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The family of quaternionic quasi-unitary Lie algebras and their central extensions



arXiv:math-ph/9811010v1 13 Nov 1998

The family of quaternionic quasi-unitary Lie algebras and their central extensions

Francisco J. Herranz? and Mariano Santander? Departamento de F? ?sica, E.U. Polit?cnica e Universidad de Burgos, E–09006 Burgos, Spain
? ?

Departamento de F? ?sica Te?rica, Universidad de Valladolid o E–47011, Valladolid, Spain

Abstract The family of quaternionic quasi-unitary (or quaternionic unitary Cayley– Klein algebras) is described in a uni?ed setting. This family includes the simple algebras sp(N + 1) and sp(p, q) in the Cartan series CN +1 , as well as many non-semisimple real Lie algebras which can be obtained from these simple algebras by particular contractions. The algebras in this family are realized here in relation with the groups of isometries of quaternionic hermitian spaces of constant holomorphic curvature. This common framework allows to perform the study of many properties for all these Lie algebras simultaneously. In this paper the central extensions for all quasi-simple Lie algebras of the quaternionic unitary Cayley–Klein family are completely determined in arbitrary dimension. It is shown that the second cohomology group is trivial for any Lie algebra of this family no matter of its dimension.

1

Introduction

This paper is devoted to a double purpose. First, it introduces and describes the structure of a family of Lie algebras, the quaternionic quasi-unitary algebras, or quaternionic unitary Cayley–Klein algebras, which include as simple members the algebras in the Cartan series CN +1 which in the standard notation are written as sp(p, q), p + q = N + 1, as well as many non-simple members which can be obtained from the former by a sequence of contractions. The description is also done in relation to the symmetric homogeneous spaces (the quaternionic hermitian spaces of rank one) where these groups act in a natural way. The second and main purpose is to investigate the Lie algebra cohomology of the algebras in this Cayley–Klein (hereafter CK) family, in any dimension. These extensions have both mathematical interest and physical relevance. Therefore, this part of the paper can be considered as a further step in a systematic study of properties of the these families of Lie algebras [1]–[8], by using a formalism which allows a clear view of the behaviour of these properties under contraction; in physical terms contractions are related to some kind of approximation. In particular, the central extensions of algebras in the two other main CK families of Lie algebras (the quasi-orthogonal algebras and the two families of quasi-unitary algebras) have been studied in two previous papers, in the general situation and for any dimension [7], [8]. We refer to these works for references and for physical motivations. The knowledge of the second cohomology group for a Lie algebra relies on the general solution of a set of linear equations, but in special cases the calculations may be bypassed by using some general results: for instance, the second cohomology group is trivial for semisimple Lie algebras. But once a contraction is made, the semisimple character disappears, and the contracted algebra might have non-trivial central extensions. Instead of ?nding the general solution for the extension equations on a case-by-case basis, our approach (as developed previously for the quasi-orthogonal algebras [7] and for the quasi-unitary algebras [8]) is to do these calculations for a whole family including a large number of algebras simultaneously. In this paper we discuss the ‘next’ family: the quaternionic quasi-unitary one. The advantages in this approach can be summed up in: a) it allows to record, in a form easily retrievable, a large number of results which can be needed in applications, both in mathematics and in physics, and b) it avoids at once and for all the case-by-case type computation of the central extensions of algebras included in each family and a?ords a global view on the interrelations between cohomology and contractions. Section 2 is devoted to the description of the family of quaternionic unitary CK algebras. We show how to obtain these as graded contractions of the compact algebra u(N + 1, H) ≡ sp(N + 1), and we provide some details on their structure. These algebras are associated to the quaternionic hermitian spaces (of rank one) with metrics of di?erent signatures and to their contractions, so we devote a part of this section to dwell upon these questions. In section 3 the general solution to the central extension problem for these algebras is given. The result obtained is 2

quite simple to state: all the extensions of any algebra in the quaternionic unitary CK family are trivial. This triviality is already known (Whitehead’s lemma) for the simple algebras u(p, q, H) ≡ sp(p, q)) in this family, but comes as a surprise for the rather large number of non-semisimple Lie algebras in this CK family, which can be obtained by contracting u(p, q, H). This is also in marked contrast with the results for the central extensions of both the orthogonal and the unitary CK families, where some algebras (particularly the most contracted one) always allow some non-trivial extensions. Finally, some remarks close the paper.

2

The family of quaternionic unitary CK algebras

To begin with we consider the compact real form of the Lie algebra in the Cartan series CN +1 . This compact real form can be realized as the Lie algebra of the complex unitary-symplectic group sometimes denoted as USp(2(N + 1)) [9] but more usually referred to shortly as the ‘symplectic’ group, Sp(N + 1). The usual convention is to denote this group without any reference to a ?eld to avoid confusion with the true symplectic groups over either the reals Sp(2(N + 1), R) or over the complex numbers Sp(2(N +1), C); in these last cases the term symplectic is properly associated to the symmetry group of an antisymmetric metric. This double use of the name ‘symplectic’ and of the symbols Sp and sp is rather unfortunate, and following Sudbery [10], we shall change the symbol for one of the families, and use Sq, sq for the unitary-symplectic groups and algebras usually denoted, without any ?eld reference, by Sp, sp. The group Sq(N + 1) ≡ USp(2(N + 1)) is the intersection of the complex unitary group U(2(N + 1), C) and the complex symplectic group Sp(2(N + 1), C): Sq(N + 1) ≡ USp(2(N + 1)) = U(2(N + 1), C) ∩ Sp(2(N + 1), C), which is a consequence of the nature of Sq(N + 1) as the group of quaternionic matrices leaving invariant a quaternionic hermitian de?nite positive metric. We recall that all other non-compact real forms in the Cartan series CN +1 are the real symplectic algebra sp(2(N + 1), R), and the quaternionic pseudo-unitary algebras sq(p, q), p + q = N + 1, which allow a realization as Sq(p, q) ≡ USp(2p, 2q) = U(2p, 2q, C) ∩ Sp(2(N + 1), C), and they are the groups of quaternionic matrices leaving invariant a quaternionic hermitian metric of signature (p, q). The Lie algebra sq(N + 1) has dimension 2(N + 1)2 + (N + 1) and is usually realized by 2(N + 1) × 2(N + 1) complex matrices [9, 11]. The alternative realization which we shall consider in this paper, in accordance to the interpretation of these groups and algebras as quaternionic unitary ones Sq(N + 1) ≡ U(N + 1, H) [12], is done by means of antihermitian matrices over the quaternionic skew ?eld H: Jab = ?eab + eba
α Mab = iα (eab + eba ) α Ea = iα eaa

