Bianchi spaces and their 3-dimensional isometries as S-expansions of 2-dimensional isometries
aa r X i v : . [ m a t h - ph ] M a r Bianchi spaces and their -dimensional isometriesas S -expansions of -dimensional isometries Ricardo Caroca , Igor Kondrashuk , Nelson Merino , , Felip Nadal , Departamento de Matem´atica y F´ısica Aplicadas, Universidad Cat´olica de la Sant´ısimaConcepci´on, Alonso de Rivera 2850, Concepci´on, Chile Departamento de Ciencias B´asicas, Univerdidad del B´ıo-B´ıo,Campus Fernando May, Casilla 447, Chill´an, Chile Departamento de F´ısica, Universidad de Concepci´on, Casilla 160-C, Concepci´on, Chile Dipartimento di Fisica, Politecnico di Torino,Corso Duca degli Abruzzi, 24, I-10129 Torino, Italia IFIC (Instituto de F´ısica Corpuscular)Edificio Institutos de Investigaci´on. c/ Catedr´atico Jos´e Beltr´an, 2. E-46980 Paterna, Spain
June 23, 2018
Abstract
In this paper we show that some 3-dimensional isometry algebras, specifically those oftype I, II, III and V ( according Bianchi’s classification), can be obtained as expansions of theisometries in 2 dimensions. It is shown that in general more than one semigroup will lead to thesame result. It is impossible to obtain the algebras of type IV, VI-IX as an expansion from theisometry algebras in 2 dimensions. This means that the first set of algebras has properties thatcan be obtained from isometries in 2 dimensions while the second set has properties that are insome sense intrinsic in 3 dimensions. All the results are checked with computer programs. Thisprocedure can be generalized to higher dimensions, which could be useful for diverse physicalapplications. Introduction
Expansions of Lie algebras are a generalization of the contraction method which were introducedsome years ago in refs. [1], [2], [3], [4], [5], [6], [7]. These methods, in particular the S -expansionprocedure developed in ref. [5], are powerful tools to find non-trivial relations between differentLie algebras which are a very interesting problem from both, physical and mathematical points ofview. In fact, many physical applications have been found in this context (see, for example, refs.[2], [6], [7], [8], [9], [10] and [11]).It is the aim of this paper to show that the S -expansion method permits us to obtain some typesof 3-dimensional isometries from the 2-dimensional isometries. We present a complete study aboutthe possibility of finding non-trivial relations between 2- and 3-dimensional isometry Lie algebras.Even when these isometries are well known in the literature (see [12]), the non-trivial relations wefind between 2- and 3-dimensional isometry algebras are new and interesting results. In fact, weidentify the 3-dimensional algebras that can be obtained from the 2-dimensional algebras by meansof an S -expansion and describe explicitly how to do it. For other 3-dimensional algebras we showthat it is impossible to obtain them from the information of the 2-dimensional algebras, and insome sense the information that they contain is intrinsic to 3 dimensions. A possible generalizationof this procedure to higher dimensions can be useful in some physical applications (see section 7).This paper is organized as follows: In section 2 we present a brief technical description of thebasic ingredients that we are going to use along this work. In section 2.1 we will review some aspectsof the S -expansion procedure while in section 2.3 we summarize the history about the enumerationand characterization of finite semigroups existing for each order n = 1 , ...,
9. In section 2.2 weintroduce a general kind of expansions that will be made in next sections and in section 2.4 webriefly describe the Bianchi classification of isometries in 2- and 3-dimensional spaces. In section4 it is shown in an instructive way how some types of isometries are related with 2-dimensionalisometries using known semigroups and also by introducing other semigroups that have not beenused earlier in the applications of the S -expansion procedure. In section 4.4 we briefly summarizethe results obtained by this iterative procedure. In section 5 it is shown why is it not possible toobtain, by expansions, the other 3-dimensional isometries from the 2-dimensional algebras. Finallyin section 6 we check the results using computer programs and solve the problem entirely. In this section we briefly describe the general abelian semigroup expansion procedure ( S -expansionfor short). We refer to ref. [5] for further details.Consider a Lie algebra G and a finite abelian semigroup S = { λ α } . According to Theorem 3.1from ref. [5], the Cartesian product G S = S × G , (1)is also a Lie algebra. The elements of this expanded algebra are denoted by X ( i,α ) = X i λ α (2) By non-trivial relations we mean that these mechanisms of contractions and expansions allow us to obtain someLie algebras starting with other algebras that have completely different properties. Also, the original algebra is notnecessarily (could be in specific cases) contained as a