This communication is focused on the study of contractions of certain types of Lie algebras of low dimension, in order to address the implementation of the results to certain physical concepts, such as the boundary process by which
quantum mechanics contracts to classical mechanics. To do this, we consider in the first place the contractions of filiform Lie algebras, which were  introduced by M. Vergné in 1966, by using the psi and phi invariant functions, introduced in 2008 by Hrivnák and Novotny. These functions are also dealt with other types of algebra, as Heisenberg algebras among others, and we study the existing contractions between these algebras.Ministerio de Ciencia e InnovaciónJunta de Andalucí

Carriazo Rubio, Alfonso (Coordinador)

Escobar Rica, José María

Fuente Benito, Daniel de la (Coordinador)

Márquez García, María del Carmen (Coordinador)

Núñez Valdés, Juan

Pérez Fernández, Pedro

Romero Sarabia, Alfonso (Coordinador)

idUS. Depósito de Investigación Universidad de Sevilla

XI Encuentro Andaluz de GeometríaIMUS (Universidad de Sevilla), 15 de mayo de 2015, págs. 5358Contractions of certain Lie algebrasJosé María Escobar • Juan Núñez • Pedro Pérez-FernándezAbstract. This communication is focused on the study of contractions of certaintypes of Lie algebras of low dimension, in order to address the implementation ofthe results to certain physical concepts, such as the boundary process by whichquantum mechanics contracts to classical mechanics. To do this, we consider inthe ﬁrst place the contractions of ﬁliform Lie algebras, which were introduced byM. Vergné in 1966, by using the psi and phi invariant functions, introduced in2008 by Hrivnák and Novotny. These functions are also dealt with other typesof algebra, as Heisenberg algebras among others, and we study the existingcontractions between these algebras.1. IntroductionAt present, the study of certain physical concepts has signiﬁcantly increaseddue to its signiﬁcance in limit processes which allow us to relate Lie algebrasbetween themselves. These processes were ﬁrst investigated by Segal [5] in 1951and two are the better known examples of them. The ﬁrst of them involvesthe connection between classical mechanic and relativistic mechanic, with theirrespective Poincaré symmetry group and Galilean symmetry group. The secondone is the limit process by which quantum mechanic is contracted to classicalmechanic, when } → 0, which actually corresponds to a contraction of theHeisenberg algebra to the abelian algebra of the same dimension.For these reasons physical or mathematical contractions are of great interestnowadays, not only for their applications but for the proper study of their alge-braic properties. So, continuing with this research, the main goal of this paper isto study the proper contractions of certain types of Lie algebras, mainly ﬁliform,although other types of Lie algebras of lower dimensions are also considered. Todo this, we use the invariant functions ψ and ϕ, introduced in 2007 by Hrivnákand Novotný [4] as a tool.José María Escobar, pinchamate@gmail.comDepartamento de Geometría y Topología, Universidad de SevillaJuan Núñez, jnvaldes@us.esDepartamento de Geometría y Topología, Universidad de SevillaPedro Pérez-Fernández, pedropf@us.esDpto. de Física Aplicada III, Esc. Téc. Sup. de Ingeniería, Universidad de Sevilla54 J.M. Escobar, J. Núñez, P. Pérez-Fernández2. PreliminariesWe show in this section some preliminaries on ﬁliform Lie algebras, invariantfunctions in Lie algebras and proper contractions of Lie algebras. A completereview on Lie algebras can be