Proceedings of Institute of Mathematics of NAS of Ukraine 2000, Vol. 30, Part 2, 544–550. The Tangent Groups of a Lie Group and Gauge Invariance in Lagrangian Dynamics Yurij YAREMKO Institute for Condensed Matter Physics, 1 Svientsitskyj Street, 290011, Lviv, Ukraine E-mail: [email protected] A tangent Lie group with elements and group operations which are tangent prolongations of those corresponding to another Lie group is examined. An action of such extended Lie group on differentiable manifold and its tangent bundle is defined by using contact transformations. It turned out that a tangent Lie symmetrical Lagrangian describes a dynamical system with first-class constraints. Geometrical aspects of the reduction procedure are considered. Introduction We propose to analyse dynamical systems with first-class constraints by using of the tangent Lie groups [1]. In our opinion, this approach explains the genesis of Dirac systems of this type and offers the conceptual clarity. 1 Second-order tangent group of a Lie group Let G be an R-dimensional Lie group. The tangent bundle TG and the second tangent bundle T 2G are also the Lie groups with elements and group operations which are tangent prolongations of those corresponding to the original Lie group G [1, 2, 3]. Let us consider the curves ν : I → G and λ : I → G where I is open neighbourhood of the zero point 0 ∈ R. We introduce the coordinate system (U, g)inG. Henceforth Greek symbols να(a):=(gα ◦ ν)(0), λβ(b):=(gβ ◦ λ)(0) etc. denote the local coordinates of group elements a, b etc. in manifold G. These coordinates are chosen so that εκ(e)=0,κ =1,...,R := 1, R,for identity element e. We also use the induced coordinate systems (U 1,g1)and(U 2,g2) on the 1-st and 2-nd order tangent bundles TG and T 2G, respectively. Greek indices are meant to run from 1to R throughout the paper; the summation convention is used for dummy indices. We denote 1 2 2 2 t a, t a etc. the elements tν(0) and t ν(0) of tangent Lie groups TG and T G, respectively. 1 1 2 i i ◦ i Their coordinates are να,να and να,να,να where να = d (gα ν)/dt 0, i =0, 1, 2. Starting with a group multiplication µ : G × G → G, (1.1) we construct the multiplication law for TG [1, 2] Tµ : TG× TG → TG defined by Tµ(tλ(0),tν(0)) = t(µ(λ, ν))(0). The Tangent Groups of a Lie Group and Gauge Invariance in Lagrangian Dynamics 545 We obtain R expressions in local coordinates 1 1 ∂µα(b, a) 1 ∂µα(b, a) ηα = dT µα(λβ(b),νγ(a)) = λβ + νγ , (1.2) ∂λβ(b) ∂νγ(a) in addition to the relations ηα = µα(λβ,νγ), (1.3) which illustrate the law (1.1) locally. Here dT is the Tulczyjew differential operator [4]. 1 As it follows from eqs.(1.2), the identity element t e has zero-valued derivative coordinates: 1 1 εκ t e =0forallκ = 1, R. In analogy with Tµ we construct the second order prolongation T 2µ : T 2G × T 2G → T 2G of the multiplication law (1.1): T 2µ t2λ(0),t2ν(0) = t2(µ(λ, ν))(0), where on the right side is the second tangent prolongation of the curve η = µ(λ, ν):I → G taken at zero point. On the local level we have R relations 2 2 ηα = dT µα(λβ,νγ), (1.4) in addition to eqs.(1.3) and (1.2). Now we consider the embedding ι1 : G → TG, locally given by (να) → (να, 0). The submani- 1 1 fold ι1(G) ⊂ TG is a slice [5] of the coordinate system U ,g . According to ref.[1], ι1 is a group homomorphism and (G, ι1) is a closed subgroup of a Lie group TG. Similarly we construct 2 2 a closed subgroup ι2(G) ⊂ T G where the inclusion map ι2 : G → T G is the embedding locally written as (να) → (να, 0, 0). 2 2 Note that the bundle projections τG : TG → G and τG : T G → G are also the group 2,1 2 homomorphisms. The projection τG is the homomorphism from group T G to group TG. Therefore, an original Lie group is a Lie subgroup and a submanifold of its own first- and second-order tangent groups [1]. (More exactly, we consider the slices ι1(G) ⊂ TG and ι2(G) ⊂ T 2G on which all the derivative coordinates are equal to zero.) Moreover, the constants of structure of these tangent Lie groups are determined by the structure constants of G.To 2 demonstrate it we study the involutive distribution XL(T G) of all left invariant vector fields on 2 ∗ 2 T G and its dual space XL(T G) of all left invariant one-forms. Taking into account an exclusive role of Tulczyjew differential operator in prolongation al- ∗ 2 gorithm (see eqs.