0) fl ∞ and if = ∑ ̸ = 0 (see the notation in (21)), we set with the convention exp(−∞) = 0. The metric space ( , ) is complete and it is an isometric completion of K( )⟨ , ⟩ (as a metric subspace); moreover it is ultrametric; that is, In the sequel we shall tacitly use the well-behaved properties of the topology of ( , ) 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 ( , ), since md( ) ≥ → ∞ as → ∞ (because ∈ ).
Since, for any ∈ + , one has md( ) → ∞ as → ∞, Remark 8 ensures that the following maps are well posed, as convergent series in the metric space : We have the following results, whose simple proofs are omitted.
(i) Each of the maps exp and exp in (93) admits an inverse function, which we, respectively, denote by log and log , from 1 + + to + . We say that log is the -logarithm.
(ii) There exists a map : It is known that has the explicit expansion (see [16,37]) Then we infer that in the associative algebra there exists one and only one formal power series ( , ) such that (6) is valid, and this is defined as in (28). The series ( , ) is referred to as the -Baker-Campbell-Hausdorff series (shortly, the -BCH series), and it will be also denoted by ⬦ . Its associative presentation is (29). Grouping together the summands of the same degree, we use the notation 16

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Remark 9 (intertwining of the deformed/undeformed BCH series). By using the map in (95) we can obtain a representation for ⬦ deriving from the following argument: exp ( ⬦ ) (6) = exp ( ) exp ( ) Since exp is injective we get the identity intertwining the undeformed and the -deformed BCH series. Due to explicit form (95) of , identity (99) can be used for computational issues to derive explicit summands of ⬦ 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 ( , ) in (96). We next claim that, apart from 1 ( , ) = + , any ( , ) (with ≥ 2) is a sum of polynomials containing − as a factor. This fact can be seen as a consequence of Schützenberger's result (7) [32]; see also Cigler [33]), where Indeed, one can easily prove (103) by induction on , using (101) and the well-known -Pascal rule (see, e.g., [4]) 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 -commuting variables (see (7)).
For example (see also [38]), one has 1,1 ( , ) = − 1 [2] [ , ] , Proof. We first equip / with the structures of a topological algebra and of a complete ultrametric space, by imitating the corresponding structures on . Then one can define a -exponential on / as well, denoted byẽxp . Since the projection [⋅] is continuous morphism of the underlying algebras one obtains On the other hand, (103) gives The injectivity ofẽxp implies that [ ⬦ ] = [ ] + [ ] or equivalently ⬦ − ( + ) ∈ , which immediately proves the theorem.
Next we describe another feature of the -BCH series.
Definition 11 (nested -mutators). One says that any element of the form with ≥ 2 and 1 , 2 , . . . , ∈ { , }, is a right-nestedmutator of and of length . Analogously, one says that any element of the form is a left-nested -mutator of and of length . For = 1, one qualifies and as the right-nested (and the left-nested) -mutators of length 1.
We use the following result.
Then R , is a basis of , as well.
Clearly one can obtain an analogous result with rightnested -mutators. For brevity, we shall refer to R , as the left-nested Reiner basis of , (the choice of the notation "R" refers to "Reiner"). Since we have an associative presentation of ⬦ , we immediately get, from Reiner's Theorem 12, the following result. Corollary 13. With the notation in Lemma 10, any , ( , ) can be expressed in a unique way as a linear combination of elements of the left-nested Reiner basis R , of , . Therefore, the -BCH series ⬦ admits a presentation as a series of leftnested -mutators of and as in (112) (with coefficients in K( )).