consulted in [3].The lower central series of a Lie algebra g is deﬁned asg1 = g, g2 = [g1, g], . . . , gk = [gk−1, g], . . . (1)If there exists m ∈ N such that gm ≡ 0, then g is called nilpotent.The nilpotency index of g if the smallest integer c such that gc ≡ 0.An n-dimensional nilpotent Lie algebra g is said to be ﬁliform if it is veriﬁedthatdim gk = n− k, for all k ∈ {2, . . . , n}. (2)The only n-dimensional ﬁliform Lie algebra for n < 3 is the abelian.For n ≥ 3, it is always possible to ﬁnd a so-called adapted basis {e1, . . . , en}of g such that [e1, e2] = 0, [e1, ej ] = ej−1, for all j ∈ {3, . . . , n}, [e2, ej ] =[e3, ej ] = 0, for all j ∈ {3, . . . , n}. All these brackets are called brackets due toﬁliformity.If all the structure constants of a ﬁliform Lie algebra, with the exceptionof those corresponding to the brackets due to the ﬁliformity of the algebra, arezeros, then that ﬁliform Lie algebra g is called model.Now, we recall the deﬁnitions and main properties of invariant functions ψand ϕ, obtained by Hrivnák and Novotný [4] in 2007.Deﬁnition 2.1. Let g be a Lie algebra. An isomorphism d of g is called a(α, β, γ)-derivation of g if there exist α, β, γ ∈ C such thatαd[X,Y ] = β [dX, Y ] + γ [X, dY ], ∀X,Y ∈ gThe set of (α, β, γ)-derivations of g is denoted by Der(α,β,γ)g.Deﬁnition 2.2. The function ψg : C 7→ {0, 1, 2, ..., (dim g)2} deﬁned as(ψg)(α) = dimDer(α,1,1)gis called the ψg invariant function corresponding to the (α, β, γ)-derivations ofg.A κ-twisted cocycle (simply κ-cocycle ) is any c ∈ Cq(g, V ), with q ∈ N,verifying 0 =∑q+1i=1 (−1)i+1κiif(xi)c(x1, . . . , xˆi, . . . , xq+1) +∑q+1i,j=1,i<j(−1)i+jκijc([xi, xj ], x1, . . . , xˆi, . . . , xˆj , . . . , xq+1).Contractions of certain Lie algebras 55Deﬁnition 2.3. The ϕ invariant function corresponding to an n-dimensionalLie algebra g is the mappingϕ : C 7→ {0, 1, . . . , n2(n− 1)2} deﬁned by (ϕg)(α) = dim coc(1,1,1,α,α,α)g.Deﬁnition 2.4. Let g = (V, [, ]) be an n-dimensional Lie algebra and U :(0, 1] 7→ gl (V ) be an one-parameter mapping. If the limit[X,Y ]0 = l´ımε→0+U−1 (ε) [U(ε)X,U(ε)Y ]exists for all X,Y ∈ g, we say that g0 = (V, [, ]0) is an one-parameter contractionof the algebra g and we write g 7→ g0.Finally, with respect to contractions of Lie algebras, let us recallDeﬁnition 2.5. Let g = (V, [, ]) be an n-dimensional Lie algebra and U :(0, 1] 7→ gl (V ) be an one-parameter mapping. If the limit[X,Y ]0 = l´ımε→0+U−1 (ε) [U(ε)X,U(ε)Y ]exists for all X,Y ∈ g, we say that g0 = (V, [, ]0) is an one-parameter contractionof the algebra g and we write g 7→ g0.The contraction g 7→ g0 is said to be proper if g is not isomorphic to g0.The following results were shown in [3].If g0 is a proper contraction of the complex Lie algebra g, then1. dim Der (g) < dim Der (g0) .2. ψg ≤ ψg0 and ψg(1) < ψg0(1).3. ϕg ≤ ϕg0 and ϕ0g ≤ ϕ0g0.Moreover, it is satisﬁed that, in dimension 3, Condition 2 is a characterizationof proper contractions of g.3. Extending the deﬁnitions of the previously de-ﬁned invariant functions to the case of ﬁliformLie algebras. New results obtained.With the objective of extending the deﬁnitions of the invariant functionsψ and ϕ introduced by Hrivnák and Novotný [4] to the case of ﬁliform Lie56 J.M. Escobar, J. Núñez, P. Pérez-Fernándezalgebras of lower dimensions, authors have obtained in a previous paper thefollowing values of these invariant functions for the case of ﬁliform Lie algebrasof dimension 3, 4 and 5 (see [2])The ψ invariant function for 3-dimensional ﬁliform Lie algebras. Accordingto the notation used in [4], we have obtained that ψf3(α) = 6, for all