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HIGHER ORDER ALGEBRAS AND GROUPS M. RAUSCH de TRAUBENBERG∗ Universit´ e de Strasbourg ´ IPHC-DRS-Physique Th´ eorique des Hautes Energies 23 rue du loess - BP28 - 67037 Strasbourg cedex 2, France ∗ E-mail:
[email protected] Cubic extensions of the Poincar´ e algebra can be constructed in consistency with physical assumptions and provide new insights for the description of phenomena. In this paper we review some recent results associated to these structures and give the salient steps to construct their associated groups. Keywords: Cubic extension of the Poincar´ e algebra; ternary algebras and groups
1. Introduction Lie algebras and Lie-superalgebras are central in the description of elementary particles. However, recently some non-quadratic structures emerged in different contexts in the description of physical phenomena. For instance, this generalised algebraic approach is given by the n-linear algebras in Quantum Mechanics,1 ternary algebras in the description of multiple M2 branes2,3 or higher order extensions of the Poincar´e algebra.4–6 2. Lie algebras of order F : definition and examples A (real or complex) vector space g = a g0 ⊕ g1 with basis Xi , i = 1, · · · , dim(g0 ) , Ya , a = 1, · · · , dim(g1 ) , is called an elementary Lie algebra of order F if it satisfies the following requirements: (1) g0 is a Lie algebra with bracket [Xi , Xj ] = fij k Xk ; (2) g1 is a representation of g0 , that is [Xi , Ya ] = Ria b Yb ; (3) the composition of F elements in g1 closes within g0 : {Ya1 , · · · , YaF } = Qa1 ···aF i Xi , where {· · · } is the fully symmetric product of F elements ; a From now on, generic element of g are denoted X, of g denoted Y and of g denoted 0 1 Z.
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(4) the following fundamental identities are [{Ya2 , · · · , YaF +1 }, Ya1 ] + · · · + [{Ya1 , · · · , YaF }, YaF +1 ] = 0.
satisfied:
There is basically two ways to obtain Lie algebras of order F : either by matrix representations or by an induction theorem. For the matrix representation,7 as an example, consider glel (m1 , m2 , m3 ) the set of (m1 + m2 + m3 ) × (m1 + m2 + m3 ) matrices of the form a0 b 1 0 glel (m1 , m2 , m3 ) = 0 a1 b2 , b 0 0 a2 with a0 ∈ gl(m1 ), a1 ∈ gl(m2 ), a2 ∈ gl(m3 ), and b1 ∈ Mm1 ,m2 (R), b2 ∈ Mm2 ,m3 (R), b0 ∈ Mm3 ,m1 (R). It is obvious to show that g = g0 ⊕ g1 = gl(m1 ) ⊕ gl(m2 ) ⊕ gl(m3 ) ⊕ Mm1 ,m2 (R) ⊕ Mm2 ,m3 (R) ⊕ Mm3 ,m1 (R) ,
g is a Lie algebra of order-three. The second types of algebras are based upon an induction theorem.5 As an example consider g0 a semi-simple Lie algebra and g1 its adjoint representation. Let {Ja , a = 1, · · · , dim g0 } be a basis of g0 and {Aa , a = 1, · · · , dim g0 } be the corresponding basis of g1 . Let gab = Tr(Aa Ab ) be the Killing form and fab c be the structure constants of g0 . Then one can endow g = g0 ⊕ g1 with a Lie algebra of order-three structure [Ja , Jb ] = fab c Jc , [Ja , Ab ] = fab c Ac , {Aa , Ab , Ac } = gab Jc + gac Jb + gbc Ja . (1) It was then realised that within the framework of Lie algebras of order F higher order extensions of the Poincar´e algebra can be defined. Historically the first higher order extensions of the Poincar´e algebra were infinite dimensional and related to anyons in (1 + 2)− dimensional space-time, (i.e., discrete series of so(1, 2)).9–11 Later on, it was observed that the cubic algebra (1) with g0 = so(1, 4) leads, by means of an In¨ on¨ u-Wigner contrac5,6,8 tion, to the cubic extension of the Poincar´e algebra g = Iso(1, 3) ⊕ R1,3 with Iso(1, 3) the Poincar´e algebra in (1 + 3)−dimensions generated by {Lµν , Pµ } and R1,3 the vector representation of so(1, 3) generated by {Vµ }.