An analogous result holds for right-nested -mutators.
The disadvantage of the above nested presentation of the -BCH series, based on Reiner's Theorem 12, is that it (necessarily) allows for summands of the form 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 Compared to our expression (107) for 2,1 , we get general identity (39) (implicitly contained in [9]). Our main task is to compound Lemma 10 and Corollary 13 and prove that ⬦ admits a presentation with -mutators (not necessarily nested) where the innermost -mutator is [ , ] , consistently with the mentioned Schützenberger's result. This Proof. We prove that , = , for every , ∈ N ∪ {0}. Since , ⊆ , , we are left to prove the reverse inclusion. From (25) (and the definition of , when or vanishes) we have , = {0} = , whenever or is 0. We can therefore suppose that , ≥ 1. If = = 1 we trivially have 1,1 = span{[ , ] } = 1,1 . We can thus suppose that + ≥ 3. Any element of , is, by definition, a linear combination (with coefficients in K( )) of where 1 , 2 , . . . , + −2 all belong to the set of maps { , , , } (see Definition 2), in such a way that appears exactly − 1 times and appears exactly − 1 times. By (9), any of the maps 1 , 2 , . . . , + −2 in (115) is a linear combination of suitable maps belonging to {ad * , ad * , ad , ad } (preserving the total number of 's and 's). Therefore (115) is a [ , ] -centered polynomial in , . This shows that , ⊆ , . In particular, since , ( , ) ∈ , (see Lemma 10-(3)), we get , ( , ) ∈ , .

A Criterion for [ , ] -Centered -Mutators
Due to our interest in [ , ] -centered -mutators, we introduce a criterion for characterizing the elements of , (or equivalently, of , ).

Any nonvanishing monomial in
can be written in a unique way as a scalar multiple of the following basis 1 monomials (we agree that 0 = 0 = 1 K , the identity of K): We denote by any of the above monomials. In the sequel, we also agree that any monomial in , has been written in the above unique way.
Definition 15. Let M = { } denote the collection of the monomials in (116a)-(116b). We set By an abuse of notation, we agree that the map is also defined on the multi-indices appearing in (116a)-(116b), so that we also write ( 1 , 1 ) = 0 if the indexes are as in (116a), and if the indexes are as in (116b).
Starting from (101), which can be rewritten as ≡ , by an inductive argument one gets (30); namely, 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 [ , ] -centrality).
With the notation in Definition 15, we consider the unique (continuous) K( )-linear map : → defined on monomials as follows: Then, given ∈ , one has ∈ (or, equivalently, ∈ ) if and only if ( ) = . Moreover, is valued in = so that is a projection onto . By using the abused notation following Definition 15, a homogeneous polynomial belonging to , , say Note that the latter is simply an identity in K( ).
Example 17. We consider the polynomial in 2,2 defined by The associated scalar as in (122) is Since this is evidently null (as one can check upon expansion), we can infer that ∈ . Actually one can verify that is equal to [[ , [ , ] ] , ] ∈ 2,2 .
Proof of Lemma 16. We split the proof into five steps.
(III) Conversely, suppose that ∈ ; we need to show that ( ) = , or equivalently we have to prove that | is the identity on . To this aim, it suffices to show that is the identity on any , . To this end, let ∈ , ; like any polynomial, can be uniquely written in the basis (116a)-(116b) as where F is a finite family of basis monomials in M and ( ) ∈ K( ) for any ∈ F . Then, by recalling that the (nonzero) monomials which span , have bidegree ( , ), we infer Moving terms around we get (∑ ∈F ( ) ( ) ) = ( ) − . Now, the right-hand term belongs to since is -valued (see part (I) of the proof) and since ∈ by assumption; so the same is true of the left-hand side, but a scalar multiple of can belong to iff the scalar factor is null. Hence, from (127) we get ( ) = .
(IV) The surjectivity of is a trivial consequence of part (III).
(V) We have to prove the last assertion of the theorem. On the one hand, let ∈ (this part of the proof does not require ∈ , ). By part (III) of the proof we know that = ( ); hence, if we write in the basis monomials as as needed (this gives precisely (122) when ∈ , ). Conversely, let ∈ , and suppose that after we have written as it is known that (122) holds true. In the preceding computations we proved that If ∈ , this is just By assumed (122), the term in the parenthesis is null, whence ( ) = . By part (II) of the proof we therefore get ∈ .
However, it can be easily checked that the definition of is unambiguous for any monomial and it leads to the same result; that is, with the convention (which we tacitly assume in the sequel) that the sum is 0 if = 1. Accordingly, the map is well posed for every monomial: The next section provides a closed formula for the terms , in the -BCH series, only depending on the coefficients of the -logarithm. The main tool is Lemma 16.