subalgebra of the algebra obtained by these processes. X i of G and the elements λ α of the semigroup S . The Lie product in G S is defined as (cid:2) X ( i,α ) , X ( j,β ) (cid:3) = λ α λ β [ X i , X j ] (3)The set (1) with the composition law (3) is called a S -expanded Lie algebra.In a nutshell, the S -expansion method can be seen as the natural generalization of the In¨on¨u-Wigner contraction, where instead of multiplying the generators by a numerical parameter, wemultiply the generators by the elements of an Abelian semigroup, for more detail see [5]. S -reduced algebra As shown in [5] smaller algebras can be extracted from the above expanded algebra: the resonantsubalgebra and the reduced algebra. Their existence depends on certain conditions expressed bythe Eqs. (23) and (34) of ref. [5].In fact the original algebra will be one of the 2-dimensional isometry algebras (9-10) both havingthe subspace structure G = V ⊕ V given by[ V , V ] ⊂ V (4)[ V , V ] ⊂ V (5)[ V , V ] ⊂ V where V , V are respectively generated by X and X . On the other hand, the semigroups weare going to construct will possess a resonant decomposition, i.e, must be of the form S = S ∪ S where S × S ⊂ S S × S ⊂ S S × S ⊂ S (6)Then, according to Theorem 4.2 of ref. [5], the resonant subalgebra is of of the form G S,R = ( S × V ) ⊕ ( S × V ) (7)Note that Eq. (6) is a particular case of Eq. (34) of Ref. ([5]).An even smaller algebra can be obtained when there is a zero element in the semigroup, i.e.,an element 0 S ∈ S such that, for all λ α ∈ S , 0 S λ α = 0 S . When this is the case, the whole 0 S × G sector can be removed from the resonant subalgebra by imposing 0 S × G = 0 (see Definition 3.3from ref. [5]). The resulting algebra continues to be a Lie algebra and here it will be denoted by G red S,R . The numbers of finite non-isomorphic semigroups of order n are given in the following table:3rder Q = [Distler, Kelsey, Mitchell ’09] (8)All the semigroups of order 4 have been classified by Forsythe in Ref. [13], of order 5 by Motzkinand Selfridge in Ref. [14], of order 6 by Plemmons in Ref. [15, 16, 17], of order 7 by J¨urgensenand Wick in Ref. [18], and of order 8 by Satoh, Yama and Tokizawa in Ref. [19], and monoids andsemigroups of order 9 by Distler and Kelsey in Ref. [20, 21] and by Distler and Mitchell in Ref.[22]. Also, for semigroups of order 9 the result can be found in Ref. [23].As shown in the table the problem of enumerating the all non-isomorphic finite semigroups of acertain order is a non-trivial problem. In fact, the number Q of semigroups increases very quicklywith the order of the semigroup.In ref. [24] a set of algorithms is given that permit us to make certain calculations with finitesemigroups. The first program, gen.f , gives all the non-isomorphic semigroups of order n for n =1 , , ..., . The input is the order, n , of the semigroups we want to obtain and the output is alist of all the non-isomorphic semigroups that exist in this order. In this work, the elements of thesemigroup are labeled by λ α with α = 1 , ..., n and each semigroup will be denoted by S a ( n ) where thesuper-index a = 1 , ..., Q identifies the specific semigroup of order n . The second program in [24], com.f , takes as input one of the mentioned list for a certain order, picks up just the symmetric tablesand generates another list with all the abelian semigroups. For example for n = 3 the elementsof the semigroup are labeled by λ , λ and λ and the program com.f gives the following list ofsemigroups: S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , The order n = 9 is non trivial and the algorithms of the mentioned reference fails. This non-trivial problem wassolved in 2009 by Andreas Distler, Tom Kelsey & James Mitchell. However, in this paper we are going to considercalculations with semigroups of at most the order 4. λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ , S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ,So in general the program com.f of [24] gives a list of tables of the all abelian non-isomorphicsemigroups of a certain order (up to order 8).In section 6 we show that each semigroup that we construct in this paper, to make the relationsbetween 2- and 3- dimensional isometries, are isomorphic to one of the semigroups of the lists givenby [24]. This is a way to check that the iterative procedure (that we shall present in sections 4, 5)to find semigroups with zero elements and resonant decompositions is working well. In ref. [12] Bianchi proposed a procedure to classify the 3-dimensional spaces that admit a 3-dimensional isometry. He had shown how to represent the generators as Killing vectors and how toget the corresponding metrics, and formulated Bianchi’s theorem.Here we are interested in study of the possibility of relating, by means of expansions, 2- and3-dimensional isometry algebras. The 2-dimensional algebras are given by[ X , X ] = 0 or (9)[ X , X ] = X (10)The 3 − dimensional algebras are given in the following table, Group Algebra