Home Search Collections Journals About Contact us My IOPscience Invariant Cones in Lie Algebras and Positive Energy Representations and Contractions of Conformal Algebras This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 J. Phys.: Conf. Ser. 462 012037 (http://iopscience.iop.org/1742-6596/462/1/012037) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 136.243.24.42 This content was downloaded on 06/02/2015 at 15:00 Please note that terms and conditions apply. 6th International Symposium on Quantum Theory and Symmetries (QTS6) IOP Publishing Journal of Physics: Conference Series 462 (2013) 012037 doi:10.1088/1742-6596/462/1/012037 Invariant Cones in Lie Algebras and Positive Energy Representations and Contractions of Conformal Algebras Patrick Moylan Department of Physics, Pennsylvania State University, The Abington College, Abington, PA 19001 USA E-mail: Abstract. We recall some important results, due to Kostant and others, about invariant convex cones in Lie algebras and positive energy representations. We apply these results to a study of positive energy representation of the conformal groups in n dimensions, and we present a proof of the converse of a theorem attributed to I.E. Segal, which relates positive energy representations to positivity of the action of the generator of time translations for representations of the n-dimensional conformal group. We also discuss related notions of deformation and contractions of Lie algebras and describe a deformation of the Poincar´e subalgebra of the conformal algebra which generalizes the usual treatment. We consider the positive energy representations of the anti-deSitter subalgebras in the physically important four dimensional case, and apply this generalization to argue that the singelton representations cannot have nontrivial contractions to representations of the Poincar´e algebra. We believe that our results represent a sharpening of the meaning of “kinematical confinement”, introduced by Flato, Fronsdal and their coworkers. 1. Invariant Convex Cones in Simple Lie Algebras Let V be a finite dimensional vector space over IR. A cone in V is a closed convex subset stable under scalar multiplication by nonnegative real numbers i.e. C is a cone if C is closed and convex and such that tC C t 0 By convexity we mean [1] λv μ where λ 0 μ 0 λ to the adjoint group i.e. if μw C if v w 1. A cone C in a Lie algebra exp adX C A causal cone is one for which C " $ C C X is called invariant if it is invariant relative C 0. The classification of invariant cones in Lie algebras was initiated by Kostant and Vinberg, and the classification of invariant cones in simple Lie algebras was accomplished by S.M. Paneitz in 1980 as part Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 6th International Symposium on Quantum Theory and Symmetries (QTS6) IOP Publishing Journal of Physics: Conference Series 462 (2013) 012037 doi:10.1088/1742-6596/462/1/012037 of his Ph.D. thesis and then independently by G. I. Olshanskii [3]. Classification schemes for arbitrary Lie algebras are reported in [4]. A main theorem in the subject is the following theorem of Kostant [2]. Theorem 1.1. (Kostant) Let G be connected semisimple Lie group acting in a real finite dimensional vector space V. Let K be a maximal compact subgroup of G. Then there exists a a closed G-invariant convex cone C in V satisfying C 0 if and only if V has a nonzero K-invariant vector which lies in , the Lie algebra of K. C For simple Lie algebras there is a corollary of this theorem, namely: there exists a non-trivial casual cone in G/K is Hermitian