Multi-operator brackets acting thrice

Thomas Curtright, Xiang Jin and Luca Mezincescu

Department of Physics, University of Miami, Coral Gables, FL 33124-8046, USA

E-mail: curtright@miami.edu

Received 9 September 2009, in final form 24 September 2009
Published 22 October 2009

1. Introduction

Nambu introduced a multilinear operator bracket in the context of a novel formulation of mechanics [13]. His N-bracket is defined by

where the sum is over all N! permutations of the operators. For example, [ABC]  =  ABC – ACB  +  BCA – BAC  +  CAB – CBA. The operator product is assumed to be associative. To avoid ambiguities when some of the entries within a bracket are themselves products, commas are often used to separate the entries. Parentheses also suffice in such cases. For example, [AD, B, C]  =  [(AD)BC]  =  ADBC – ADCB  +  BCAD – BADC  +  CADB – CBAD.

The same construction independently appeared in the mathematical literature [10, 11]. The theory of such multi-operator products, as well as their `classical limits' in terms of multivariable Jacobians, has been studied extensively [1–4, 6–9, 12, 14–18].

From an algebraic point of view, it is natural to seek the analogue of the Jacobi identity for N-brackets. For the case of even N-brackets, the obvious generalization where one N-bracket acts on another leads to a true identity if all entries are totally antisymmetrized (see (5) below). But for odd N-brackets this procedure does not work [1, 4]—the total antisymmetrization over all entries of one odd N-bracket acting on another does not vanish, but rather yields a higher order (2N – 1)-bracket.

Nevertheless, an interesting generalization of the Jacobi identity was discovered by Bremner for 3-brackets acting three times [1]. He showed

where A is fixed, but it is implicitly understood that lowercase entries are totally antisymmetrized by summing over all 6! signed permutations of them. The point of this short communication is to show that Bremner's identity generalizes to all odd N-brackets.

Before discussing the general case, we anticipate and indicate a proof for the case of 3-brackets. The Bremner identity can be proved through a resolution of both left- and right-hand sides as a series of canonically ordered words. By direct calculation we find

where all lowercase entries are implicitly totally antisymmetrized. Precisely the same expansion holds for [[Abc][def]g], again by direct calculation. Hence, the identity is established.

That is to say, both [[A[bcd]e]fg] and [[Abc][def]g] can be rendered as a 7-bracket plus another 3-bracket containing 3-brackets, when antisymmetrized over lowercase entries:

Thus, the Bremner identity amounts to the combinatorial statement, as written, that there are two distinct ways to present a 7-bracket in terms of nested 3-brackets.

2. Results for any N

As known, and previously mentioned, even brackets need only act twice to yield an identity. Namely [4, 9]

Total antisymmetrization of all the Bs is understood^Note1. When N  =  2 this is the familiar Jacobi identity. The proof is by direct calculation and follows as a consequence of associativity.

However, for odd N, [B₁ · · · B_N–1[B_N · · · B_2N–1]] ≠ 0, but instead produces the (2N – 1)-bracket [B₁ · · · B_2N–1] upon total antisymmetrization [3, 4]. Apparently, the simplest identity obeyed by odd brackets of only one type, that does not introduce higher order brackets, requires that they act at least three times. For any odd N  =  2L  +  1, a valid relation is the immediate generalization of that found by Bremner for the case of 3-brackets. To show this, we present two easily established lemmata, followed by our main theorem and its proof. Firstly,

Lemma 1. 

Total antisymmetrization of the Bs is understood. Here we have also used the convention that an empty product equals 1. Explicitly, B₁ · · · B₀  =  1  =  B_J + 1 · · · B_J, so that the first and last terms in the sum are and , respectively. It is a simple exercise to use this first lemma to prove (5). Similarly,

Lemma 2. 

Finally, it is rather tedious but fairly straightforward to use both lemmata to prove the following.

Theorem 1. For associative products, with implicit total antisymmetrization of the Bs,

Proof. The result (8) follows from resolving the left- and right-hand sides into sums of canonically ordered words, as illustrated above for the case of 3-brackets. We have

All the coefficients m^(1,2)_n in these two resolutions are manifestly positive integers. The theorem is established by showing that m⁽¹⁾_n  =  m⁽²⁾_n for all n.

By direct calculation, through the use of the two lemmata, we find

The determination of the m_ns is just a matter of enumerating the ways to obtain a particular intercalation of A among the Bs.