(1.2) and (1.4)), we deal with XL(T G). We write the local expressions for ∗ 2 2 canonical left invariant one-forms [5, 6] which constitute the basis for XL(T G)atapointt a: γk ∂ k i θ = i 2 dT µγ(b, a) dνα. (1.5) ∂να(t a) t2b=t2a−1 Small roman indices run from 0 to 2. The exterior derivatives of the θγk are given by the Maurer–Cartan equation γk 1 Γ αi βj dθ = − CABθ ∧ θ . (1.6) 2 Γ 2 We use multi-index notation in structure constants {CAB} of T G, where multi-indices A, B 2 and Γ are the 2-tuples of natural numbers, e.g. A = (αi). Particularly, for subgroup ι2(G) ⊂ T G we have 1 γ dθγ = − c θα ∧ θβ, 2 αβ 546 Yu. Yaremko γ where {cαβ} are structure constants of the original Lie group G (zero-valued roman indices are omitted). Left-invariant one-forms θα are given by eqs.(1.5) if integer k is equal to zero. Tulczyjew operator dT is the derivation of type d∗ of zero degree which acts on the 0-forms as a total time derivative [4]. Having used the commutation d ◦ dT = dT ◦ d and the expressions i i+1 dT νβ = νβ , after short calculations we establish the following relations between the “higher- order” one-forms (1.5) and the original ones: γ1 γ γ2 2 γ θ = dT θ ,θ= dT θ . Thanks to commutation of Tulczyjew operator with an exterior derivative and positively signed Leibniz’ rule for wedge product [3] we arrive at 1 γ 1 γ dθγ1 = − c θα1 ∧ θβ − c θα ∧ θβ1, 2 αβ 2 αβ 1 γ γ 1 γ dθγ2 = − c θα2 ∧ θβ − c θα1 ∧ θβ1 − c θα ∧ θβ2. 2 αβ αβ 2 αβ When comparing these expressions with the Maurer–Cartan equation (1.6) we deduce the con- stants of structure {CΓ }. It is convenient to write them as the following block matrices: AB cˆγ 00 0ˆcγ 0 00ˆcγ Cˆ(γ0) = 000 , Cˆ(γ1) = cˆγ 00 , Cˆ(γ2) = 02ˆcγ 0 . 000 000 cˆγ 00 γ γ Here symbolc ˆ denotes the skew-symmetric matrix cαβ with fixed integer γ. Similarly we obtain the structure constants of a Lie group TG: cˆγ 0 0ˆcγ Cˆ(γ0) = , Cˆ(γ1) = . 00 cˆγ 0 2 (2) The basis for XL(T G) consists of the left invariant vector fields [2, 5, 6], say XB , locally given by (2) α ∂ α ∂ 2 α ∂ X(β0) = Lβ + dT (Lβ ) 1 + dT (Lβ ) 2 , ∂gα ∂gα ∂gα (2) α ∂ α ∂ X(β1) = Lβ 1 +2dT (Lβ ) 2 , ∂gα ∂gα (2) α ∂ X(β2) = Lβ 2 . ∂gα α Here Lβ (g) are the components of the left invariant vector fields Xβ which form the basis for XL(G). (2) 2 (2) Vector field X(β0) is the 2-lift of corresponding one Xβ to tangent bundle T G, i.e. X(β0) = (2,2) Xβ (see refs. [1, 9]). The former belongs to the basis of the sub distribution XL(ι2(G)) ⊂ 2 (2) (2) XL(T G). The others X(β1) and X(β2) are intimately connected with the 1-st and 0-th lifts [1, 9] 2 (2) (2,2) (2) 2 (2,2) of Xβ to T G, respectively. Namely, we have X(β1) = J1Xβ and X(β2) =(1/2)(J1) Xβ , 2 where J1 is the canonical almost tangent structure [10] of order 2 on T G. (1) 2 2 Let X(βi), i =0, 1, be the canonical left-invariant vector fields on XL(TG). If τ1 : T G → TG (2) (1) 2 2 (2) (1) is the canonical projection, then X(βi) and X(βi) are τ1 -related, i.e. Tτ1 (X(βi))=X(βi). Each of homomorphism of groups, mentioned in this Section, corresponds the Lie algebra homomor- phism which describes its effect on left invariant vector fields, as well as the mapping which relates the dual algebras. The former are then nothing but the differential of originating group homomorphism and the latter is precisely the transpose of this differential [5]. The Tangent Groups of a Lie Group and Gauge Invariance in Lagrangian Dynamics 547 2 An action of the tangent Lie group on a smooth manifold When considering the group parameters as constants, an action r : Q × G → Q of a Lie group G 1 on an N -dimensional smooth manifold Q lifts to an action r :TQ×G → TQof G on the tangent 1 → 1 ∈ bundle TQ as follows [6, 11, 9]: r a : TQ TQ, where r a = Tra for any fixed a G.