type I [ X , X ] = [ X , X ] = [ X , X ] = 0type II [ X , X ] = [ X , X ] = 0 , [ X , X ] = X type III [ X , X ] = [ X , X ] = 0 , [ X , X ] = X type IV [ X , X ] = 0 , [ X , X ] = X , [ X , X ] = X + X type V [ X , X ] = 0 , [ X , X ] = X , [ X , X ] = X type VI [ X , X ] = 0 , [ X , X ] = X , [ X , X ] = hX , where h = 0 , . type VII [ X , X ] = 0 , [ X , X ] = X , [ X , X ] = − X type VII [ X , X ] = 0 , [ X , X ] = X , [ X , X ] = − X + hX , where h = 0 (0 < h < . type VIII [ X , X ] = X , [ X , X ] = 2 X , [ X , X ] = X type IX [ X , X ] = X , [ X , X ] = X , [ X , X ] = X (11) The whole paper is dedicated to demonstrate that the following proposition is valid. As the number of non-isomorphic semigroups increases very quickly with the order n (see table (8)), the mentionedlists are very large for higher orders. Note also that the semigroups S , S , S , S , S and S are not givenin the list for n = 3 because they are not abelian (non-commutative). roposition 1 Some but not all of the Bianchi algebras can be obtained as S -expansions of the2-dimensional algebras (9-10). This proposition will be proven by performing the S -expansion of (9-10) and by applying twoknown procedures which permit to extract a smaller algebra from the expanded algebra. As wemention in section 2.2, those are the construction of resonant subalgebra and the 0 S -reduction ofthe resonant subalgebra. -dimensional isometries Theorem 2
An abelian semigroup of four elements S = { λ , λ , λ , λ } with decomposition S = { λ , λ , λ } (12) S = { λ , λ } compatible with resonance condition (6) and possessing the following constraints in the multiplica-tion table λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ (13) produces, after S -reduction of the resonant subalgebra of the S -expanded algebra of the -dimensionalisometry (10), an algebra which coincides with -dimensional Bianchi type III isometry. Proof.
Let’s begin with an unknown semigroup S = { λ , λ , λ , λ } where we impose the followingconditions:( i ) λ is a zero of the semigroup, so the table of multiplication law is restricted to the form λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ where the empty spaces must be filled in a way such that it is in fact an abelian semigroup, i.e.,closed, associative and commutative. In order to get a smaller algebra we also demand( ii ) that it contains a decomposition like that given in (12) which is resonant, i.e., that satisfiesequation (6). Note that we start with a 2-dimensional algebra and we are looking for a 3-dimensional one. The S -expandedalgebra with a semigroup of order 4 is 8-dimensional and the reduced is 6-dimensional. So to obtain a 3-dimensionalalgebra we need to extract an even smaller algebra. This is only possible by extracting a resonant subalgebra. G S,R = S × G is given by G S,R = ( S × V ) ⊕ ( S × V ) (14)= { λ X , λ X , λ X } ⊕ { λ X , λ X } = { λ X , λ X , λ X , λ X , λ X } Now we have to extract an even smaller algebra by means of a 0 S -reduction. This is done by justtaking off from (14) the elements that contain the zero element, λ . Therefore, the reduction of theresonant subalgebra is given by G red S,R = { λ X , λ X , λ X } with the following commutation relations[ λ X , λ X ] = 0[ λ X , λ X ] = − λ λ X [ λ X , λ X ] = − λ λ X In order for this algebra to be closed we should choose:( a ) λ λ = λ or λ λ = λ and( b ) λ λ = λ or λ λ = λ .So we are lead with four possibilities to construct a closed algebra:( i ) λ λ = λ λ = λ ,( ii ) λ λ = λ and λ λ = λ ,( iii ) λ λ = λ and λ λ = λ ,( iv ) λ λ = λ λ = λ .The case ( i ) will lead to translations in 3 dimensions, i.e., to the type I algebra . It canbe checked that the case ( iv ) it is not useful, because in this case the multiplication law is nonassociative. On the other hand, it can be seen that both ( ii ) and ( iii ) will lead to the type IIIalgebra. In fact, in case ( ii ) we have[ λ X , λ X ] = 0[ λ X , λ X ] = − λ X [ λ X , λ X ] = − λ X = 0 ( λ is a zero element)renaming the generators as Y = λ X Y = λ X Y = λ X we immediately recognize the type III algebra (see table (11)). In case ( iii ) we would have[ λ X , λ X ] = 0[ λ X , λ X ] = − λ X = 0 ( λ is a zero element)[ λ X , λ X ] = − λ X Note that we have an abuse of notation here. The last equation must be read as: the resonant subalgebra isgenerated by the set of generators that appears in the right hand side. This case will be analyzed later. Y = λ X Y = λ X Y = λ X We choose to study case ( iii ) to construct a semigroup that leads to Type III algebra, althoughthe case ( ii ) could also be studied to generate other semigroups leading to the same result. So, thetable describing the multiplication law case ( iii ) is given by λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ where the empty spaces must be filled in such a way that satisfy associativity and the decomposition(12) satisfies the resonant condition (6). Proposition 3
There are semigroups that fit multiplication table (13) and resonant condition (12).