(2.1)

3

where a < b, a, b = 0, 1, . . . , N, α = 1, 2, 3; i1 = i, i2 = j, i3 = k are the usual quaternionic units, and eab is the (N + 1) × (N + 1) matrix with a single 1 entry in α row a, column b. Notice that the matrices Jab and Mab are traceless, but the trace α of Ea is a non-zero pure imaginary quaternion, so the realization is by antihermitian quaternionic matrices whose trace has a zero real part. When quaternions are realized as 2 × 2 complex matrices (see e.g. [13]) then (2.1) reduces to the usual realization of sq(N + 1) by complex matrices 2(N + 1) × 2(N + 1) which are at the same time complex unitary and complex symplectic; we remark that all these matrices are traceless. The multiplication of quaternionic units is encoded in iα iβ = ?δαβ + 3 εαβγ iγ γ=1 where εαβγ is the completely antisymmetric unit tensor with ε123 = 1. This relation allows to derive the expression for the Lie bracket of two pure quaternionic matrices X α = iα X, Y β = iβ Y , where X, Y are real matrices, as
3

[X , Y ] = ?δαβ [X, Y ] +
γ=1

α

β

εαβγ iγ {X, Y }

(2.2)

where both the commutator and the anticommutator {X, Y } = XY +Y X of the real matrices X, Y appear. Using this formula, the commutation relations of sq(N + 1) in the basis (2.1) read [Jab , Jac ] = Jbc [Jab , Jbc ] = ?Jac [Jac , Jbc ] = Jab α α α α α α [Mab , Mac ] = Jbc [Mab , Mbc ] = Jac [Mac , Mbc ] = Jab α α α α α α [Jab , Mac ] = Mbc [Jab , Mbc ] = ?Mac [Jac , Mbc ] = ?Mab α α α α α α [Mab , Jac ] = ?Mbc [Mab , Jbc ] = ?Mac [Mac , Jbc ] = Mab α α α [Jab , Jde ] = 0 [Mab , Mde ] = 0 [Jab , Mde ] = 0 α α α α [Jab , Ed ] = (δad ? δbd )Mab [Mab , Ed ] = ?(δad ? δbd )Jab α α α α α [Jab , Mab ] = 2(Eb ? Ea ) [Ea , Eb ] = 0
γ β β γ α β α γ α [Mab , Mac ] = εαβγ Mbc [Mab , Mbc ] = εαβγ Mac [Mac , Mbc ] = εαβγ Mab β γ β α α γ [Mab , Mab ] = 2εαβγ (Ea + Eb ) [Mab , Mde ] = 0 β γ β α α γ [Mab , Ed ] = (δad + δbd )εαβγ Mab [Ea , Eb ] = 2δab εαβγ Ea

(2.3)

(2.4)

where hereafter the following notational conventions are assumed: ? Whenever three indices a, b, c appear, they are always assumed to verify a < b < c. ? Whenever three indices a, b, d appear, a < b is assumed but the index d is arbitrary, and it might coincide with either a or b. ? Whenever four indices a, b, d, e appear, a < b, d < e and all of them are assumed to be di?erent. ? Whenever three quaternionic indices α, β, γ appear, they are also assumed to be di?erent (so they are always some permutation of 123). 4

? There is no any implied sum over repeated indices; in particular there is no sum in γ in expressions like εαβγ X γ . This matrix realization of the Lie algebra sq(N + 1) displays clearly the existence of several subalgebras. By one hand, the 1 N(N + 1) generators Jab (a, b = 2 0, 1, . . . , N) close an orthogonal algebra so(N +1) whose non-zero commutation rules are written in the ?rst row of (2.3). On the other hand, for each ?xed α = 1, 2, 3, the α α (N + 1)2 generators {Jab , Mab , Ea } (a, b = 0, 1, . . . , N; a < b) give rise to an algebra isomorphic to the unitary algebra u(N + 1) with commutators given by (2.3); these subalgebras we denote as uα (N + 1). Hence sq(N + 1) contains three subalgebras isomorphic to u(N + 1), whose intersection is a subalgebra so(N + 1). The family of algebras we study in this paper can be obtained as graded contractions [14, 15] from sq(N + 1). The algebra sq(N + 1) can be endowed with a grading by a group Z?N constituted by 2N involutive automorphisms SS de?ned by 2 SS Jab = (?1)χS (a)+χS (b) Jab α α SS Mab = (?1)χS (a)+χS (b) Mab
α α SS Ea = Ea

α = 1, 2, 3;