Consider in more detail some of the calculations involved. As a first step, with the implicit antisymmetrization^Note2, the internal brackets [B₁ · · · B_2L + 1] or [B_2L + 1 · · · B_4L + 1] may be supplanted by products: [B₁ · · · B_2L + 1]  =  (2L  +  1)!(B₁ · · · B_2L + 1) or [B_2L + 1 · · · B_4L + 1]  =  (2L  +  1)!(B_2L + 1 · · · B_4L + 1). Then we may write, on the one hand,

In this expression, we may now rename indices, bearing in mind the antisymmetrization:

Next, we apply lemma 2 for J  =  2L – 1, and identify [AB_2L · · · B_4L–1] with , and (B_4L · · · B_6L) with :

To continue, consider first the coefficients m⁽¹⁾_n where n ≤ 2L.

For the determination of m⁽¹⁾_n≤2L, since consists of (2L  +  1) Bs, it must be placed to the right of in the application of lemma 2. Otherwise there would be too many Bs to the left of A. Thus, for m⁽¹⁾_n≤2L we need keep only the first line in the last relation, (14a). To place a total of n Bs to the left of the A contained in , with k Bs already to the left as in (14a), we then need only the terms in with an additional (n – k) Bs to the left of A. That is to say, from lemma 1, with J  =  2L and all B indices shifted up by 2L – 1,

and from this we need only the term with l  =  n – k. The net result for m⁽¹⁾_n≤2L is

On the other hand, with similar steps, we have

We again apply lemma 2 for J  =  2L – 1, but to [AB₁ · · · B_2L–1(B_2L · · · B_4L)], so now we identify A with , and (B_2L · · · B_4L) with . As before, consider first only m⁽²⁾_n coefficients where n ≤ 2L. For the determination of m⁽²⁾_n≤2L, must once again be placed to the right of , so we need keep only the line (14a). We pick up an additional (n – k) Bs by applying again lemma 1, only this time to the remaining outside bracket in (18). The net result for m⁽²⁾_n≤2L is

with exactly the same expression for c_n≤2L as before, (17). Thus, we have shown m⁽¹⁾_n≤2L  =  m⁽²⁾_n≤2L.

Next, consider the coefficients where 2L  +  1 ≤ n ≤ 3L. There are still contributions to either m⁽¹⁾_n or m⁽²⁾_n from the line (14a), as above, of the form (2L  +  1)!(2L)!(2L – 1)! × c_n, and these contributions to either m⁽¹⁾_n or m⁽²⁾_n still turn out to be the same. But in this case the sums contributing to c_n give

Moreover, from applying lemma 2, there are now contributions to either m⁽¹⁾_n or m⁽²⁾_n from the second line, (14b), where the respective s are placed to the left of the s. Following steps similar to those above, it is not difficult to see that these other terms contribute the same amount to either m⁽¹⁾_n or m⁽²⁾_n, for 2L  +  1 ≤ n ≤ 3L. Namely, (2L  +  1)!(2L)!(2L – 1)!×

Thus, the net result is m⁽¹⁾_{2L + 1≤n≤3L}  =  m⁽²⁾_{2L + 1≤n≤3L}  =  (2L  +  1)!(2L)!(2L – 1)! × c_{2L + 1≤n≤3L} with

Finally, consider the coefficients for 3L  +  1 ≤ n ≤ 6L. These are given by an elementary reflection symmetry: m⁽¹⁾_n  =  m⁽¹⁾_6L–n and m⁽²⁾_n  =  m⁽²⁾_6L–n. Thus, m⁽¹⁾_n  =  m⁽²⁾_n  =  (2L  +  1)!(2L)!(2L – 1)! × c_6L–n for 3L  +  1 ≤ n ≤ 6L.        

As a check, the coefficients must sum to give the number of generic terms that appear in three nested (2L  +  1)-brackets (i.e. in either [[[ · · · ] · · · ] · · · ] or [[ · · · ] · · · [ · · · ]]). That is ∑^6L_n = 0 m_n  =  ((2L  +  1)!)³. Equivalently,

This condition is indeed satisfied by the c_n given in (11).

3. Conclusion

Perhaps N-brackets and algebras have an important role to play in physics, as originally suggested by Nambu. Recently there has been considerable interest in N-brackets, especially 3-brackets, as expressed in the physics literature (see [2] and references therein). These ideas await further development.