We give several examples of semigroups of this type. S (3) K Consider the semigroup S ( n ) K , with the multiplication law defined by λ α λ β = λ min { α,β } , α + β > n (15) λ α λ β = λ n +1 , α + β ≤ nα, β = 1 , , ..., n It is directly seen that for n = 3 the table of multiplication law λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ fits with the form of the table (13), where we have filled those empty spaces that appears in (13).Therefore, an expansion with the semigroup S (3) K reproduces the type III algebra after a reductionof the resonant subalgebra. S N Let’s consider the following table of multiplication λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ S N , also reproduces the type III algebra and it is not isomorphic to theprevious semigroup, S (3) K . S (2) E , another way to obtain the type III algebra In order to show that there are even other semigroups that lead to the type III algebra we considerthe semigroup S (2) E introduced in ref. [5] for n = 2. Its multiplication law is given by the followingtable λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ and its resonant partition is S = { λ , λ , λ } S = { λ , λ } The 0 S -reduction of the resonant subalgebra is given by G red S,R = { λ X , λ X , λ X } with commutation relations [ λ X , λ X ] = 0[ λ X , λ X ] = − λ X [ λ X , λ X ] = 0Renaming the generators as Y = λ X Y = λ X Y = λ X we obtain again the type III algebra[ Y , Y ] = [ Y , Y ] = 0 , [ Y , Y ] = Y . The natural question here is: is it possible to generate other type of Bianchi algebras from theisometries in 2 dimensions? To answer this question we will continue the procedure of section4.1, considering a semigroup S = { λ , λ , λ , λ } where λ is a zero element, but modify resonantdecomposition (12). The decomposition can be chosen in another way, as can be seen in the following9 emma 4 A resonant decomposition S = { λ , λ } S = { λ , λ , λ } (16) satisfying resonant condition (6) can produce, after S -reduction of resonant subalgebra of the S -expanded algebra of 2-dimensional isometry (10), a 3-dimensional algebra. Proof.
The reduction of the resonant subalgebra is given by G red S,R = { λ X , λ X , λ X } and the commutation relations are given by[ λ X , λ X ] = − λ λ X (17)[ λ X , λ X ] = − λ λ X [ λ X , λ X ] = 0Resonant condition (6) guarantees that (17) is a closed algebra.Here we have different possibilities in order to make this algebra closed. S N semigroupTheorem 5 An abelian semigroup of four elements S = { λ , λ , λ , λ } satisfying the conditionsof Lemma 5 and possessing the following constraints in the multiplication table λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ (18) produces, after S -reduction of the resonant subalgebra of the S -expanded algebra of the -dimensionalisometry (10), an algebra which coincides with -dimensional Bianchi type II isometry. Proof.
To reproduce the type II algebra we have to choose, for example, λ λ = λ and λ λ = λ (19)In that case the commutation relations (17) take the form[ λ X , λ X ] = − λ X [ λ X , λ X ] = 0[ λ X , λ X ] = 0and renaming the generators as Y = λ X Y = λ X Y = λ X
10e obtain the type II algebra [ Y , Y ] = [ Y , Y ] = 0 , [ Y , Y ] = Y But in order for this result to be true, we must provide an explicit semigroup that satisfies theconditions (19). Until now our table has the form λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ and the empty spaces must be filled in such a way that this table defines an associative, commutativeproduct and such that the decomposition (16) satisfies the resonant condition (6). Proposition 6
There are semigroups that fit multiplication table 18 and resonant condition (16).
After looking for different possibilities we have found one way to fill the table (18). The proposedsemigroup is λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ (20)That this multiplication table represents in fact an abelian semigroup can be checked directly.That it is also commutative is seen from the table. The associativity is proved by a tedious butdirect calculation.Note that there may be other semigroups that can also lead to the type II algebra. Thosecorrespond to other ways to fill the empty spaces in table (18). S N semigroupTheorem 7 An abelian semigroup of four elements S = { λ , λ , λ , λ } satisfying the conditionsof Lemma 5 and possessing the following constraints in the multiplication table λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ (21) produces, after S -reduction of the resonant subalgebra of the S -expanded algebra of the -dimensionalisometry (10), an algebra which coincides with -dimensional Bianchi type V isometry. Proof.