symmetric [2]. A list of hermitian symmetric algebras may be found in [5]. It suffices for us to note that they include so 2 n (n 3), sp n IR and su p q (p q 1). Let be the (one-dimensional) center of . Choose Z suitably nomalized. Note that for so 2 n , Z = X0 is the Einstein energy [6]. We let CMin be the closed cone generated by Ad G Z. It is a theorem that any (invariant) cone in contains CMin . Self-dual cones are those for which CMin CMax CMin . If sl n IR , then sp 2 IR [5], we have for so 2 3 the existence of a minimal, CMin is self-dual [2]. Since so 2 3 invariant self-dual cone in . 2. Lie Algebra Deformations and Contractions Given V , a finite dimensional vector space of dimension n over IR, let on V i.e. the space of all equivalence classes of bilinear mappings μ :V such that μ x y V be the space of all Lie structures V n x y μ x y μ y x and ∑ μ x μ y z cycl x y z 0 Jacobi identity where two such maps are equivalent if they give isomorphic Lie algebras. Let Λ IR , then a deformation of a given Lie algebra over IR is a continuous mapping ψ: 0Λ V ψ 0 i.e. ψ 0 is the Lie structure of , where continuous means n with 3 continuous in the topology n inherits from the inclusion of the space of structure constants in IR n (i.e. in the Segal topology)[7], [8]. ψ is a trivial deformation if ψ t for all t 0 Λ is equivalent to ψ 0 . The process of deformation of a Lie algebra may be viewed, at least in special circumstances, as the inverse of contraction of a Lie algebra, which idea goes back to Segal [9] and Inon ¨ u¨ and Wigner [10] and formalized by Saletan [11]. We now turn to this notion of contraction of a Lie algebra. Let V be an n-dimensional Lie algebra over IR with an underlying n-dimensional vector space V over IR and . Consider a continuous family φ λ λ IR of surjective mappings φλ Hom V W a Lie bracket where W is another vector space of the same dimension as V . If the φ λ are injective, we may define a new bracket on V as follows: λ V x yλ Set φλ 1 φλ x φλ y λ IR x y V 1 λ and assume that 1 , so that φ1 defines an isomorphism of lim x y λ : λ onto its image. If 0 x y0 2 exists for all x y V , then it defines a (possibly) new Lie algebra and we call this new Lie algebra, V , a contraction of the Lie algebra or (simply) the contracted of . For the case when 0 0 2 6th International Symposium on Quantum Theory and Symmetries (QTS6) IOP Publishing Journal of Physics: Conference Series 462 (2013) 012037 doi:10.1088/1742-6596/462/1/012037 V W and so φλ GL V our definition reduces to the usual definition given in the literature c.f. [8], [12], [13]. We now make use of the family of mappings φλ λ IR to define contractions of representations of Lie algebras [14], [15]. We start with a given Lie algebra W and an infinitesmally of defined on a fixed Hilbert space, . Given a continuous unitarizable representation d π with Π 1 Id (Id the identity on family Πλ λ IR of closed invertible linear transformations of .) it is easy to see that the map λ V λ X d πλ X Πλ 1 d π φλ X Πλ 3 defines a representation of λ on . In order to assure infinitesimal unitarizability of the contracted representation, the family Πλ λ IR should be chosen so that d πλ X is skew-symmetric. To check the representation condition we have [15]: d πλ X d π λ Y Πλ 1 d π φλ X d πλ φ Y Πλ 1 d π φλ X φλ Y Πλ Πλ 1 d π φλ X Y Πλ λ d πλ X Y Πλ λ 4 where in the last line we have made use of eqn. (1). We define the representation of the contracted Lie algebra, 0 , as d π0 X lim d πλ X provided this limit exists, and we call it the contracted of the λ representation d π 0 . 