(2.5)

where S denotes any subset of the set of indices {0, 1, . . . , N}, and χS (a) denotes the characteristic function over S. A particular solution of the Z?N graded contractions 2 of sq(N + 1) leads to a family of Lie algebras which are called quaternionic unitary CK algebras or quaternionic quasi-unitary Lie algebras [2, 3]. This family comprises the simple quaternionic unitary and pseudo-unitary algebras sq(p, q) (p + q = N + 1) in the Cartan series CN +1 as well as many non-simple real Lie algebras which can be obtained from the former by contractions. Collectively, all these algebras preserve some properties related to simplicity, so they belong to the class of so-called ‘quasisimple’ Lie algebras [16, 17], which explains the use of the pre?x quasi in their name. Overall this is very similar to the situation of the families of quasi-orthogonal algebras (with so(N + 1) as the initial Lie algebra [1, 4]) or to the families of quasiunitary or quasi-special unitary algebras over the complex numbers (starting from either u(N + 1) or su(N + 1) [8]). The quaternionic unitary CK algebras can be described by means of N real coe?cients ωa (a = 1, . . . , N) and are denoted collectively as sqω1 ,...,ωN (N + 1), or in an abbreviated form, as sqω (N +1) where ω stands for ω = (ω1 , . . . , ωN ). Introducing the two-index coe?cients ωab as ωab := ωa+1 ωa+2 · · · ωb a, b = 0, 1, . . . , N a<b ωaa := 1 (2.6) then the commutation relations of the generic CK algebra in the family sqω (N + 1) are given by [2] [Jab , Jac ] = ωab Jbc [Jab , Jbc ] = ?Jac [Jac , Jbc ] = ωbc Jab α α α α α α [Mab , Mac ] = ωab Jbc [Mab , Mbc ] = Jac [Mac , Mbc ] = ωbc Jab α α α α α α [Jab , Mac ] = ωab Mbc [Jab , Mbc ] = ?Mac [Jac , Mbc ] = ?ωbc Mab α α α α α α [Mab , Jac ] = ?ωab Mbc [Mab , Jbc ] = ?Mac [Mac , Jbc ] = ωbc Mab α α α [Jab , Jde ] = 0 [Mab , Mde ] = 0 [Jab , Mde ] = 0 α α α α [Jab , Ed ] = (δad ? δbd )Mab [Mab , Ed ] = ?(δad ? δbd )Jab α α α α α [Jab , Mab ] = 2ωab (Eb ? Ea ) [Ea , Eb ] = 0 5

(2.7)

γ β β γ α β α γ α [Mab , Mac ] = ωab εαβγ Mbc [Mab , Mbc ] = εαβγ Mac [Mac , Mbc ] = ωbc εαβγ Mab β β γ α α γ [Mab , Mde ] = 0 [Mab , Mab ] = 2ωab εαβγ (Ea + Eb ) β γ β α α γ [Mab , Ed ] = (δad + δbd )εαβγ Mab [Ea , Eb ] = 2δab εαβγ Ea (2.8) where we adhere to the notational conventions given after (2.4).

The pattern of subalgebras previously discussed for the compact form sq(N + 1) clearly holds for any member of the complete family. The quaternionic unitary CK algebra sqω (N + 1) contains also as Lie subalgebras an orthogonal CK algebra soω (N +1) [2, 7] and three unitary CK algebras uα (N +1) [2, 8] where α = 1, 2, 3; the ω commutation relations of the former correspond to the ?rst row of (2.7) and those of the latter are given by (2.7) (for an index α ?xed). Hence we ?nd the sequence soω (N + 1) ? uα (N + 1) ? sqω (N + 1). ω (2.9)

2.1

The quaternionic unitary CK groups

The matrix realization (2.1) allows a natural interpretation of the quaternionic unitary CK algebras as the Lie algebras of the motion groups of the homogeneous symmetric spaces with a quaternionic hermitian metric (the two-point homogeneous spaces of quaternionic type and rank one). Let us consider the space HN +1 endowed with a hermitian (sesqui)linear form . | . ω : HN +1 × HN +1 → H de?ned by
N

a|b

ω

:= a0 b0 + a1 ω1 b1 + a2 ω1 ω2 b2 + . . . + aN ω1 · · · ωN bN = ? ? ? ?
i=0

ai ω0i bi ?

(2.10)

where a, b ∈ HN +1 and ai means the quaternionic conjugation of the component ai . ? For the moment, we assume that we are in the generic case with all ωa = 0. The underlying metric is provided by the matrix Iω = diag (1, ω01 , ω02 , . . . , ω0N ) = diag (1, ω1 , ω1 ω2 , . . . , ω1 · · · ωN ) (2.11)

and the CK group Sqω1 ,...,ωN (N + 1) ≡ Sqω (N + 1) is de?ned as the group of linear isometries of this hermitian metric over a quaternionic space. Thus the isometry condition for an element U of the Lie group Ua|Ub leads to the following relation U ? Iω U = Iω which for the Lie algebra implies X ? Iω + Iω X = 0 ?X ∈ sqω (N + 1). (2.14) ?U ∈ Sqω (N + 1) (2.13)
ω

= a|b

ω

? a, b ∈ HN +1 ,

(2.12)

From this equation, it is clear that the quaternionic unitary CK algebra is generated by the following (N + 1) × (N + 1) Iω -antihermitian matrices over H (cf. (2.1)) Jab = ?ωab eab + eba
α Mab = iα (ωab eab + eba ) α Ea = iα eaa .