In fact, if we choose in the commutation relations (17), for example, λ λ = λ and λ λ = λ (22)11n that case the commutation relations (17) take the form[ λ X , λ X ] = − λ X [ λ X , λ X ] = − λ X [ λ X , λ X ] = 0and renaming the generators as Y = λ X Y = λ X Y = λ X we obtain the type V algebra[ Y , Y ] = 0 , [ Y , Y ] = Y , [ Y , Y ] = Y But again, in order for this result to be true we must provide of an explicit semigroup thatsatisfies the conditions (22). Until now our table has had the form λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ and the empty spaces must be filled in a way that respects the required conditions. Note that thereare 4 = 256 possibilities to fill this table in a closed form. This number is reduced by imposingassociativity, commutativity and the resonant condition for the decomposition (16). This numbercan be even reduced by the associativity condition. Proposition 8
There are semigroups that fit multiplication table 21 and resonant condition (16).
After studying different possibilities we found one way to fill the table (21). The proposedsemigroup is λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ That this multiplication represents an abelian semigroup can be checked again by direct calcu-lation.We point out again that there may be other semigroups that can also lead to the type V algebra.Those correspond to other ways to fill the empty spaces in table (21).12 .3 The type I algebra
Starting from the abelian 2-dimensional algebra[ X , X ] = 0 (23)we note that it also possesses the subspace structure (5) where V = { X } and V = { X } . So, forexample, by choosing the semigroup S (3) K or S N both having a resonant decomposition of the form S = { λ , λ , λ } (24) S = { λ , λ } we obtain that the reduction of the resonant subalgebra G red S,R = { λ X , λ X , λ X } (25)will have the following commutation relations:[ λ X , λ X ] = λ λ [ X , X ] = 0 (26)[ λ X , λ X ] = λ λ [ X , X ] = 0[ λ X , λ X ] = λ λ [ X , X ] = 0what means that it doesn’t matter if we use the semigroup S (3) K or S N , the result will be alwaysan abelian algebra in 3 dimensions because the original algebra is abelian. The same result can bereached with the semigroup S (2) E whose semigroup decomposition is similar to (24).Also, by using the semigroups S N , S N and probably others that have a resonant decompositionof the form S = { λ , λ } (27) S = { λ , λ , λ } we obtain to a reduction of the resonant subalgebra G red S,R = { λ X , λ X , λ X } whose commutation relations [ λ X , λ X ] = λ λ [ X , X ] = 0[ λ X , λ X ] = λ λ [ X , X ] = 0[ λ X , λ X ] = λ λ [ X , X ] = 0are again no more than the 3-dimensional abelian algebra.So we conclude that starting from (23) whatever semigroup with a zero element and that havea resonant decomposition of the form (24) or (27) will lead to the type I algebra. Moreover, thisresult can be generalized: Proposition 9
An abelian algebra in d dimensions can be obtained as an expansion of the abelianalgebra in -dimensions by using a semigroup with probably a zero element and a suitable resonantdecomposition. Note that a crucial property to relate a 3-dimensional algebra (whichever of the type I, II, III andV) with a 2-dimensional algebra is the existence of the resonant subalgebra and the 0 S -reduction.This is the only way to obtain three generators starting from two.13 .4 Brief summary Starting from [ X , X ] = 0 (28)it is possible to obtain the type I abelian algebra in three dimensions using many semigroups as forexample S (2) E , S (3) K , S N , S N , S N and probably others. Now starting from[ X , X ] = X (29)it is also possible to obtain the type I abelian algebra in three dimensions using for example asemigroup whose multiplication satisfies the condition ( i ) of section 4.1, i.e., whose table has theform λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ where the empty spaces must be filled with the corresponding conditions of associativity, resonantcondition, reduction condition, etc.The semigroups with which it is possible to generate the type I, II, III and V algebra startingfrom the 2-dimensional algebra (29) appear in the following table: Algebra Semigroup used
Type I whatever semigroup with 0-elementand a resonant decompositionType II S N and probably othersType III S (2) E , S (3) K , S N and probably othersType V S N and probably others (30)14here the mentioned semigroups are described in the following tableSemigroup Table of multiplicarion Resonant decomposition 0 S -element S (2) E λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ S = { λ , λ , λ } S = { λ , λ } λ S (3) K λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ S = { λ , λ , λ } S = { λ , λ } λ S N λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ S = { λ , λ , λ } S = { λ , λ } λ S N λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ S = { λ , λ } S = { λ , λ , λ } λ S N λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ S = { λ , λ } S = { λ , λ , λ } λ (31) -dimensional isome-tries , VIII and IX algebras Let’s consider for example the type IV algebra[ Y , Y ] = 0 , (32)[ Y , Y ] = Y , (33)[ Y , Y ] = Y + Y . (34)As the S -expansion method uses an induced bracket[ λ α X i , λ β X j ] = λ α λ β [ X i , X j ] = λ γ ( α,β ) [ X i , X j ]for the expanded algebra and considering that for our original algebra i, j = 1 , X , X ] = X ,
