Expanding Lie (super)algebras through abelian semigroups
Abstract
We propose an outgrowth of the expansion method introduced by de Azcárraga et al. [Nucl. Phys. B 662 (2003) 185]. The basic idea consists in considering the direct product between an abelian semigroup and a Lie algebra . General conditions under which relevant subalgebras can systematically be extracted from are given. We show how, for a particular choice of semigroup , the known cases of expanded algebras can be reobtained, while new ones arise from different choices. Concrete examples, including the M algebra and a D’Auria–Frélike Superalgebra, are considered. Finally, we find explicit, nontrace invariant tensors for these expanded algebras, which are essential ingredients in, e.g., the formulation of Supergravity theories in arbitrary spacetime dimensions.
Contents
I Introduction
The rôle played by Lie algebras and their interrelations in physics can hardly be overemphasized. To mention only one example, the Poincaré algebra may be obtained from the Galilei algebra via a deformation process. This deformation is one of the ways in which different Lie algebras can be related.
The purpose of this paper is to shed new light on the method of expansion of Lie algebras (for a thorough treatment, see the seminal work Azcarraga Et Al and references therein; early work on the subject is found in Hatsuda Sakaguchi ). An Expansion is, in general, an algebra dimensionchanging process. For instance, the M algebra Towsend1 ; Towsend2 ; CECS M , with 583 Bosonic generators, can be regarded as an expansion of the orthosymplectic algebra , which possesses only 528. This vantage viewpoint may help better understand fundamental problems related to the geometrical formulation of 11dimensional Supergravity. Some physical applications of the expansion procedure have been developed in Sak06 ; Azc05 ; Ban04 ; Ban04b ; Hat04 ; Ban03 ; Hat03 ; Mee03 ; Iza06c .
The approach to be presented here is entirely based on operations performed directly on the algebra generators, and thus differs from the outset with the one found in Azcarraga Et Al , where the dual Maurer–Cartan formulation is used. As a consequence, the expansion of free differential algebras lies beyond the scope of our analysis.
Finite abelian semigroups play a prominent rôle in our construction. All expansion cases found in Azcarraga Et Al may be regarded as coming from one particular choice of semigroup in the present approach, which is, in this sense, more general. Different semigroup choices yield, in general, expanded algebras that cannot be obtained by the methods of Azcarraga Et Al .
The plan of the paper goes as follows. After some preliminaries in section II, section III introduces the general procedure of abelian semigroup expansion, expansion for short, and shows how the cases found in Azcarraga Et Al can be recovered by an appropriate choice of semigroup . In section IV, general conditions are given under which relevant subalgebras can be extracted from an expanded algebra. The case when satisfies the WeimarWoods conditions WW ; WW2 and the case when is a superalgebra are studied. Section V gives three explicit examples of expansions of : (i) the M algebra Towsend1 ; Towsend2 ; CECS M , (ii) a D’Auria–Frélike Superalgebra DAuriaFre and (iii) a new Superalgebra, different from but resembling aspects of the M algebra, and D’Auria–Fré superalgebras. In section VI, the remaining cases of expanded algebras shown in Azcarraga Et Al are seen to also fit within the current scheme. The fivebrane Superalgebra vanHo82 ; deAz89 is given as an example. Section VII deals with the crucial problem of finding invariant tensors for the expanded algebras. General theorems are proven, allowing for nontrivial invariant tensors to be systematically constructed. We close in section VIII with conclusions and an outlook for future work.
Ii Preliminaries
Before analyzing the Expansion procedure itself, it will prove convenient to introduce some basic notation and definitions.
ii.1 Semigroups
Definition II.1
Let be a finite semigroup ^{1}^{1}1There does not seem to be a unique, universallyaccepted definition of semigroup. Here it is taken to be a set provided with a closed associative product. It does not need to have an identity., and let us write the product of as
(1) 
The selector is defined as
(2) 
Since is associative, the selector fulfills the identity
(3) 
Using this identity it is always possible to express the selector in terms of 2selectors, which encode the information from the multiplication table of .
An interesting way to state the same is that selectors provide a matrix representation for . As a matter of fact, when we write
(4) 
then we have
(5) 
We will restrict ourselves from now on to abelian semigroups, which implies that the selectors will be completely symmetrical in their lower indices.
The following definition introduces a product between semigroup subsets which will be extensively used throughout the paper.