3. Positive Energy Representations of the Conformal Group in n dimensions Now we want to appy the above to obtain some physically useful results about representations of the conformal group in n dimensions and some of its subgroups. Consider the quadratic form Q x defined on IRn 2 by x2 1 x20 x21 x22 x2n 5 Qx IRn 2 . IRn 2 where x x 1 x0 x1 xn Thus equipped with the metric defined by Q x is n 2 SO0 2 n dimensional Minkowski space. We denote n dimensional Minkowski space by M 0 . Let G denote the connected component of the group of linear transformations of IR n 2 preserving the symmetric bilinear form which is associated to Q x by polarization. We shall call G the n-dimensional conformal group, and we denote the universal cover of G by G SO0 2 n . Let be the Lie algebra of G. is identified with the set of all matrices ai j 1 i j n such that aii 0 0 i n, ai j a ji 1 i j n , a0 j aj 0 1 j n , a 1j aj 1 1 j n and a 1 0 a0 1 . We define subalgebras , , , and as follows. Let Ei j be the matrix such that the i j component is equal to 1 and the other components are all equal to 0. Let i j n , L0i Ei o Eo i 1 i n ,L 1i Ei 1 E 1i 1 i n Li j Ei j E ji 1 and L 10 E 10 E0 1 . Let be the subalgebra spanned by: L i j 1 i j n and be the subalgebra spanned by: Li j 1 i j n 1 , L0i 1 i n 1 , L 1i L 10 ; 1 i n 1 and L 1 0 ; be the subalgebra spanned by L 1 n ; and ( ) be the subalgebra 1 1 ˜ L L 0 i n 1 ( P L L 0 i n 1 ). Denote the spanned by Pi ni 1i i ni 1i 2 2 and by K, H, A, N and N , respectively. analytic subgroups of G corresponding to , , , H SO0 2 n 1 is the anti-de Sitter subgroup, and we have an Iwasawa like decomposition of G i.e. A N G is an injective diffeomorphism onto an open, dense subset of G, where the map H H is SO 2 n 1 [16]. Consider the n 1 dimensional isotropic cone in IR n 2 defined by C x IRn 2 Qx 0 6 Let IRn 2 and C be the sets of nonzero elements in IR n 2 and C, respectively. Let P G be the n 2 stabilizer subgroup of e e 1 en where ei 00 0 1 0 0 IR (i.e. ei is the vector 3 6th International Symposium on Quantum Theory and Symmetries (QTS6) IOP Publishing Journal of Physics: Conference Series 462 (2013) 012037 doi:10.1088/1742-6596/462/1/012037 with 1 in the ith slot and zeros elsewhere). P SO0 1 n 1 s N , where s denotes semi-direct G P. product, i.e. P is the n dimensional Poincar´e group. The orbit of e under G is C . Hence C Now we consider a representation π of SO 2 n and d π the associated representation of so 2 n which lifts to a representation of the enveloping algebra. For brevity, write X for d π X with X IC , the enveloping algebra of the complexification IC of . We let be a Cartan subalgebra of IC and . Define λ X hX λ hX h and λ Δ . let Δ be the root system of λ λ ˜ Set Σλ 0 and Σλ 0 Let V be a IC module with action d π of IC on V . Let: a) ˜ ); v V Hi v λ Hi v ; b) v0 V i X X v0 0 with X (X V ∑λ Vλ with Vλ ii Hi v0 Λ0 Hi v0 ; iii V If d π comes from a unitary representation of G , then it is IC v0 an infinitesmally unitarizable lowest (highest) weight representation with lowest (highest) weight Λ 0 . An irreducible, infinitesmally unitarizable representation d π of on a Hilbert space is a representation with positive energy if Cπ X id π X is a nonnegative self adjoint operator is a non-zero and proper cone in . Theorem 3.1. Let d π be an irreducible, infinitesmally unitarizable representation of energy if and only if d π is a highest (lowest) weight representation. . Then d π is positive on To prove the theorem we refer the reader to [17] where it is shown that if id π X 0 a positive, selfis a lowest (highest) weight