(2.15)

6

These matrices can be checked to satisfy the commutation relations (2.7) and (2.8). When any of the constants ωa are equal to zero, then the set of linear isometries of the hermitian metric over the quaternions (2.12) is larger than the group generated by (2.15), though in these cases there exists additional geometric structures in HN +1 , which are related to the existence of invariant foliations, and the proper de?nition of the automorphism group for these structures leads again to the matrix Lie algebra generated by (2.15) with commutation relations (2.7) and (2.8). The action of the group Sqω (N + 1) in HN +1 is not transitive, and the ‘sphere’ with equation
N

x|x

ω

:=
i=0

xi ω0i xi = 1 ?

(2.16)

is stable. However, if we take O = (1, 0, . . . , 0) as a reference point in this sphere, the realization (2.15) shows that the isotropy subgroup of O is Sqω2 ,ω3 ,...,ωN (N), and the isotropy subgroup of the ray of O is Sq(1) ? Sqω2 ,ω3 ,...,ωN (N) (note that the quaternions being non-commutative, a choice for left or right multiplication for scalars is required). Here the algebra sq(1) of the subgroup Sq(1) can be identi?ed with the Lie algebra of automorphisms of the quaternions, generated by the three matrices
N

I = iα
a=0

α

eaa

α = 1, 2, 3

(2.17)

which can be identi?ed to the three quaternionic units. We note in passing that these are the elements of the Lie algebra which are unavoidably realized by matrices α with non-zero pure imaginary trace, as all the generators Ea can be expressed in α α α terms of zero trace combinations (say Bl ≡ El?1 ? El , l = 1, . . . , N) and the three I α . In this way we ?nd the quaternionic hermitian homogeneous spaces as associated to the quaternionic unitary family of CK groups: Sqω1 ,ω2 ,ω3 ,...,ωN (N + 1)/(Sq(1) ? Sqω2 ,ω3 ,...,ωN (N)), (2.18)

For ?xed ω1 , ω2 , ω3 , . . . , ωN this space, which has real dimension 4N, has a natural real quadratic metric (either riemannian, pseudoriemannian or degenerate ‘riemannian’), coming from the real part of the quaternionic hermitian product in the ambient space. At the origin and in an adequate basis, this metric is given by the diagonal matrix with entries (1, ω2 , ω2 ω3 , . . . , ω2 · · · ωN ), each entry repeated four times. The three well known hermitian elliptic, euclidean and hyperbolic quaternionic spaces, of constant holomorphic curvature 4K (either K > 0, K = 0 and K < 0 respectively) appear in this family as associated to the special values ω1 = K and ω2 = ω3 = . . . = ωN = 1, where the metric is riemannian (de?nite positive). All CK hermitian spaces of quaternionic type with ω1 = K have constant holomorphic curvature 4K and the signature (and/or the eventual degeneracy) of the metric is determined by the remaining constants ω2 , ω3 , . . . , ωN . When all these constants are di?erent from zero, but some are negative, the metric is pseudoriemannian (inde?nite and not degenerate), and when some of the constants ω2 , ω3 , . . . , ωN vanish the metric is degenerate. 7

2.2

Structure of the quaternionic unitary CK algebras

As each real coe?cient ωa can be positive, negative or zero, the quaternionic unitary CK family sqω (N + 1) includes 3N Lie algebras. Semisimple algebras appear when all the coe?cients ωa are di?erent from zero: these are the algebras sq(p, q) in the Cartan series CN +1 , where p and q (p + q = N + 1) are the number of positive and negative terms in the matrix Iω (2.11). If we set all ωa = 1 we recover the initial compact algebra sq(N + 1). When one or more coe?cients ωa vanish the CK algebra turns out to be a non-semisimple Lie algebra; the vanishing of one (or several) coe?cient ωa is equivalent to performing an (or series of) In¨n¨ –Wigner o u contraction [18, 19]. Some of the quaternionic unitary CK algebras are isomorphic; for instance, the isomorphism sqω1 ,ω2 ,...,ωN?1 ,ωN (N + 1) ? sqωN ,ωN?1 ,...,ω2 ,ω1 (N + 1) (that interchanges ωab ? ωN ?b,N ?a ) is provided by the map
′ Jab → Jab = ?JN ?b,N ?a 1 ′1 2 Mab → Mab = ?MN ?b,N ?a 2 ′2 1 Mab → Mab = ?MN ?b,N ?a 3 ′3 3 Mab → Mab = ?MN ?b,N ?a 1 ′1 2 Ea → Ea = ?EN ?a 2 ′2 1 Ea → Ea = ?EN ?a 3 ′3 3 Ea → Ea = ?EN ?a .

(2.19)

(2.20)

Each algebra in the family of quaternionic unitary CK algebras has many subalgebras isomorphic to orthogonal, unitary, or special unitary CK algebras, as well as many subalgebras isomorphic to quaternionic unitary algebras in the family sqω (M + 1) with M < N. A clear way to describe this is to denote by Xab the α α four generators {Jab , Mab } (α = 1, 2, 3), by Ea the set of three generators Ea , and arrange the basis generators of sqω (N + 1) as follows: E0 X01 E1 X02 X12 .. . ... ... Ea?2 X0 a?1 X1 a?1 . . . Xa?2 a?1 Ea?1 X0a X1a . . . Xa?2 a Xa?1 a Ea X0 a+1 X1 a+1 . . . Xa?2 a+1 Xa?1 a+1 Xa a+1 .. . ... ... ... ... ... EN ?1 X0N X1N . . . Xa?2 N Xa?1 N XaN . . . XN ?1 N EN

3 3 3 A Cartan subalgebra is made up of the N + 1 generators E0 , E1 , . . . , EN (in the outermost diagonal). In this arrangement the generators to the left and below the rectangle span subalgebras sqω1 ,...,ωa?1 (a) and sqωa+1 ,...,ωN (N + 1 ? a) respectively, while the generators inside the rectangle do not span a subalgebra unless ωa = 0