Definition II.2
Let and be two subsets of . The product is defined as
(6) 
In other words, is the set which results from the product of every element of with every element of . Since is abelian, .
Let us emphasize that, in general, , and need not be semigroups by themselves.
The abelian semigroup could also be provided with a unique zero element, . This element is defined as the one for which
(7) 
for each .
ii.2 Reduced Lie Algebras
The following definition introduces the concept of reduction of Lie algebras.
Definition II.3
Consider a Lie (super)algebra of the form , with being a basis for and a basis for When , i.e., when the commutation relations have the general form
(8)  
(9)  
(10) 
then it is straightforward to show that the structure constants satisfy the Jacobi identity by themselves, and therefore corresponds by itself to a Lie (super)algebra. This algebra, with structure constants , is called a reduced algebra of and symbolized as
The reduced algebra could be regarded in some way as the “inverse” of an algebra extension, but does not need to be an ideal. Note also that a reduced algebra does not in general correspond to a subalgebra.
Iii The Expansion Procedure
iii.1 Expansion for an Arbitrary Semigroup
The following theorem embodies one of the main results of the paper, the concept of expanded algebras.
Theorem III.1
Let be an abelian semigroup with selector and a Lie (super)algebra with basis and structure constants . Denote a basis element of the direct product by and consider the induced commutator . Then, is also a Lie (super)algebra with structure constants
(11) 
Proof. Starting from the form of the induced commutator and using the multiplication law (1) one finds
The definition of the 2selector [cf. eq. (2)],
now allows us to write
(12) 
Therefore, the algebra spanned by closes and the structure constants read
(13) 
Since is abelian, the structure constants have the same symmetries as , namely
(14) 
and for this reason, , where denotes the degree of (1 for Fermi and 0 for Bose).
In order to show that the structure constants satisfy the Jacobi identity, it suffices to use the properties of the selectors [cf. eq. (3)] and the fact that the structure constants satisfy the Jacobi identity themselves. This concludes the proof.
The following definition is a natural outcome of Theorem III.1.
Definition III.2
Let be an abelian semigroup and a Lie algebra. The Lie algebra defined by is called Expanded algebra of .
When the semigroup has a zero element , it plays a somewhat peculiar rôle in the expanded algebra. Let us span in nonzero elements and a zero element . Then, the selector satisfies
(15)  
(16)  
(17)  
(18) 
Therefore, can be split as
(19)  
(20)  
(21) 
Comparing (19)–(21) with (8)–(10), one sees that the commutation relations
(22) 
are those of a reduced Lie algebra of (see Def. II.3). The reduction procedure in this particular case is equivalent to imposing the condition
(23) 
Notice that in this case the reduction abelianizes large sectors of the algebra; for each and satisfying (i.e., ) we have .
The above considerations motivate the following definition:
Definition III.3
Let be an abelian semigroup with a zero element , and let be an expanded algebra. The algebra obtained by imposing the condition on (or a subalgebra of it) is called reduced algebra of (or of the subalgebra).
The algebra (22) appears naturally when the semigroup’s zero matches the (algebra) field’s zero. As we will see in the next section, this is the way Maurer–Cartan forms powerseries expanded algebras fit within the present scheme. It is also possible to extract other reduced algebras from ; as will be analyzed in Sec. VI, the reduced algebra turns out to be a particular case of Theorem VI.1.
iii.2 Maurer–Cartan Forms Power Series Algebra Expansion as an Expansion
The Maurer–Cartan forms power series algebra expansion method is a powerful procedure which can lead, in stark contrast with contraction, deformation and extension of algebras, to algebras of a dimension higher than the original one. In a nutshell, the idea consists of looking at the algebra as described by the associated Maurer–Cartan forms on the group manifold and, after rescaling some of the group parameters by a factor , in expanding the Maurer–Cartan forms as a power series in . Finally this series is truncated in a way that assures the closure of the algebra. The subject is thoroughly treated by de Azcárraga and Izquierdo in Ref. LibroAzcarraga and de Azcárraga, Izquierdo, Picón and Varela in Ref. Azcarraga Et Al .
Theorem 1 of Ref. Azcarraga Et Al shows that, in the more general case, the expanded Lie algebra has the structure constants
(24) 
where the parameters correspond to the order of the expansion, and is the truncation order.
These structure constants can also be obtained within the expansion procedure. In order to show this, one must consider the reduction of an expanded algebra where corresponds to the semigroup defined below.