representation. The adjoint operator then the representation d π so 2 n must contain C Min . theorem then follows from the observation that any cone in Next consider the 3 dimensional (simple) subalgebra (TDS) spanned by: L 10 X0 L0 n L 1n S 7 X0 is the Einstein energy, and S is the generator of scale. Recall P0 i L 2 i T0 10 L0 n 8 We have S X0 L0n S L0 n X0 X0 L0 n S 9 Additionally L 10 α eα S L 10 e αS L 1i α Li0 L 10 α These Lab α ’s are a basis for a conjugate copy of so 2 n Lμν α Lμν μ ν 012 n 1 10 1 , conjugate under so 2 n , and Pμ i lim e 2 α ∞ α L 1μ α Lμν where Pμ and L μν are the generators of the n-dimensional Poincar´e Lie algebra, 4 . 11 6th International Symposium on Quantum Theory and Symmetries (QTS6) IOP Publishing Journal of Physics: Conference Series 462 (2013) 012037 doi:10.1088/1742-6596/462/1/012037 Theorem 3.2. d π is a positive energy representation of so 2 n d π P0 ( d π P0 ) is a positive, self-adjoint operator. direction is due to Segal and is given in [17]. It uses the conformal inversion. The proof in the Although the other direction (the converse of Segal’s theorem) seems to be known to experts in the field, to our knowledge an explicit proof of it, at least in the physics literature, seems to be lacking, and so we record it here: we observe that positivity of the Einstein energy implies ψ d π iX0 α ψ where ψ e X0 α α Sψ . ψ id π eα S X0 e αS ψ ψ d π iX0 ψ 0 α 0 12 Now eqn. (9) implies eα S X0 e αS cosh α X0 sinh α Lon 2iP0 eα as α ∞ 13 The desired result is easily seen to be a consequence of these two equations. 4. A Deformation of the Poincar´e Subalgebra and the Contracted of the Di and Rac Denote by and be the universal enveloping algebras of so 2 n n 1 respectively. We introduce the following element of : Q2 i 1 1 and n 1 1 2 ∑ L20i ∑ L2i j . Let , i j 1 be the skew field of and λ be the skew field of commutative algebraic extensions of by (see below for definition of λ ext ). Define a λ a bY a b Y a 0 a Y2 P2 where Y commutes with all elements of and P 2 a mapping τλ from ∑kn 1 0 Pk Pk . Now define to λ by τλ L˜ μν Lμν τλ L˜ 1μ iλ Q2 P μ 2Y Pμ 14 ˜ 1μ and τλ L ˜ μν satisfy the commutation relations of the generators of . The τλ L ˜ 1μ The λ 1 τλ L and τλ L˜ μν are a basis for an isomorphic copy λ of . For the remainder of paper we take n = 4, but it seems clear that much of the following holds for the higher dimensions [18] and even for other groups like SL n IR [20], especially regarding contractions of the positive energy representations of the anti-deSitter group [15]. Let τ λ 1 Y˜ Y , then τ τλ 1 and define a commutative algebraic extension of by λ ext λ a b Y˜ c Y˜ 2 d Y˜ 3 a b c d λ where Y˜ commutes with all elements of λ . Then τλ can be extended to a homomorphism of ext into ext in an obvious way, which, because of Lemma 4.2, is actually surjective in the ext have case n = 4 which we now consider. Denote this extension also by τ τ λ 1 . Elements of ext ext and , and we introduce structures on a tilde to keep them distinct from elements of ext . Lemma 4.1. Let τ λ be the isomorphic copy of having basis elements Li j defined by eqns. (14). Then (for λ 1) the following holds: ext and L 1μ C2 Y2 W Y2 9 I 4 C4 Y2 5 1 W 4 Y2 15 6th International Symposium on Quantum Theory and Symmetries (QTS6) IOP Publishing Journal of Physics: Conference Series 462 (2013) 012037 doi:10.1088/1742-6596/462/1/012037 (I is the identity in defined in [18], [19].) λ and C2 and and C4 are the 2nd and 4th order Casimir operators of Lemma 4.2. Solutions P μ λ to eqns. (14) are given by P μ λ D A μ ν Lν 1 as with 1 Aμ ν Cλ4 