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The algebra of order-three g is defined by the following brackets [Lµν , Lρσ ] = ηνσ Lρµ − ηµσ Lρν + ηνρ Lµσ − ηµρ Lνσ , [Lµν , Pρ ] = ηνρ Pµ − ηµρ Pν , [Pµ , Pν ] = 0, [Lµν , Vρ ] = ηνρ Vµ − ηµρ Vν , [Pµ Vν ] = 0, {Vµ , Vν , Vρ } = ηµν Pρ + ηνρ Pµ + ηρµ Pν , with ηµν = diag(1, −1, −1, −1) the Minkowski metric. This algebra was intensively studied12,13 and invariant actions were explicitly constructed. It was also noted that it admits a natural generalisation to any spacetime dimensions. Subsequently higher dimensional invariant actions were constructed, and in this case it turns out that the higher dimensional cubic extensions of the Poincar´e algebra induce a symmetry on generalised gauge fields or p−forms14 from the natural operations on p−forms (wedge product, inner product, etc). Next we have studied several other possibilities to extend the Poincar´e algebra along the lines of cubic algebras: (i) we have studied in a systematic way all the possible cubic extensions of the Poincar´e algebra in (1 + 3)−dimensions,6 (ii) we have defined an adapted superspace associated to the cubic extensions of the Poincar´e algebra,15 (iii) we have studied kinematical algebras of order-three and shown that they are related by deformations and/or contractions.8 It was also shown that quartic extensions of the Poincar´e algebra can be defined: these extensions further turned out to be strongly related to N = 2 supersymmetry.16
3. Groups associated to Lie algebras of order F A priori, the structure of group seems to be incompatible with the structure of ternary algebra. Indeed, for groups the product of two elements is always defined although for ternary algebras only the product of three elements is defined. However, groups associated to Lie algebras of order-three can be constructed using Hopf algebras.6 We recall here the basic steps to obtain a group associated to a Lie algebra of order-three (see6 for more details). (1) Consider g = g0 ⊕ g1 a real Lie algebra of order-three. (2) The universal enveloping algebra U(g) is the quotient of the tensor algebra T (g) by the two sided-ideal generated by
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X1 ⊗ X2 − X2 ⊗ X1 − [X1 , X2 ], X1 ⊗ Y2 − Y2 ⊗ X1 − [X1 , Y2 ], Y1 ⊗ Y2 ⊗ Y3 + perm. − {Y1 , Y2 , Y3 } . Then, as a vector space, we have the following isomorphism U(g) ∼ = U(g0 ) ⊕ Λ3 (g1 ) , with U(g0 ) the usual universal enveloping algebra of the Lie algebra g0 , Λ3 (g1 ) be Roby (or three-exterior) algebra.11,17 The three-exterior algebra Λ3 (Rm ) is generated by the elements θi , i = 1, · · · , m subjected to the relation {θi , θj , θk } = 0. The three-exterior algebra is Z3 −graded and has the following decomposition Λ3 (Rm ) = Λ3 (Rm )0 ⊕ Λ3 (Rm )1 ⊕ Λ3 (Rm )2 , where Λi ≡ Λ3 (Rm )i corresponds to the set of polynomials of degree i mod. 3. However, in contrast to the usual Grassmann algebra, the Roby algebra is infinitely generated. A basis of Λ3 (Rm ) were identified in.17 We call the elements of the basis of Λ3 (Rn ) the Roby elements. (3) A Poincar´e-Birkhoff-Witt theorem is proven and a basis of U(g) is identified. (4) Endow U(gC ) (gC is the complexified of g) with a Hopf algebra structure ˆ ∆Z = Z ⊗1 ˆ + 1⊗Z ˆ ,Z ∈ g , co-product ∆1 = 1⊗1, antipode S(1) = 1, S(Z) = −Z , Z ∈ g , co-unity ǫ(1) = 1, ǫ(Z) = 0 , Z ∈ g , ˆ the Z3 −twisted tensor product (the elements of g0 (resp. g1 ) with ⊗ are of grade zero (resp. one)). Extend the definition of the co-product, antipode and co-unit to any elements of U(g). (5) Consider the dual of U(g) and endow it with a Hopf algebra structure. The following vector space isomorphism can be proved U(g)∗ ∼ = C[g0 ] ⊕ Λ3 (g1 ) . Furthermore, one can show that g = αX + βY, α ∈ C, β ∈ Λ1 is grouplike ˆ eg , S(eg ) = e−g , ǫ(eg ) = 1 . ∆eg = eg ⊗ (6) To define a group associated to g some care is needed. Indeed, we have to a take care with (i) Baker-Campbell-Hausdorff formulæ since there