Definition III.4
Let us define as the semigroup of elements
(25) 
provided with the multiplication rule
(26) 
where is defined as the function
(27) 
The selectors for read
(28) 
where is the Kronecker delta. From eq. (26), we have that is the zero element in , i.e., .
Using eq. (11), the structure constants for the expanded algebra can be written as
(29) 
with . When the extra condition is imposed, eq. (29) reduces to
(30) 
which exactly matches the structure constants (24).
The above arguments show that the Maurer–Cartan forms power series expansion of an algebra , with truncation order , coincides with the reduction of the expanded algebra .
This is of course no coincidence. The set of powers of the rescaling parameter , together with the truncation at order , satisfy precisely the multiplication law of . As a matter of fact, we have
(31) 
and the truncation can be imposed as
(32) 
It is for this reason that one must demand in order to obtain the MC expansion as an expansion: in this case the zero of the semigroup is the zero of the field as well.
The expansion procedure is valid no matter what the structure of the original Lie algebra is, and in this sense it is very general. However, when something about the structure of is known, a lot more can be done. As an example, in the context of MC expansion, the rescaling and truncation can be performed in several ways depending on the structure of , leading to several kinds of expanded algebras. Important examples of this are the generalized İnönü–Wigner contraction, or the M algebra as an expansion of (see Refs. Azcarraga Et Al ; AzcarragaSuperspace ). This is also the case in the context of expansions. As we will show in the next section, when some information about the structure of is available, it is possible to find subalgebras of and other kinds of reduced algebras. In this way, all the algebras obtained by the MC expansion procedure can be reobtained. New kinds of expanded algebras can also be obtained by considering semigroups different from .
Iv Expansion Subalgebras
An expanded algebra has a fairly simple structure. In a way, it reproduces the original algebra in a series of “levels” corresponding to the semigroup elements. Interestingly enough, there are at least two ways of extracting smaller algebras from . The first one, described in this section, gives rise to a “resonant subalgebra,” while the second, described in section VI, produces reduced algebras (in the sense of Def. II.3).
iv.1 Resonant Subalgebras for an Arbitrary Semigroup
The general problem of finding subalgebras from an expanded algebra is a nontrivial one, which is met and solved (in a particular setting) in this section (see theorem IV.2 below). In order to provide a solution, one must have some information on the subspace structure of . This information is encoded in the following way.
Let be a decomposition of in subspaces , where is a set of indices. For each it is always possible to define such that
(33) 
In this way, the subsets store the information on the subspace structure of .
As for the abelian semigroup , this can always be decomposed as , where . In principle, this decomposition is completely arbitrary; however, using the product from Def. II.2, it is sometimes possible to pick up a very particular choice of subset decomposition. This choice is the subject of the following
Definition IV.1
Let be a decomposition of in subspaces, with a structure described by the subsets , as in eq. (33). Let be a subset decomposition of the abelian semigroup such that
(34) 
where the subset product is the one from Def. II.2. When such subset decomposition exists, then we say that this decomposition is in resonance with the subspace decomposition of ,
The resonant subset decomposition is crucial in order to systematically extract subalgebras from the expanded algebra as is proven in the following
Theorem IV.2
Now, it is clear that for each , one can write
(38) 
Then,
(39) 
and we arrive at
(40) 
Therefore, the algebra closes and is a subalgebra of .
Definition IV.3
The algebra obtained in Theorem IV.2 is called a Resonant Subalgebra of the expanded algebra .
The choice of the name resonance is due to the formal similarity between eqs. (33) and (34); eq. (34) will be also referred to as “resonance condition.”
Theorem IV.2 translates the difficult problem of finding subalgebras from an expanded algebra into that of finding a resonant partition for the semigroup . As the examples from section V help make clear, solving the resonance condition (34) turns out to be an easily tractable problem. Theorem IV.2 can thus be regarded as a useful tool for extracting subalgebras from an expanded algebra.
Using eq. (11) and the resonant subset partition of it is possible to find an explicit expression for the structure constants of the resonant subalgebra . Denoting the basis of by , one can write
(41) 
An interesting fact is that the expanded algebra “subspace structure” encoded in is the same as in the original algebra , as can be observed from eq. (40).