8

(and in this case this is an abelian subalgebra). The unitary subalgebras uα (N + 1) ω α appear in a similar way by keeping only Jab , a single Mab out of each Xab and a single α Ea out of each set Ea (for a ?xed α). By keeping only Jab we get the soω (N + 1) subalgebra. If a coe?cient ωa = 0, then the contracted algebra has a semidirect structure sqω1 ,...,ωa?1 ,ωa =0,ωa+1,...,ωN (N +1) ≡ t⊙(sqω1 ,...,ωa?1 (a)⊕sqωa+1 ,...,ωN (N +1?a)) (2.21) where t is spanned by the generators inside the rectangle (it is an abelian subalgebra of dimension 4a(N + 1 ? a)), while sqω1 ,...,ωa?1 (a) and sqωa+1 ,...,ωN (N + 1 ? a) are two quaternionic unitary CK subalgebras spanned by the generators in the triangles to the left and below the rectangle. When there are several coe?cients ωa = 0 the contracted algebra has simultaneously several semidirect structures (2.21). Notice that when ω1 = 0 the contracted algebra has the structure sq0,ω2 ,...,ωN (N + 1) ≡ t4N ⊙ (sq(1) ⊕ sqω2 ,...,ωN (N)) (2.22)

and here the subindex 4N in t is the real dimension of the ?at homogeneous space (2.18) which can be identi?ed with HN endowed with a ?at metric given, over H, by the diagonal matrix (1, ω2 , ω2 ω3 , . . . , ω2ω3 · · · ωN ); when all these are positive this Lie algebra can be called inhomogeneous quaternionic unitary algebra isq(N).

3

Central extensions

After having described the structure of the quaternionic unitary CK algebras, we now turn to the second goal of this paper: to give a complete description of all possible central extensions of the algebras in the quaternionic unitary CK family. The outcome of this study is simple to state: in any dimension, and for all quaternionic unitary CK algebras –no matter of how many ωa are equal or di?erent from zero–, there are no non-trivial central extensions. For any r-dimensional Lie algebra with generators {X1 , . . . , Xr } and structure k constants Cij , a generic central extension by the one-dimensional algebra generated by Ξ will have (r + 1) generators (Xi , Ξ) with commutation relations given by
r

[Xi , Xj ] =
k=1

k Cij Xk + ξij Ξ

[Ξ, Xi ] = 0.

(3.1)

The extension coe?cients or central charges ξij must be antisymmetric in the indices i, j, ξji = ?ξij and must ful?l the following conditions coming from the Jacobi identities for the generators Xi , Xj , Xl in the extended Lie algebra:
r k k k Cij ξkl + Cjl ξki + Cliξkj = 0. k=1

(3.2)

9

If for a set of arbitrary real numbers ?i we perform a change of generators: Xi → Xi′ = Xi + ?i Ξ, (3.3)

the commutation rules for the generators {Xi′ } are given by the expressions (3.1) k k ′ with a new set of ξij = ξij ? r Cij ?k , where δ?(Xi , Xj ) = r Cij ?k is the twok=1 k=1 coboundary generated by ?. Extension coe?cients di?ering by a two-coboundary correspond to equivalent extensions; and those extension coe?cients which are a k two-coboundary ξij = ? r Cij ?k correspond to trivial extensions; the classes of k=1 equivalence of non-trivial two-cocycles determine the second cohomology group of the Lie algebra.

3.1

Central extensions of the unitary CK subalgebras

In order to simplify further computations, we ?rst state the result about the structure of the central extensions of the unitary CK algebra uω (N + 1)[8], which will naturally appear when studying the extensions of the quaternionic unitary CK algebras, because each sqω (N + 1) contains three such unitary CK subalgebras. Theorem 3.1. The commutation relations of any central extension uα (N + 1) of the unitary CK ω α α algebra uα (N +1) with generators {Jab , Mab , Ea } (a, b = 0, 1, . . . , N and quaternionic ω index α ?xed) by the one-dimensional algebra generated by Ξ are
α α [Jab , Jac ] = ωab (Jbc + hα Ξ) [Mab , Mac ] = ωab (Jbc + hα Ξ) bc bc α α [Jab , Jbc ] = ?(Jac + hα Ξ) [Mab , Mbc ] = Jac + hα Ξ ac ac α α [Jac , Jbc ] = ωbc (Jab + hα Ξ) [Mac , Mbc ] = ωbc (Jab + hα Ξ) ab ab α α [Jab , Jde ] = 0 [Mab , Mde ] = 0 α α α α α α [Jab , Mac ] = ωab (Mbc + gbc Ξ) [Mab , Jac ] = ?ωab (Mbc + gbc Ξ) α α α α α α [Jab , Mbc ] = ?(Mac + gac Ξ) [Mab , Jbc ] = ?(Mac + gac Ξ) α α α α α α [Jac , Mbc ] = ?ωbc (Mab + gab Ξ) [Mac , Jbc ] = ωbc (Mab + gab Ξ) α α α α [Jab , Ed ] = (δad ? δbd )(Mab + gab Ξ) [Jab , Mde ] = 0 α α α [Mab , Ed ] = ?(δad ? δbd )(Jab + hab Ξ) α α α α [Jab , Mab ] = 2ωab (Eb ? Ea ) + fab Ξ α α [Ea , Eb ] = eα Ξ a,b

(3.4)

a<b

(3.5)

where
α fab =

b α ωa s?1 ωsbfs?1 s . s=a+1

(3.6)

The extension is characterized by the following types of extension coe?cients:
α Type I: N(N + 1)/2 coe?cients gab and N(N + 1)/2 coe?cients hα (a < b and ab a, b = 0, 1, . . . , N). α Type II: N coe?cients fa?1 a (a = 1, . . . , N).