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is no quadratic relations upon the Y ’s, (ii) the fact that Λ3 (Ck ) is infinitely generated. One can show that m n i Y ka b G(Λ) = eα Xi eβ b θ Ya , m ∈ N, αi , β ka b ∈ C, k=1
o with β ka b θb Ya nilpotent matrices
is a group. If we apply this construction to the Lie algebra of order three gel (m1 , m2 , m3 ) we obtain a group that we denote GLel (m1 , m2 , m3 , Λ). Differently, and in a straight analogy with Lie super-groups, one can construct a group associate to Lie algebras of order three. This construction has the advantage not to invoke the heavy machinery of Hopf algebra. First of all we introduce some set of matrices with coefficients in Λ3 (Rm ): Mk (Λ0 ) = Mk (R) ⊗ Λ0 , Mk,ℓ (Λ1 ) = Mk,ℓ (R) ⊗ Λ1 , Mk,ℓ (Λ2 ) = Mk,ℓ (R) ⊗ Λ2 . A matrix, e.g. in Mk (Λ0 ) is given by A(Λ0 ) = A(0) + A(3)ijk θi θj θk + A(6)ijkℓmn θi θj θk θℓ θm θn + · · ·
(2)
where the sum above is finite and taken only over the Roby elements, i.e., upon the basis vectors. We can show that the matrix given in (2) is invertible if and only if A(0) is invertible.7 We denote now by GL(m1 , Λ0 ) the set of invertible matrices of Mm1 (R, Λ0 ) such that the matrix A(Λ0 ) and its inverse A(Λ0 )−1 have finite expansions. We then introduce A0 ∈ GL(m1 , Λ0 ), A1 ∈ GL(m2 , Λ0 ), A2 ∈ GL(m3 , Λ0 ), B0 ∈ Mm3 ,m1 (Λ1 ), B1 ∈ Mm1 ,m2 (Λ1 ), B2 ∈ Mm2 ,m3 (Λ1 ) and C0 ∈ Mm2 ,m1 (Λ2 ), C1 ∈ Mm3 ,m2 (Λ2 ), C2 ∈ Mm1 ,m3 (Λ2 ), and define the (m1 + m2 + m3 ) × (m1 + m2 + m3 ) matrices ( A0 B1 C2 ) GL(m1 , m2 , m3 ) = M = C0 A1 B2 . B0 C1 A2
Then, GL(m1 , m2 , m3 ) is a group. It is obvious to show that the product of two elements in GL(m1 , m2 , m3 ) is in GL(m1 , m2 , m3 ). The associativity is simply a consequence of the associativity of the matrix algebra
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and of the 3−exterior algebra. Finally one can obtain explicit formulæ for the inverse matrix.7 To end, we would like to notice that the two groups constructed so far are not equal and we have GLel (m1 , m2 .m3 , Λ) ⊂ GL(m1 , m2 , m3 ) . References 1. Y. Nambu Phys. Rev. D7 2405 (1973); T. Curtright and C. K. Zachos Phys. Rev. D68 085001 (2003) [arXiv:hep-th/0212267]. 2. J. Bagger and N. Lambert Phys. Rev. D7 065008 (2008) [arXiv:0711.0955 [hep-th]]; J. Bagger and N. Lambert Phys. Rev. D79 025002 (2009) [arXiv:0807.0163 [hep-th]]; A. Gustavsson Nucl. Phys. B811 66 (2009) [arXiv:0709.1260 [hep-th]]. 3. J. A. de Azc´ arraga and J. M. Izquierdo J. Phys. A: Math. Theor. A43 293001 (2010) [arXiv:math-ph/1005.1028v1]. 4. M. Rausch de Traubenberg and M. J. Slupinski J. Math. Phys. 41 4556 (2000) [arXiv:hep-th/9904126]. 5. M. Rausch de Traubenberg and M. J. Slupinski J. Math. Phys. 43 5145 (2002) [arXiv:hep-th/0205113]. 6. M. Goze, M. Rausch de Traubenberg and A. Tanasa J. Math. Phys. 48 093507 (2007) [arXiv:math-ph/0603008]. 7. M. Rausch de Traubenberg J. Phys. Conf. Ser. 128 012060 (2008) [arXiv:0710.5368 [math-ph]]; M. Goze and M. Rausch de Traubenberg J. Math. Phys. 50 063508 (2009) [arXiv:0809.4212 [math-ph]]. 8. R. Campoamor-Stursberg and M. Rausch de Traubenberg, J. Math. Phys. 49 063506 (2008) [arXiv:0801.2630 [hep-th]]. 9. M. Rausch de Traubenberg and M. J. Slupinski, Mod. Phys. Lett. A12 3051 (1997) [hep-th/9609203]. 10. M. Rausch de Traubenberg, J. Phys. Conf. Ser. 175 012003 (2009) [arXiv:0811.1465 [hep-th]]. 11. M. Rausch de Traubenberg, hep-th/9802141. 12. N. Mohammedi, G. Moultaka and M. Rausch de Traubenberg Int. J. Mod. Phys. A19 5585 (2004) [arXiv:hep-th/0305172]. 13. G. Moultaka, M. Rausch de Traubenberg and A. Tanasa Int. J. Mod. Phys. A20 5779 (2005) [arXiv:hep-th/0411198]. 14. G. Moultaka, M. Rausch de Traubenberg and A. Tanasa, in Proceedings of the XIth International Conference Symmetry Methods in Physics, Prague 21-24 June 2004 [arXiv:hep-th/0407168]. 15. R. Campoamor-Stursberg and M. Rausch de Traubenberg, J. Phys. A; Math. Theor. 42 495202 (2009) [arXiv:0907.2149 [hep-th]]. 16. R. Campoamor-Stursberg and M. Rausch de Traubenberg, J. Phys. A43 455201 (2010) [arXiv:1005.4994 [hep-th]]. 17. N. Roby Bull. Sc. Math. 94 49 (1970).