δμν 1 4 Q2 i 2 1 δν 4 μ Q2 Cλ2 δμν 3 ν L 2 μ Lνμ and Lμρ Lρν Lμρ Lρν Y2 λ2 D Q4 1 Q2 4 Cλ4 3 I 16 1 ν δ 2 μ Lνμ 1 2 Y λ Q2 Cλ2 1 2 λ 2 Cλ2 Y 2 λ 4 Cλ4 3 3 W 3 5 2 I and C4 1 Pρ Pρ Lμν Lν μ 2 ˜ μν L ˜† L 4μ Y3 λ3 17 C4 1 4 C2 9 16 I Y˜ , then P˜ μ 18 ˜ ρμ L˜ ˜ 1A D 1ρ and † 1ρ ˜ 4μ and if Y˜ † L ˜ ρμ L ˜ 1A ˜ D P˜ †μ 2i C2 Pμ Pν Lνρ Lρ μ μ 0ν 0ρ 0 Theorem 4.1. ˜ †μν Let † be such that L 16 ∑∑∑ 0 with C2 Also Y 2 satisfies Y 4 and Y λ Y3 λ3 Y2 λ2 i Q2 ν Q4 εμρτ Lρτ ˜† ˜† ˜† A L 1ρ μρ D 1 P˜ μ 19 Furthermore P˜ μ P˜ ν 0 and ker τ 0 Now let the vector space V be the underlying vector space of the Lie algebra of ext by define a map φλ : V 0 and φλ Lμν τλ L˜ μν ; φλ Pμ τλ L˜ 1μ 20 Due to ker τ 0 in the just stated theorem, φλ must be a vector space isomorphism onto its image and thus we can define a new Lie bracket on by: φλ 1 φλ a φλ b a b λ a b in 21 Thus, if we let ψ : I IR with λ I, then ψ is a (nontrivial) n be defined as ψ λ λ deformation of , with ψ 1 being the Lie structure of . Since τλ is an isomorphism, we can apply it in the other direction to obtain contractions of representations of . A classification of the infinitesmally unitarizable, irreducible lowest weight representations of so 2 3 using a modern representation theoretic approach involving Verma modules and singular vectors is given in [21]. The results there are: let D E 0 s0 be a given such lowest weight representation s0 and Λ H2 E0 . Then we have: i) D E0 s0 D 12 0 (Rac); ii) with lowest weight Λ Λ H1 1 1 D E0 s0 D 1 12 (Di); iii) D E0 0 ; iv) D E0 1 s0 s0 1 s0 1. 2 s0 2 ; v) D E0 s0 1 s0 1. The massless representations are: D E 0 All of the representations given there have nontrivial contractions to representations of , except for the case of the singelton representations (cf. [15] for details on the contractions of the massless representations in n dimensions). For the singletons, i.e. the Di and the Rac (which is the minimal 6 6th International Symposium on Quantum Theory and Symmetries (QTS6) IOP Publishing Journal of Physics: Conference Series 462 (2013) 012037 doi:10.1088/1742-6596/462/1/012037 ˜ 1 and d π D ˜ † 1 representation [22] of SO 2 3 ), what goes wrong is that it is necessary that both d π D of Theorem 4.1 must exist on a suitable, dense domains in , in order that there exists a representation on D 1 2 0 (e.g. for the Rac). For both the Di and Rac is Y˜ 2 1 14 which d π˜ 1 of 2 ˜ implies not only that d π P˜ μ are not skew-symmetric operators, but, e.g. for the Rac: d π D 0 ˜ i)! Thus the d π˜ Pμ cannot exist in any sense as operators on the Hilbert space of the (with Y ˜ 1μ being the action of L ˜ 1μ in the Rac representation. Using this fact it representation with d π τλ L where d π is the Di or Rac representation, which is is straighforward to see that the contracted of d π induced by the isomorphism τλ cannot exist except trivially i.e. as the identity representation of . 5. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] Eggleston 1958 Convexity (Cambridge: Cambridge Univ. Press) Paneitz S M 1981 Jour. Funct. Anal. 43 313-359 Olshanskii G I 1981 Funct. Anal. and Its Applications 15, 4, 53-66 Hilgert J, Hoffman K H 1988 Semigroup Forum 37 241-252 Helgason S 1978 Differential Geometry, Lie Groups and Symmetric Spaces’ (New York: Academic Press) Paneitz S M, Segal I E 1982 Jour. Funct. Anal., 47, 78-142; Paneitz S M 1983 Jour. Funct. Anal., 54, 18-112. Ritter W G 2003 Quantum Field Theory and the Space of All Lie Algebras Preprint: Harvard Univ., (arXiv: mathph/0304031) Levy-Nahas M 1967 J. Math. 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