An Explicit Formula for the -BCH Series
We already showed the basic associative presentation of ⬦ in (29), where the coefficients (from the expansion of log ) are used: they can be derived, for example, by the recursion formula (27). In this section we provide an explicit formula for , in terms of iterated -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 -BCH series with an example: this is given in Section 7. For obtaining an explicit formula for , in terms of iteratedmutators, we first need some lemmas whose proofs (mainly, some inductive arguments) are omitted.

Lemma 19. For any ≥ 1 one has
where ( ) denotes the classical (undeformed) binomial coefficient, and the notation ad denotes the left -commutation map in (42).
Note that the sum over in the right-hand side of (137) is an element of , the bilateral ideal generated by [ , ] . A direct application of formula (62) proves the following result, starting from (137).

(138)
Remark 21. Formula (138) could be written in an even more explicit form: indeed, the operators ad * and ad commute, for every ∈ , as identity (39) proves. Hence one can apply Newton's binomial formula to obtain Moreover, since any right multiplication commutes with any left multiplication (by associativity), we infer that any map ad * + ad commutes with any map ad We now obtain a formula, generalizing the above lemma, which also expresses in a "quantitative" way the congruence ≡ (mod ).

Lemma 22.
For any , ≥ 1 one has As stated in Remark 21, this formula can be made even more explicit by unraveling the powers of ad * + ad , ad + ad * , and ad * + ad by means of (139).
Note that the right-hand side of identity (141) is a linear combination of iterated -mutators, centered at [ , ] , since − > 0. In other words it is an element of = . Our final prerequisite is to find an explicit form (in terms of iterated [ , ] -centered -mutators) for the projection defined in Lemma 16, when it acts on a generic monomial . This is given in the next result.
Again (see Remark 21) the above formula can be made more explicit (although more cumbersome) by unraveling the powers of ad * + ad and of ad + ad * (with = , ), by means of (139). We are ready for the proof of Theorem 1.

An Example of the Rearrangement Technique
We know from (101) that 3 is congruent to 2 4 modulo . By means of a repeated application of the trivial identity we can write 3 − 2 4 as an element of (i.e., as a sum of polynomials factorizing [ , ] ); subsequently, we can apply basic procedure (9) to write it as an element of (i.e., as a linear combination of [ , ] -centered -mutators). This is done in the next computation: 22

Advances in Mathematical Physics
In Lemma 23 we provided a formula for the above procedure for any monomial This procedure allows us to write any , as a [ , ]centered -mutator. For example, a direct computation based on (143) and (27) gives out the associative presentation of 1,3 : We then use the identity = [ , ] + on each summand of 1,3 (other than 3 ) to eventually produce, modulo , the monomial 3 . For instance, (These formulas can be improved by shifting all the factors on the right as in Lemma 19.) Inserting these identities in the expansion of 1,3 we get Obviously, the cancelation of the summand 3 is not sheer chance, but it derives from the fact that 1,3 belongs to . We next apply the technique exemplified above, based on (9), thus obtaining the presentation 2  for any , ∈ and any ∈ N, where ad and ad * have the obvious meanings. We let > 0 be small (it will be conveniently chosen in due course) and we take any , ∈ such that ‖ ‖, ‖ ‖ < . Then, by the triangle inequality and a repeated application of (A. 4 In the last inequality we also used | | < 1 and ≥ 1. In order to estimate (⋆) we observe the following facts: (i) the inner sum on ℎ equals +1 and is therefore bounded above by ≤ ; (ii) setting = | |/|1− 2 |, the sum over is majorized by In order to estimate (2⋆) we observe that |1 − 2 | < 1 (since | | < 1 by assumption), so that the inner sum over is bounded above by Here we used the fact that the second sum in (2⋆) specifies that 1 + ⋅ ⋅ ⋅ + = . Furthermore, from the first sum in (2⋆) we know that ≤ + , and the latter is obviously ≤ + (since ≥ 1). Therefore we get To end the proof, we recall that (as is well known, see, e.g., [39]), the complex series exp ( ) = ∑ ∞ =0 ( /[ ] !) has a positive radius of convergence (depending on ), say ∈ ]0, ∞]; for | | < one obviously also has This ends the proof. 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 secondnamed author wishes to thank ISA for its hospitality. Part of the paper was prepared during the visit period of the firstnamed 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.