15e have that the first two relations (32,33) can easily be reproduced with some semigroup product,but to reproduce (34) we need a non-zero result as the first requirement. This means that we musthave a relation like [ λ α X , λ β X ] = λ α λ β [ X , X ] = λ γ ( α,β ) X . And here we can see that no matter which semigroup we choose, λ γ ( α,β ) will always be an element ofthe semigroup (it is closed) and therefore we will never be able to reproduce a sum of two generators.Now consider the type VI algebra.[ Y , Y ] = 0 , [ Y , Y ] = Y , [ Y , Y ] = hY , h = 0 , . Again the first two brackets could be reproduced by a certain semigroup, but for the third one wewould have something like [ λ α X , λ β X ] = λ γ ( α,β ) X and again, no matter which semigroup we choose, λ γ ( α,β ) will always be an element of the semigroupand we will never be able to reproduce semigroup element multiplied by a numeric factor. A similarargument can be used to show that type VII algebra cannot be obtained by the S -expansionprocedure.A mix of the above arguments can be applied to explain why it is impossible to obtain the typeVII and VIII algebras as an expansion of a 2-dimensional isometry.Finally, to show why it is also impossible to reproduce the type IX algebra[ Y , Y ] = Y , [ Y , Y ] = Y , [ Y , Y ] = Y we have to realize that the candidate for being the expanded algebra will have three commutationrelations of the form [ λ α X i , λ β X j ] = λ γ ( α,β ) [ X i , X j ]but where i, j takes the values 1 and 2. Therefore, in one of the three commutation relations oneindex will always be repeated leading to a vanishing bracket. So it is impossible to generate, bymeans of an S -expansion, a 3-dimensional algebra with the three brackets having a non- zero value.Thus, we conclude that these types of algebra, that cannot be obtained by an expansion of the2-dimensional isometries, are in some sense intrinsic in 3 dimensions. A common question when working with semigroups in section 4 is that of the existence of diversesemigroups given a multiplication table with some elements already chosen. We have, for example,a table like this one S ? λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ
16n principle there are 256 different symmetric matrices which fill this template, but not all ofthem will be semigroups because the multiplication table will not always be associative. Moreover,we have to select only those that satisfy a certain resonant condition . Finally, many of theseassociative tables will be isomorphic, so we only have to select those that are not.In what follows we find all the non isomorphic forms to fill the tables (13), (18) and (21) withthe mentioned conditions and show that all the semigroups given in table (30) (those semigroupsthat we have constructed by hand) are isomorphic to one of the semigroups given by the computerprogram com.f of [24].
The template is: S ? λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ By using computer programs, we have found that there are two non isomorphic ways of filling thistemplate such that: a ) the resulting table is an abelian semigroup and b ) the resonant decompositionis given by S = { λ , λ } , S = { λ , λ , λ } . (35)Those ways are: S II λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ; S II λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ . (36)Each of them is isomorphic to one of the semigroups of the list given by the program com.f of [24]for n = 4. We give this information in the following table,isomorphic to isomorphism S II ⇐⇒ S ( λ λ λ λ ) S II ⇐⇒ S ( λ λ λ λ ) (37)where the isomorphism denoted by ( λ a λ b λ c λ d ) means: change λ by λ a , λ by λ b , λ by λ c and λ by λ d . The semigroups S and S of the list given by the program com.f for n = 4 are: S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ; S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ . (38)It can be checked directly that applying the isomorphism ( λ λ λ λ ) to S obtains S II and applying the isomorphism ( λ λ λ λ ) to S obtains S II .17 .2 Type III The template is: S ? λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ We have found that there are 7 non-isomorphic ways of filling this template such that: a ) theresulting table is an abelian semigroup and b ) the resonant decomposition is given by S = { λ , λ , λ } , S = { λ , λ } . (39)Those ways are: S III λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ; S III λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ; S III λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ S III λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ; S III λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ; S III λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ S III λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ As before, each of these forms are isomorphic to one of the semigroups of the list given by theprogram com.f of [24] for n = 4. Those semigroups and the corresponding isomorphisms are givenin the following table: isomorphic to isomorphism S III ⇐⇒ S ( λ λ λ λ ) S III ⇐⇒ S ( λ λ λ λ ) S III ⇐⇒ S ( λ λ λ λ ) S III ⇐⇒ S ( λ λ λ λ ) S III ⇐⇒ S ( λ λ λ λ ) S III ⇐⇒ S ( λ λ λ λ ) S III ⇐⇒ S ( λ λ λ λ ) (40)where the semigroups S , S , S , S , S , S y S , of the list generated by the program com.f for n = 4, are explicitly given in the Appendix.18 .3 Type V The template is: S ? λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ In this case we have found that there is just one way of filling this template such that: a ) theresulting table is an abelian semigroup and b ) the resonant decomposition is given by S = { λ , λ } , S = { λ , λ , λ } . (41)This is: S V λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ This table is isomorphic to the semigroup S given in Appendix. The isomorphism is given by( λ λ λ λ ) . (42)Note that the semigroup S also permits us to obtain type III algebra. So we can ask, howcan the same semigroup lead at the same time to type III and V algebras? The reason is that thissemigroup has two different resonant decompositions, (35) and (39). Each of them permits us toextract different kinds of resonant subalgebras leading, after the reduction, to completely differentalgebras.