Resonant Subalgebras play a central rôle in the current scheme. It is interesting to notice that most of the cases considered in Azcarraga Et Al can be reobtained using the above theorem for [recall eqs. (25)–(26)] and reduction, as we will see in the next section. All remaining cases can be obtained as a more general reduction of a resonant subalgebra (see sec. VI).
iv.2 Resonant Subalgebras and MC Expanded Algebras
In this section, some results presented for algebra expansions in Ref. Azcarraga Et Al are recovered within the expansion approach. To get these algebras one must proceed in a threestep fashion:

Perform an expansion using the semigroup ,

Find a resonant partition for and construct the resonant subalgebra ,

Apply a reduction (or a more general one, see sec. VI) to the resonant subalgebra.
Choosing a different semigroup or omitting the reduction procedure one finds algebras not contained within the Maurer–Cartan forms power series expansion of Ref. Azcarraga Et Al . Such an example is provided in sec. V.3.
iv.2.1 Case when , with being a Subalgebra and a Symmetric Coset
Let be a subspace decomposition of , such that
(42)  
(43)  
(44) 
Let , with arbitrary, be a subset decomposition of , with ^{2}^{2}2Here denotes the integer part of .
(45)  
(46) 
This subset decomposition of satisfies the resonance condition (34), which in this case explicitly reads
(47)  
(48)  
(49) 
Using eq. (41), it is straightforward to write the structure constants for the resonant subalgebra,
Imposing , the reduced structure constants are obtained as
(53) 
In order to compare with the MC Expansion, let us observe that, with the notation of Azcarraga Et Al , the reduction of the expanded algebra corresponds to for the symmetric coset case, with
(54)  
(55) 
The structure constants (53) correspond to the structure constants (3.31) from Ref. Azcarraga Et Al (the notation is slightly different though).
A more intuitive idea of the whole procedure of expansion, resonant subalgebra and reduction can be obtained by means of a diagram, such as the one depicted in Fig. 1. This diagram corresponds to the case , with a subalgebra and a symmetric coset, and the choice .
The subspaces of are represented on the horizontal axis, while the semigroup elements occupy the vertical one. In this way, the whole expanded algebra corresponds to the shaded region in Fig. 1 (a). In Fig. 1 (b), the gray region represents the resonant subalgebra with
(56)  
(57) 
Let us observe that each column in the diagram corresponds to a subset of the resonant partition. Finally, Fig. 1 (c) represents the reduced algebra, obtained after imposing . This figure actually corresponds to the case .
As is evident from the above discussion, the case , reproduces the İnönü–Wigner contraction for . More on generalized İnönü–Wigner contractions is presented in section VI.
iv.2.2 Case when fulfills the WeimarWoods Conditions
Let be a subspace decomposition of . In terms of this decomposition, the WeimarWoods conditions WW ; WW2 on read
(58) 
Let
(59) 
be a subset decomposition of , where the subsets are defined by
(60) 
with .
This subset decomposition is a resonant one under the semigroup product (26), because it satisfies [compare eq. (61) with eq. (58)]
(61) 
Using eq. (41), we get the following structure constants for the resonant subalgebra:
Imposing , this becomes
(63) 
This reduced algebra corresponds to the case of Theorem 3 from Azcarraga Et Al with for every . The structure constants (63) correspond to the ones of eq. (4.8) in Ref. Azcarraga Et Al (with a slightly different notation). The more general case,
can be also obtained in the context of an expansion, from a resonant subalgebra and applying a kind of reduction more general than the reduction (see sec. VI and app. A).
The resonant subalgebra for the WeimarWoods case with , and its reduction are shown in Fig. 2 (a) and (b), respectively.
iv.2.3 Case when is a Superalgebra
A superalgebra comes naturally split into three subspaces , and , where corresponds to the Fermionic sector and to the Bosonic one, with being a subalgebra. This subspace structure may be written as
(64)  
(65)  
(66)  
(67)  
(68)  
(69) 
Let be a subset decomposition of , where the subsets are given by the general expression
(70) 
This subset decomposition is a resonant one, because it satisfies [compare eqs. (71)–(76) with eqs. (64)–(69)]
(71)  
(72)  
(73)  
(74)  
(75)  
(76) 
Theorem IV.2 assures us that , with , , is a resonant subalgebra.
Using eq. (41), it is possible to write down the structure constants for the resonant subalgebra as
(77) 
Imposing , the structure constants for the reduction of the resonant subalgebra are obtained:
(78) 
This reduced algebra corresponds to the algebra