10

Type III: N(N + 1)/2 coe?cients eα (a < b and a, b = 0, 1, . . . , N), satisfying a,b ωab eα = 0 a,b ωab (eα ? eα ) = 0 a,c b,c ωbc (eα ? eα ) = 0 a,b a,c a < b < c. (3.7)

This theorem expresses the results previously obtained in [8] but in a di?erent basis (we are using here a di?erent set of Cartan generators) so that we use another notation for the extension coe?cients. The extension coe?cients are classed into types according as their behaviour under contraction. All type I coe?cients correspond to central extensions which are trivial for all the unitary CK algebras, no matter of how many coe?cients ωa are equal to zero, since they can be removed at once by means of the rede?nitions Jab → Jab + hα Ξ ab
α α α Mab → Mab + gab Ξ.

(3.8)

α Each type II coe?cient fa?1 a gives rise to a non-trivial extension if ωa = 0 and to a trivial one otherwise. That is, these extensions become non-trivial through the contraction and they behave as pseudoextensions [20, 21]. On the other hand, when a type III coe?cient eα is non-zero, the extension that it determines is always a,b non-trivial so that it cannot appear through a pseudoextension process. Therefore, the only extensions which can be non-trivial for a given algebra in the CK family uω (N + 1) are those appearing in the Lie brackets (3.5).

We also recall that the dimension of the second cohomology group of a unitary CK algebra uω (N + 1) with n coe?cients ωa equal to zero is dim (H 2 (uω (N + 1), R) = n + n(n + 3) n(n + 1) = 2 2 (3.9)

α where the ?rst term n corresponds to the extension coe?cients fa?1 a and the second term n(n+1) to the extensions determined by eα . a,b 2

3.2

Central extensions of the quaternionic unitary CK algebras

In the sequel we determine the non-trivial extension coe?cients ξij for a generic centrally extended quaternionic unitary CK algebra sq ω (N + 1) (3.1) by solving the Jacobi identities (3.2). First, we consider a generic extended unitary CK subalgebra, say u1 (N + 1), ω 1 1 spanned by the generators {Jab , Mab , Ea , Ξ} (a, b = 0, 1, . . . , N; a < b) with pure quaternionic index equal to 1. It is clear that the set of Jacobi identities involving only these generators lead to the results given in the theorem 3.1. Hence, we ?nd 1 the commutation relations (3.4) and (3.5) with extension coe?cients denoted gab , 1 h1 , fab and e1 ; we apply the rede?nitions (cf. (3.8)) ab a,b Jab → Jab + h1 Ξ ab 11
1 1 1 Mab → Mab + gab Ξ

(3.10)

and the Lie brackets of u1 (N + 1) ? sqω (N + 1) turn out to be ω [Jab , Jac ] = ωab Jbc 1 1 [Mab , Mac ] = ωab Jbc 1 1 [Jab , Mac ] = ωab Mbc 1 1 [Mab , Jac ] = ?ωab Mbc [Jab , Jde ] = 0 1 1 [Jab , Ed ] = (δad ? δbd )Mab [Jab , Jbc ] = ?Jac [Jac , Jbc ] = ωbc Jab 1 1 1 1 [Mab , Mbc ] = Jac [Mac , Mbc ] = ωbc Jab 1 1 1 1 [Jab , Mbc ] = ?Mac [Jac , Mbc ] = ?ωbc Mab 1 1 1 1 [Mab , Jbc ] = ?Mac [Mac , Jbc ] = ωbc Mab 1 1 1 [Mab , Mde ] = 0 [Jab , Mde ] = 0 1 1 [Mab , Ed ] = ?(δad ? δbd )Jab
1 1 [Ea , Eb ] = e1 Ξ a,b

(3.11)

1 1 1 1 [Jab , Mab ] = 2ωab (Eb ? Ea ) + fab Ξ

a < b.

(3.12)

The two remaining extended unitary CK subalgebras uλ (N + 1) ? sq ω (N + 1) ω λ λ with λ = 2, 3 are generated by {Jab , Mab , Ea , Ξ} (hereafter we shall reserve λ to stand exclusively for the quaternionic indices λ = 2, 3, whereas α, β, γ are allowed to take on any value 1, 2, 3). The subalgebras uλ (N + 1) have generic extended Lie brackets ω (as (3.1)) except for the common orthogonal CK subalgebra soω (N + 1) spanned by the generators {Jab } which is non-extended and whose Lie brackets are already written in (3.11). For the two remaining unitary subalgebras, we have already used up the rede?nition concerning to the common generators in soω (N + 1), so we cannot apply directly the results of the theorem 3.1 and we have to compute their corresponding Jacobi identities by taking into account that initially both contain a non-extended soω (N + 1). As the equations so obtained are similar to those written in detail in [8] we omit them and give the ?nal result. The extension coe?cients λ λ that appear are denoted gab , hλ a+1 , fab and eλ , for λ = 2, 3; the Lie brackets of a a,b uλ (N + 1) read ω
λ λ λ λ λ λ [Mab , Mac ] = ωab Jbc [Mab , Mbc ] = Jac [Mac , Mbc ] = ωbc Jab λ λ λ λ λ λ [Jab , Mac ] = ωab (Mbc + gbc Ξ) [Mab , Jac ] = ?ωab (Mbc + gbc Ξ) λ λ λ λ λ λ [Jab , Mbc ] = ?(Mac + gac Ξ) [Mab , Jbc ] = ?(Mac + gac Ξ) λ λ λ λ λ λ [Jac , Mbc ] = ?ωbc (Mab + gab Ξ) [Mac , Jbc ] = ωbc (Mab + gab Ξ) λ λ λ [Jab , Mde ] = 0 [Mab , Mde ] = 0 λ λ λ [Jab , Ed ] = (δad ? δbd )(Mab + gab Ξ) λ λ [Mab , Ed ] = ?(δad ? δbd )Jab b>a+1 λ λ [Ma a+1 , Ed ] = ?(δad ? δa+1 d )(Ja a+1 + hλ a+1 Ξ) a λ λ λ λ [Jab , Mab ] = 2ωab (Eb ? Ea ) + fab Ξ λ λ [Ea , Eb ] = eλ Ξ a,b