α ∈ C.The ϕ invariant function for 3-dimensional ﬁliform Lie algebras. We haveobtained that ϕf3(λ) = 9, if λ = 0, and ϕf3(λ) = 8, if λ ∈ C \ {0}.The invariant functions ψ for the 4-dimensional ﬁliform Lie algebra f4.Wehave obtained ψf4(α) = 7, for all α ∈ C.The ϕ invariant functions for 4-dimensional ﬁliform Lie algebra. We havethat ϕf4(α) = 16, if α = 0, and ϕf4(α) = 18, if λ ∈ C \ {0}.Finally, in the case of the ψ invariant function for ﬁliform Lie algebras ofdimension 5, we have found that ψf5(α) = 9, for all α ∈ C.4. Some examples of proper contractions betweendiﬀerent types of algebrasIn this section we study proper contractions from ﬁliform Lie algebras oflower dimensions to diﬀerent types of algebras.4.1. Proper contractions of 3-dimensional ﬁliform Lie al-gebrasNow, by using some results got by Novotný and Hrivnák in [3], we ha-ve obtained the invariant function ψ for some particular Lie algebras. Indeed,Contractions of certain Lie algebras 57g1: Abelian Lie algebraα ∀α ∈ Cψ3g1(α) 9g3,1: [e2, e3] = e1α ∀α ∈ Cψg3,1(α) 6g2,1 ⊕ g1 : [e1, e2] = e2 α ∀α ∈ C \ {0} 0ψg2,1⊕g1(α) 4 6g3,2 : [e1, e3] = e1, [e1, e3] = e1 + e2α 1 ∀α ∈ C \ {1}ψg3,2(α) 4 3g3,3 : [e1, e3] = e1, [e2, e3] = e2α 1 ∀α ∈ C \ {1}ψg3,3(α) 6 34.2. The invariant function ψH5The computation of the invariant functions of ﬁliform Lie algebras of dimen-sion odd allows us to compare these algebras with Heisenberg algebras (whichare only deﬁned for these dimensions).With respect to the invariant function ψH5 we have obtained that ψH5(α) =15, for all α ∈ C.5. ConclusionsAs ψf3 ≤ ψg1 and ψf3(1) < ψg1(1), a previous result assures the existen-ce of a proper contraction from f3 to g1. Analogously, the same occursbetween g3,2 and f3 since ψg3,2 ≤ ψf3 and ψg3,2(1) < ψf3(1).Note that ψf3(1) = 6 and ψg3,1(1) = 6. Therefore, the same result assuresthat there is not any proper contraction between g3,1 and f3. Similarly, thesame occurs between g3,3 and f3 due to that ψf3(1) = 6 and ψg3,3(1) = 6.We have also computed the invariant function ψ for the 5-dimensionalHeisenberg algebra and have proved that there is not a contraction fromthe 5-dimensional ﬁliform Lie algebra to the Heisenberg algebra of thisdimension.Therefore, we can conclude that the 5-dimensional classical-mechanicalmodel built upon a 5-dimensional ﬁliform Lie algebra cannot be obtai-58 J.M. Escobar, J. Núñez, P. Pérez-Fernándezned as a limit process of a quantum-mechanical model based on a 5-dimensional Heisenberg algebra.AcknowledgementThe authors gratefully acknowledge ﬁnancial support by the Spanish Minis-terio de Ciencia e Innovación and Junta de Andalucía via grants No. MTM2013-40455-P and No. FQM-326 (J.N.), respectively, and No. FIS2011-28738-C02-01,FIS2014-53448-C2-1-P and No. FQM160, respectively, (P.P.-F.).References[1] D. Burde, Degenerations of 7-dimensional nilpotent Lie algebras Comm.Algebra 33 (2005), 12591277.[2] J. M. Escobar, J. Núñez, P. Pérez-Fernández, Invariant functions and con-tractions of certain types of Lie algebras of lower dimensions. Submitted.[3] J. E. Humphreys. Introduction to lie algebras and representation theory.Springer-Verlag, New York, 1972.[4] P. Novotný, J. Hrivnák, On (α, β, γ)-derivations of Lie algebras and co-rresponding invariant functions Journal of Geometry and Physics 58(2008), no. 3, 208217.[5] I. E. Segal, A class of operator algebras which are determined by groupsDuke Math. J. 18 (1951), 221265.

Contractions of certain Lie algebras

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Contractions of certain Lie algebras

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