In the following table we summarize our results by specifying all the non-isomorphic semigroupsthat permit us to generate the type I, II, III and the V algebra starting from the 2-dimensionalalgebra (29):
Algebra Semigroup used
Type I many semigroups (see section 4.3)Type II S , S Type III S , S , S , S , S , S and S Type V S (43)For consistency we should prove that each semigroup of the table (30) (which we have constructedby hand in section 4) is isomorphic to one of the semigroups of table (43) that we have found byusing computer programs. This information is given in the following table:19somorphic to isomorphism S N ⇐⇒ S ( λ λ λ λ ) S N ⇐⇒ S ( λ λ λ λ ) S N ⇐⇒ S ( λ λ λ λ ) S (2) E ⇐⇒ S ( λ λ λ λ ) S (3) K ⇐⇒ S ( λ λ λ λ ) So, in this work we present a complete study about the possibility of relate, by means of anexpansion, the isometry algebras that act transitively in 2 and 3 dimensions. It was found thatsome isometries in 3 dimensions, specifically those of type I, II, III and V (according Bianchi’sclassification), can be obtained as expansions of the isometries in 2 dimensions. In general, there ismore than one possibility to obtain these results, i.e., it can happen that different semigroups willlead to the same expanded algebra. Also, it is shown that the other Bianchi the type IV, VI-IXalgebras cannot be obtained as an expansion from the isometry algebras in 2 dimensions. Thismeans that the first isometry algebras have properties that can be obtained from isometries in 2dimensions but the second set have properties that are in some sense intrinsic in 3 dimensions.The results obtained in this work are interesting, because even when 2 and 3-dimensional isome-try algebras are well known in the literature, the non-trivial relations we have found are somethingcompletely new. To perform extensions of Lie algebras by adding generators and where the originalalgebra is a subalgebra of the resulting ones it could be thought as a simple problem. This is forexample what happens with the algebras type III and V (see table (30) and (11)) where they aresimply two ways to extend the algebra (10) to a 3 dimensional one (it is direct to see that the algebra(10) is contained as a subalgebra in the algebras type III and V of table (11)). In the present workwe have obtained these results by means of the expansion procedure but, even further, new kindsof relations were found. This is the case of the algebras ”type II” (see table (30)) that was obtainedas an expansion of the algebra (10) and where this original algebra is not present as a subalgebra.This is why we refer to these relations between 2 and 3-dimensional as non-trivial relations. Theseresults are completely new and can just be reached by means of the S -expansion method .It is also interesting that to face this problem we had to look for other semigroups that havenot been used yet in the applications of the S -expansion method (the semigroups S N , S N , S N and S (3) K ). Their principal properties, as the problem of finding a resonant decomposition, werestudied for each of them. By using computing programs we have checked our results and solved theproblem in a complete way.In general, to understand whether two sets of algebras can be related by means of an expansionis very interesting problem from both, physical and mathematical point of view. In fact, manyphysical applications have been found in this context: for example, in [2] the M -algebra is obtained This in some sense tell us that the S -expansion method includes not only the contraction methods: contains alsosome (and probably all) kind of the extensions procedures of a Lie algebra. A formal study about this theoreticalresult is a work in progress (See ref. [26]). Note that the expansion method by using a parameter is equivalent to an S -expansion but using just one specificsemigroup, the semigroup S ( n ) E introduced in ref. [5]. Then using another semigroups will lead us to more generalexpansions and that’s why effectively this results can just be obtained via the S -expansion procedure.
