##
**\(n\)-ary algebras: a review with applications.**
*(English)*
Zbl 1202.81187

In this quite impressive review, the authors present a detailed, modern and complete overview of the theory of \(n\)-ary algebras and their most relevant and current applications to physical problems. The review is structured in various steps, starting from the case of classical Lie algebras and their main properties, and then gradually introducing and studying the different generalizations of these algebras.

The paper is divided into 16 sections of varying length, providing the most relevant structural properties of \(n\)-ary algebras, as well as their significance in the context of physical problems. The exposition is unified and self contained, containing a huge number of references that allow the reader to look for more details on specific questions. The first section presents an overview of the work, as well as its physical motivation (like recent developments in brane theory). Some general notations and conventions to be used through the review are fixed at the end of the first section.

The second and third sections summarize some of the fundamental properties of Lie algebras that are of use in the analysis of \(n\)-ary algebras, such as the existence of compatible metric tensors, the differential-geometric description of Lie algebras in terms of the Maurer-Cartan equations and various cohomological tools related to the central extensions and deformations. The relation between cocycles and Casimir operators is also briefly outlined. In Section 4 the main facts concerning Leibniz algebras and their cohomology are given. Similarities and main differences of their properties when compared to the classical Lie algebraic case are analyzed in detail.

The fifth and sixth sections are devoted to higher order or generalized Lie algebras (GLAs). A construction based on cohomological methods is given, where the relation to compact simple Lie algebras is emphasized. With these higher order brackets, the notions of higher order coderivatives and external derivatives are developed, resulting in a natural generalization of the Maurer-Cartan equations, which are described using the (generalized) BRST operators. The section finishes with a brief review of GLAs and strongly homotopy Lie algebras. Section 7 begins with the study of an important class of GLAs, the Filippov algebras. The important derivation property that determines the characteristic identity of this class of algebras also shows their different character from previously discussed structures. The general properties of Filippov algebras are given, with special emphasis on their inner derivations algebra. It is shown that various notions from the classical Lie theory do not generalize in full for Filippov algebras. The main differences arise when studying the simple algebras and the possible notions of solvability. Various examples of finite and infinite dimensional Filippov algebras are given, like the simple and Nambu Filippov algebras.

In Section 8 the problem of Lie algebras associated to \(n\)-Lie algebras is considered, taking into account the important case of Lie algebras associated to 3-Lie algebras, of great importance in high energy physics and notably in the Bagger-Lambert-Gustavsson (BLG) model. With this precedent, the general case is obtained by means of considering the inner derivations. This leads to an analogue of the Whitehead lemmas for Filippov algebras in quite natural way. Finally, the representation theory of Filippov algebras (in the Kasymov approach) is considered. Skipping the antisymmetry constraint, the \(n\)-Leibniz algebras are considered in Section 9. The particular case of of Lie-triple systems (3-Leibniz algebras) is considered, prior to the discussion of the homology and cohomology theories of Filippov algebras in Section 11. It is shown that the cohomology can also be used, although with important differences concerning the cohomology complexes, to characterize the central extensions and the (infinitesimal) deformations of Filippov algebras. The appropriate choice for the complexes arises from the study of the central extensions of 3-Lie algebras, and thus do not correspond to the well known case of classical Lie algebras. This poses the problem of finding a corresponding geometrical frame, i.e., the obtainment of equations that replace the Maurer-Cartan equations but have similar properties. These so-called Maurer-Cartan-like equations play an important role in many specific physical problems, like the description of M-branes. In direct analogy with the Lie algebraic case, it is shown that semisimple Filippov algebras are stable, i.e., they do not admit non-trivial deformations. It is at this point where the analogy of the method with the classical Whitehead lemmas emerges clearly.