Acknowledgments

We thank David Fairlie and Cosmas Zachos for sharing their thoughts about Nambu brackets. We also thank the referee for comments which significantly improved the exposition of this communication. One of us (TC) further thanks the Lago Mar Resort for providing the beautiful and stimulating surroundings where portions of this work were completed. This work was supported by NSF Awards 0555603 and 0855386.

[1] 

Bremner M R 1998 Identities for the ternary commutator J. Algebra 206 615–23 
CrossRef
Bremner M R and Hentzel I 2000 Identities for generalized Lie and Jordan products on totally associative triple systems J. Algebra 231 387–405 
CrossRef
Bremner M R and Peresi L A 2006 Ternary analogues of Lie and Malcev algebras Linear Algebra Appl. 414 1–18 
CrossRef
Nuyts J 2008  (private communication)

[2] 

Curtright T, Fairlie D, Jin X, Mezincescu L and Zachos C 2009 Classical and quantal ternary algebras Phys. Lett. B 675 387–92 (arXiv:0903.4889 [hep-th]) 
CrossRefPreprint

[3] 

Curtright T L and Zachos C K 2003 Classical and quantum Nambu mechanics Phys. Rev. D 68 085001 (arXiv:hep-th/0212267) 
CrossRefPreprint

[4] 

de Azcárraga J A and Pérez Bueno J C 1997 Higher-order simple Lie algebras Commun. Math. Phys. 184 669–81 (arXiv:hep-th/9605213) 
CrossRefPreprint

[5] 

Devchand C, Fairlie D, Nuyts J and Weingart G 2009 Ternutator Identities  arXiv:0908.1738v2 [hep-th]
Preprint

[6] 

Dito G and Flato M 1997 Generalized Abelian deformations: application to Nambu mechanics Lett. Math. Phys. 39 107–25 (arXiv:hep-th/9609114) 
CrossRefPreprint

[7] 

Filippov V T 1986 n-Lie algebras Sib. Math. J. 26 879–91 
CrossRef
Filippov V T 1998 On n-Lie algebra of Jacobians Sib. Math. J. 39 573–81 
CrossRef

[8] 

Gautheron P 1996 Some remarks concerning Nambu mechanics Lett. Math. Phys. 37 103–16 
CrossRef

[9] 

Hanlon P and Wachs M 1995 On Lie k-algebras Adv. Math. 113 206–36 
CrossRef

[10] 

Higgins P 1956 Groups with multiple operators Proc. Lond. Math. Soc. 6 366–416 
CrossRef

[11] 

Kurosh A G 1969 Multioperator rings and algebras Russ. Math. Surveys 24 1–13 
IOPscience

[12] 

Lada T and Stasheff J 1993 Introduction to SH Lie algebras for physicists Int. J. Theor. Phys. 32 1087–103 
CrossRef

[13] 

Nambu Y 1973 Generalized Hamiltonian dynamics Phys. Rev. D 7 2405-2412 
CrossRef

[14] 

Pojidaev A P 2003 Enveloping algebras of Filippov algebras Commun. Algebra 31 883–900 
CrossRef

[15] 

Schlesinger M and Stasheff J D 1985 The Lie algebra structure of tangent cohomology and deformation theory J. Pure Appl. Algebra 38 313–22 
CrossRef

[16] 

Takhtajan L 1994 On foundation of the generalized Nambu mechanics Commun. Math. Phys. 160 295–315 (arXiv:hep-th/9301111) 
CrossRefPreprint

[17] 

Vainerman L and Kerner R 1996 On special classes of n-algebras J. Math. Phys. 37 2553–65 
CrossRef

[18] 

Vaisman I 1999 A survey on Nambu–Poisson brackets Acta. Math. Univ. Comenianae 68 213–241 (arXiv:math/9901047) 
Preprint

Notes

Note1

 We will not discuss all possible symmetrizations of the bracket entries, but just particular choices that work to give identities for general N. For a thorough study of other symmetrizations, particularly in the 3-bracket case, see [1], as well as the more recent work of Nuyts et al [5].

Note2

 To avoid any misunderstanding, by implicit total antisymmetrization of the Bs in the expression [[A[B₁ · · · B_2L + 1]B_2L + 2 · · · B_4L]B_4L + 1 · · · B_6L], we mean

where the sum is over all (6N)! permutations of the indices, 1,  ..., 6N. Similar meanings apply to the other implicitly antisymmetrized expressions in the communication.