(3.13)

a < b.

(3.14)

λ The coe?cients fab and eλ (λ = 2, 3) are characterized by the theorem 3.1 (see a,b (3.6) and (3.7)), while the extensions hλ a+1 are subjected to the relations a

ωa hλ a+1 = 0 a

ωa+2 hλ a+1 = 0. a

(3.15)

Notice that now the coe?cients hλ with b > a + 1 are zero (this is a direct conab sequence of the presence of the non-extended soω (N + 1)). We now apply the rede?nitions given by λ λ λ Mab → Mab + gab Ξ λ = 2, 3 (3.16) 12

and a glance to (3.13) shows that the corresponding extensions are always trivial, λ so the extension coe?cients gab are eliminated. At this point the complete set of Lie brackets of sq ω (N + 1) turn out to be [Jab , Jac ] = ωab Jbc [Jab , Jbc ] = ?Jac [Jac , Jbc ] = ωbc Jab α α α α α α [Mab , Mac ] = ωab Jbc [Mab , Mbc ] = Jac [Mac , Mbc ] = ωbc Jab α α α α α α [Jab , Mac ] = ωab Mbc [Jab , Mbc ] = ?Mac [Jac , Mbc ] = ?ωbc Mab α α α α α α [Mab , Jac ] = ?ωab Mbc [Mab , Jbc ] = ?Mac [Mac , Jbc ] = ωbc Mab α α α [Jab , Jde ] = 0 [Mab , Mde ] = 0 [Jab , Mde ] = 0 α α 1 1 [Jab , Ed ] = (δad ? δbd )Mab [Mab , Ed ] = ?(δad ? δbd )Jab λ λ [Mab , Ed ] = ?(δad ? δbd )Jab b>a+1
λ λ [Ma a+1 , Ed ] = ?(δad ? δa+1 d )(Ja a+1 + hλ a+1 Ξ) a α α α α [Jab , Mab ] = 2ωab (Eb ? Ea ) + fab Ξ α α [Ea , Eb ] = eα Ξ a,b

(3.17)

(3.18) a<b (3.19)

γ α β [Mab , Mac ] = ωab εαβγ Mbc + εαβγ mα,β Ξ ab,ac β α γ [Mab , Mbc ] = εαβγ Mac + εαβγ mα,β Ξ ab,bc β γ α [Mac , Mbc ] = ωbc εαβγ Mab + εαβγ mα,β Ξ ac,bc β α [Mab , Mde ] = εαβγ mα,β Ξ ab,de β γ α γ [Mab , Mab ] = 2ωab εαβγ (Ea + Eb ) + εαβγ mα,β Ξ ab β γ α [Mab , Ed ] = (δad + δbd )εαβγ Mab + εαβγ meα,β Ξ ab,d β α γ [Ea , Eb ] = 2δab εαβγ Ea + εαβγ eα,β Ξ. a,b

(3.20)

Therefore the Lie brackets (3.17) are non-extended, the extension coe?cients hλ a+1 a appearing in (3.18) satisfy (3.15), the coe?cients of the commutators (3.19) are characterized by the theorem 3.1, and the extension coe?cients in the commutators (3.20) are still generic, the rede?nitions (3.10) and (3.16) having been already incorporated in the brackets (3.20). The list of all remaining extension coe?cients is hλ a+1 a
α fab

eα a,b

mα,β ab,de

mα,β ab

meα,β ab,d

eα,β a,b

(3.21)

where the two quaternionic indices α, β are always di?erent. We sort the coe?cients mα,β , meα,β and eα,β into several subsets as follows: ab,de ab,d a,b ? Coe?cients mα,β involving four di?erent indices a, b, d, e. If we rename and sort ab,de these four di?erent indices as a < b < c < d these coe?cients are mα,β ab,cd mα,β ac,bd mα,β . ad,bc (3.22)

? Coe?cients mα,β involving three di?erent indices. If we write the indices as ab,de a < b < c the coe?cients are mα,β ab,ac mα,β ab,bc mα,β . ac,bc (3.23)

13

? Coe?cients meα,β with two di?erent indices a < b and a third one d ∈ {a, b}: ab,d meα,β ab,a meα,β . ab,b (3.24)

? Coe?cients meα,β with two di?erent indices a < b and a third index d ∈ {a, b}: / ab,d meα,β . ab,d ? Coe?cients eα,β with two di?erent indices a < b: a,b eα,β . a,b ? Coe?cients eα,β with a single index a: a,b eα,β . a,a (3.27) (3.26) (3.25)