Section 12 discusses the analogue tools for defining a consistent cohomology theory of \(n\)-Leibniz algebras, and their fundamental differences with the Filippov case. Once the study of the main generalizations of Lie algebras has been reviewed, the authors consider the case of generalizations of the Poisson bracket and the standard Poisson structure. Two cases are discussed in detail, namely, the generalized Poisson structure and the Nambu-Poisson structure, and their differences pointed out (like the fact that Nambu-Poisson structures are generalized Poisson structures, while the converse does generally not hold). Main properties of these two classes of algebras are nicely illustrated in tabular form. The authors also devote a paragraph to the problem of quantizing the Nambu-Poisson mechanics, which presents serious difficulties, like the associativity of the quantum operators (with the understanding that the Manin properties of the Nambu-Poisson structure are preserved).

Section 14 is fully devoted to the Bagger-Lambert-Gustavsson model and its Nambu bracket extension, using more the geometrical than the purely physical approach. The main topics discussed in this paragraph are the BLG-Lagrangian, the gauge and supersymmetry transformations, the relation with the Basu-Harvey equation, among others. All these properties are developed having in mind the original proposal of the model, in order to simplify the presentation. Along similar lines of development, Section 15 analyzes the BLG model based on the Nambu bracket Filippov algebra. The last section discusses these structures in the context of physics, briefly commenting on their possibilities to be the adequate object to describe modern developments and the constraints that these algebras present to faithfully reproduce the studied phenomena. Two appendices giving details on the two forms of the Filippov identity and the Nijenhuis-Schouten brackets are included.

The review provides also a very extensive, complete and up-dated reference list with 318 entries. They cover the topics discussed, as well as many other related ones that were not considered specifically. Although the references are obviously focused on the physical literature dealing with \(n\)-ary algebras, they also provide a long list of pure mathematical papers that explore the more formal aspects of the theory.

Summarizing, this review fills an important gap in the physical literature concerning generalized Lie algebras, their structure and representation theory with physical background, and will certainly be of great use for both the specialist and the non-specialist.

The paper is divided into 16 sections of varying length, providing the most relevant structural properties of \(n\)-ary algebras, as well as their significance in the context of physical problems. The exposition is unified and self contained, containing a huge number of references that allow the reader to look for more details on specific questions. The first section presents an overview of the work, as well as its physical motivation (like recent developments in brane theory). Some general notations and conventions to be used through the review are fixed at the end of the first section.

The second and third sections summarize some of the fundamental properties of Lie algebras that are of use in the analysis of \(n\)-ary algebras, such as the existence of compatible metric tensors, the differential-geometric description of Lie algebras in terms of the Maurer-Cartan equations and various cohomological tools related to the central extensions and deformations. The relation between cocycles and Casimir operators is also briefly outlined. In Section 4 the main facts concerning Leibniz algebras and their cohomology are given. Similarities and main differences of their properties when compared to the classical Lie algebraic case are analyzed in detail.

The fifth and sixth sections are devoted to higher order or generalized Lie algebras (GLAs). A construction based on cohomological methods is given, where the relation to compact simple Lie algebras is emphasized. With these higher order brackets, the notions of higher order coderivatives and external derivatives are developed, resulting in a natural generalization of the Maurer-Cartan equations, which are described using the (generalized) BRST operators. The section finishes with a brief review of GLAs and strongly homotopy Lie algebras. Section 7 begins with the study of an important class of GLAs, the Filippov algebras. The important derivation property that determines the characteristic identity of this class of algebras also shows their different character from previously discussed structures. The general properties of Filippov algebras are given, with special emphasis on their inner derivations algebra. It is shown that various notions from the classical Lie theory do not generalize in full for Filippov algebras. The main differences arise when studying the simple algebras and the possible notions of solvability. Various examples of finite and infinite dimensional Filippov algebras are given, like the simple and Nambu Filippov algebras.