20s an expansion of the osp (32/1) algebra. In fact, in [6] this result was re-obtained but via the S -expansion method which gives in addition the invariant tensors of the expanded algebra. Inthis way, in the mentioned reference, an eleven-dimensional gauge theory for the M -algebra wasconstructed. Another interesting application is [7] where (2+1)-dimensional Chern-Simons AdSgravity is obtained from the so-called ”‘exotic gravity”’ and [8] where Standard General Relativityis obtained from Chern-Simons Gravity. Finally, a generalization of the results presented herecan be useful to study isometries in higher dimensions, particularly, in applications related withisometries of black holes solutions. A first step on this direction is done in [25].We conclude by remarking that what remains in common in all the physical applications men-tioned above is the question: given two symmetry algebras, can they be related by means of somecontraction or expansion procedure? The method presented in this paper is very instructive toanswer on this question. If the answer is yes , there is a way to construct the semigroup thatgives the relation (as made in section 4). On the contrary, if the answer is no then it should beshown explicitly that there exist no expansion method that can reproduce this relation (as madein section 5). This mechanism can be developed to a general algorithm to study more complicatedcases, where the construction of the semigroups by hand (as we made it here) would be impossible.General criteria and the mentioned algorithm is a work in progress (see [26]). We are grateful to Patricio Salgado for many valuable discussions and for the establishing the task.I.K. was supported by Fondecyt (Chile) grants 1050512, 1121030 and by DIUBB grant (UBB,Chile) 102609. N. M. & F.N. thank R. D’Auria, M. Trigiante and L. Andrianopoli for their kindhospitality at Dipartamento di Fisica of Politecnico di Torino, where part of this work was done.N.M. is also grateful to the Comisi´on Nacional de Investigaci´on Cient´ıfica y Tecnol´ogica Conicyt(Chile) for financial support through a Becas-Chile grant. F.N. thanks CSIC for a JAE-predocgrant, cofunded by the European Social Fund.
In this appendix we give explicitly the multiplication tables of the semigroups that we have usedin this paper and that belongs to the list generated by the program com.f of [24] for n = 4. Thosesemigroups are: S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ; S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ; S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ; S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ; S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ; S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ ; S λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ λ References [1] M. Hatsuda and M. Sakaguchi, “Wess-Zumino term for the AdS superstring and generalizedInonu-Wigner contraction,” Prog. Theor. Phys. (2003) 853 [arXiv:hep-th/0106114].[2] J. A. de Azcarraga, J. M. Izquierdo, M. Picon and O. Varela, “Generating Lie and gauge freedifferential (super)algebras by expanding Maurer-Cartan forms and Chern-Simons supergrav-ity,” Nucl. Phys. B (2003) 185 [arXiv:hep-th/0212347].[3] J. A. de Azcarraga, J. M. Izquierdo, M. Picon and O. Varela, “Extensions, expansions,Lie algebra cohomology and enlarged superspaces,” Class. Quant. Grav. (2004) S1375[arXiv:hep-th/0401033].[4] J. A. de Azcarraga, J. M. Izquierdo, M. Picon and O. Varela, “Expansions of alge-bras and superalgebras and some applications,” Int. J. Theor. Phys. (2007) 2738[arXiv:hep-th/0703017].[5] F. Izaurieta, E. Rodriguez and P. Salgado, “Expanding Lie (super)algebras through Abeliansemigroups,” J. Math. Phys. (2006) 123512 [arXiv:hep-th/0606215].[6] F. Izaurieta, E. Rodriguez and P. Salgado, “Eleven-dimensional gauge theory for the M al-gebra as an Abelian semigroup expansion of osp(32/1),” Eur. Phys. J. C (2008) 675[arXiv:hep-th/0606225].[7] F. Izaurieta, E. Rodriguez, A. Perez and P. Salgado, “Dual Formulation of the Lie AlgebraS-expansion Procedure,” J. Math. Phys. (2009) 073511 [arXiv:0903.4712 [hep-th]].[8] F. Izaurieta, E. Rodriguez, P. Minning, P. Salgado and A. Perez, “Standard General Relativityfrom Chern-Simons Gravity,” Phys. Lett. B (2009) 213 [arXiv:0905.2187 [hep-th]].[9] R. Caroca, N. Merino and P. Salgado, “S-Expansion of Higher-Order Lie Algebras,” J. Math.Phys. , 013503 (2009) [arXiv:1004.5213 [math-ph]].[10] R. Caroca, N. Merino, A. Perez and P. Salgado, “Generating Higher-Order Lie Algebras byExpanding Maurer Cartan Forms,” J. Math. Phys. , 123527 (2009) [arXiv:1004.5503 [hep-th]].[11] R. Caroca, N. Merino, P. Salgado and O. Valdivia, “Generating infinite-dimensional algebrasfrom loop algebras by expanding Maurer-Cartan forms,” J. Math. Phys. Handbook of finite semigroup programs . Preprint[25] R. Caroca, M. Cataldo, I. Kondrashuk, N. Merino and P. Salgado, ”Four-dimensional metricas S-expansion of two-dimensional metric of negative curvature” Work in progress.[26] L. Andrianopoli, R. D’Auria, N. Merino, F. Nadal, M. Trigiante,