In the sequel we proceed to compute the Jacobi identities involving the above coe?cients; the results obtained in any equation will be automatically introduced in any further equation, so the order we consider for enforcing the Jacobi identities is an integral part of the derivation, and should be respected. We denote the Jacobi identity (3.2) of the generators Xi , Xj and Xl by {Xi , Xj , Xl }. The following equations imply the vanishing of some coe?cients:
2 1 3 {Ma a+1 , Ea , Ea+1 } : 2 1 3 {Ma a+1 , Ea , Ea+1 } : α β γ {Ea , Ea , Eb } : α α γ {Mab , Mac , Ec } : γ α α {Mab , Mbc , Ec } : γ α α {Mac , Mbc , Eb } : β α {Jab , Mcd , Eb } : α β {Jbc , Mad , Ec } : β α {Jab , Mbc , Mbd } : β γ β {Mab , Ea , Eb } : β β γ {Mab , Eb , Ea } : β γ β {Mab , Ea , Ed } : γ α α {Ea , Eb , Eb } :

h2 a+1 = 0 a h3 a+1 = 0 a eα = 0 a,b mα,β = 0 ab,ac mα,β = 0 ab,bc mα,β = 0 ac,bc

(3.28) (3.29) (3.30)

mα,β = 0 ab,cd mα,β = 0 ad,bc mα,β ? mβ,α = 0 ac,bd ad,bc meα,β = 0 ab,a meα,β = 0 ab,b meα,β = 0 ab,d eα,β = 0 a,b

(3.31)

(3.32) (3.33) (3.34)

α so that the only remaining coe?cients are fab , mα,β and eα,β . The Jacobi identities a,a ab α β {Jab , Mab , Ea } : β α {Jab , Mab , Eb } : γ 2ωab eα,β ? mα,β + fab = 0 a,a ab α,β α,β γ 2ωab eb,b ? mab ? fab = 0

(3.35)

14

α allows us to express the coe?cients fab , mα,β in terms of the eα,β as follows a,a ab γ fab = ωab (eα,β ? eα,β ) a,a b,b mα,β = ωab (eα,β + eα,β ). a,a ab b,b

(3.36)

Notice that the ?rst equation is consistent with the relation (3.6). Hence the only Lie brackets of sq ω (N + 1) (3.17)–(3.20) which still involve extension coe?cients are
γ γ 1 γ [Jab , Mab ] = 2ωab (Eb + 2 eα,β Ξ) ? (Ea + 1 eα,β Ξ) b,b 2 a,a γ β 1 α γ [Mab , Mab ] = 2ωab εαβγ (Ea + 2 eα,β Ξ) + (Eb + 1 eα,β Ξ) a,a 2 b,b α β γ [Ea , Ea ] = 2εαβγ (Ea + 1 eα,β Ξ). 2 a,a

(3.37)

These equations clearly suggest to introduce the rede?nition 1 γ γ Ea → Ea + eα,β Ξ 2 a,a (3.38)

which explicitly shows the triviality of all the extensions determined by the coe?α cients eα,β (and consequently, by all the fab and mα,β ). Therefore it is not necessary a,a ab to compute more Jacobi identities and we can conclude that the most general central extension sq ω (N + 1) of any algebra in this family is always trivial. This result can be summed up in the following statement: Theorem 3.2. The second cohomology group H 2 (sq ω (N + 1), R) of any Lie algebra belonging to the quaternionic unitary CK family is always trivial, for any N and for any values of the set of constants ω1 , ω2 , . . . , ωN : dim (H 2 (sqω (N + 1), R)) = 0. (3.39)

4

Concluding remarks

This paper completes the study of cohomology of the quasi-simple or CK Lie algebras in the three main series (orthogonal, unitary and quaternionic unitary), as associated to antihermitian matrices over R, C or H. In contrast to the quasi-orthogonal or quasi-unitary cases, where the dimension of the second cohomology group of a generic algebra in the CK family ranges between 0 for the simple algebras and a maximum positive value for the most contracted algebra (with all ωa = 0), all the central extensions of any of the algebras in the quaternionic quasi-unitary family are always trivial, even for the most contracted algebra. Therefore from the three types of extensions found in the quasi-orthogonal or quasi-unitary cases, only the ?rst type (extensions which are trivial for all the algebras in the family) is present here. However we should remark the suitability of a CK approach to the study of the central extensions of a complete family, because a case-by-case study (for any 15

given algebra in the family) would be not more easy than the general analysis we have performed. In addition to these three main families of CK algebras, whose simple members so(p, q), su(p, q), sq(p, q) can be realised as antihermitian matrices over either R, C or H, there are other CK families. In the CN +1 Cartan series, the remaining real Lie algebra is the real symplectic sp(2(N + 1), R), which can be interpreted in terms of CK families either as the single simple member of its own CK family spω1 ,...,ωN (2(N +1), R), or alternatively and more like the interpretation in this paper, as the unitary family uω1 ,...,ωN ((N + 1), H′ ) over the algebra of the split quaternions H′ (a pseudo-orthogonal variant of quaternions, where i1 , i2 , i3 still anticommute, but their squares are i2 = ?1, i2 = 1, i2 = 1; this is not a division algebra). The 1 2 3 cohomology properties of algebras in this CK family could be studied using an approach similar to that made in this paper for the quaternionic unitary CK algebras. This study, a well as the study of the central extensions of the CK series of the real Lie algebras su? (2r) ≈ sl(r, H), so? (2N), sl(N + 1, R) ≈ su(N + 1, C′) not included in the three main ‘signature’ series is worth of a separate consideration.

Acknowledgments
This work was partially supported by DGICYT (project PB94–1115) from the Ministerio de Educaci?n y Cultura de Espa? a and by Junta de Castilla y Le?n o n o (Projects CO1/396 and CO2/297).

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