In Section 8 the problem of Lie algebras associated to \(n\)-Lie algebras is considered, taking into account the important case of Lie algebras associated to 3-Lie algebras, of great importance in high energy physics and notably in the Bagger-Lambert-Gustavsson (BLG) model. With this precedent, the general case is obtained by means of considering the inner derivations. This leads to an analogue of the Whitehead lemmas for Filippov algebras in quite natural way. Finally, the representation theory of Filippov algebras (in the Kasymov approach) is considered. Skipping the antisymmetry constraint, the \(n\)-Leibniz algebras are considered in Section 9. The particular case of of Lie-triple systems (3-Leibniz algebras) is considered, prior to the discussion of the homology and cohomology theories of Filippov algebras in Section 11. It is shown that the cohomology can also be used, although with important differences concerning the cohomology complexes, to characterize the central extensions and the (infinitesimal) deformations of Filippov algebras. The appropriate choice for the complexes arises from the study of the central extensions of 3-Lie algebras, and thus do not correspond to the well known case of classical Lie algebras. This poses the problem of finding a corresponding geometrical frame, i.e., the obtainment of equations that replace the Maurer-Cartan equations but have similar properties. These so-called Maurer-Cartan-like equations play an important role in many specific physical problems, like the description of M-branes. In direct analogy with the Lie algebraic case, it is shown that semisimple Filippov algebras are stable, i.e., they do not admit non-trivial deformations. It is at this point where the analogy of the method with the classical Whitehead lemmas emerges clearly.

Section 12 discusses the analogue tools for defining a consistent cohomology theory of \(n\)-Leibniz algebras, and their fundamental differences with the Filippov case. Once the study of the main generalizations of Lie algebras has been reviewed, the authors consider the case of generalizations of the Poisson bracket and the standard Poisson structure. Two cases are discussed in detail, namely, the generalized Poisson structure and the Nambu-Poisson structure, and their differences pointed out (like the fact that Nambu-Poisson structures are generalized Poisson structures, while the converse does generally not hold). Main properties of these two classes of algebras are nicely illustrated in tabular form. The authors also devote a paragraph to the problem of quantizing the Nambu-Poisson mechanics, which presents serious difficulties, like the associativity of the quantum operators (with the understanding that the Manin properties of the Nambu-Poisson structure are preserved).

Section 14 is fully devoted to the Bagger-Lambert-Gustavsson model and its Nambu bracket extension, using more the geometrical than the purely physical approach. The main topics discussed in this paragraph are the BLG-Lagrangian, the gauge and supersymmetry transformations, the relation with the Basu-Harvey equation, among others. All these properties are developed having in mind the original proposal of the model, in order to simplify the presentation. Along similar lines of development, Section 15 analyzes the BLG model based on the Nambu bracket Filippov algebra. The last section discusses these structures in the context of physics, briefly commenting on their possibilities to be the adequate object to describe modern developments and the constraints that these algebras present to faithfully reproduce the studied phenomena. Two appendices giving details on the two forms of the Filippov identity and the Nijenhuis-Schouten brackets are included.

The review provides also a very extensive, complete and up-dated reference list with 318 entries. They cover the topics discussed, as well as many other related ones that were not considered specifically. Although the references are obviously focused on the physical literature dealing with \(n\)-ary algebras, they also provide a long list of pure mathematical papers that explore the more formal aspects of the theory.

Summarizing, this review fills an important gap in the physical literature concerning generalized Lie algebras, their structure and representation theory with physical background, and will certainly be of great use for both the specialist and the non-specialist.

Reviewer: Rutwig Campoamor-Stursberg (Madrid)

### MSC:

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |

17A42 | Other \(n\)-ary compositions \((n \ge 3)\) |

22E70 | Applications of Lie groups to the sciences; explicit representations |

58A15 | Exterior differential systems (Cartan theory) |

19C09 | Central extensions and Schur multipliers |

17B37 | Quantum groups (quantized enveloping algebras) and related deformations |

81R15 | Operator algebra methods applied to problems in quantum theory |

53D17 | Poisson manifolds; Poisson groupoids and algebroids |

17B63 | Poisson algebras |