Realizations of real semisimple low-dimensional Lie algebras Realizations of real semisimple low-dimensional Lie algebras

A complete set of inequivalent realizations of threeand four-dimensional real unsolvable Lie algebras in vector fields on a space of an arbitrary (finite) number of variables is obtained. Representations of Lie algebras by vector fields are widely applicable e.g. in integrating of ordinary differential equations, group classification of partial differential equations, the theory of differential invariants, general relativity and other physical problems. There exist many papers devoted to the problem of construction of realizations of Lie algebras. All possible realizations of Lie algebras in vector fields on the two-dimensional complex and real spaces were first classified by S.Lie himself [1, 2]. In this paper we construct a complete set of inequivalent faithful realizations of unsolvable real Lie algebras of dimension no greater than four in vector fields on a space of an arbitrary (finite) number of variables. A necessary step to classify realizations of Lie algebras is classification of these algebras, i.e. classification of possible commutative relations between basis elements. Complete classification of Lie algebras of dimension up to and including six can be found in the papers of V.V. Morozov [3], G.M. Mubarakzyanov [4, 5, 6] and P. Turkowski [7]. The problem of classification of Lie algebras of higher orders is solved only for some classes e.g. the simple and semi-simple algebras. There are exist four unsolvable real Lie algebras of dimension no greater than four (here q = 1, 2, 3): sl(2,R): [e1, e2] = e1, [e1, e3] = 2e2, [e2, e3] = e3; so(3): [e1, e2] = e3, [e3, e1] = e2, [e2, e3] = e1; sl(2,R)⊕A1: [e1, e2] = e1, [e1, e3] = 2e2, [e2, e3] = e3, [eq, e4] = 0; so(3)⊕A1: [e1, e2] = e3, [e3, e1] = e2, [e2, e3] = e1, [eq, e4] = 0. Remark 1. Notations and conventions. Below ∂a = ∂/∂xa, x = (x1, . . . , xn), x̌ = (x3, . . . , xn), x̂ = (x4, . . . , xn), a = 1, n, j, k = 4, n. We use convention on summation over repeat indexes. We denote the N -th realization of an algebra A as R(A,N). Realizations of real semisimple low-dimensional Lie algebras To classify realizations of a m-dimensional Lie algebra A in the most direct way, we have to take m linearly independent vector fields of the general form es = ξ (x)∂a, s = 1,m, and require them to satisfy the commutation relations of A. As a result, we obtain a system of first-order PDEs for the coefficients ξ and then we integrate it, considering all the possible cases. For each case we transform the solution into the simplest form, using either local diffeomorphisms of the space of x and automorphisms of A if we looking for the weakly inequivalent classification or only local diffeomorphisms of the space of x if the strong inequivalence is meant. A disadvantage of this method is the necessity to solve a complicated nonlinear system of PDEs. Another way is to classify sequentially realizations of a series of nested subalgebras of A, starting with a one-dimensional subalgebra or other subalgebra with known realizations and ending up with A. Thus, to prove the following theorem, we apply the above method, starting from the algebra A2.1 formed by e1 and e2. Theorem 1. Let first-order differential operators satisfy the commutation relations of sl(2,R). Then there exist transformations reducing these operators to one of the forms: 1) ∂1, x1∂1 + x2∂2, x 2 1 ∂1 + 2x1x2∂2 + x2∂3; 2) ∂1, x1∂1 + x2∂2, (x 2 1 + x 2 )∂1 + 2x1x2∂2; 3) ∂1, x1∂1 + x2∂2, (x 2 1 − x 2 )∂1 + 2x1x2∂2; 4) ∂1, x1∂1 + x2∂2, x 2 1 ∂1 + 2x1x2∂2; 5) ∂1, x1∂1, x 2 1 ∂1. Theorem 2. A complete list of inequivalent realizations of sl(2,R) ⊕ A1 is exhausted by the following ones: 1) ∂1, x1∂1 + x2∂2, x 2 1 ∂1 + 2x1x2∂2 + x2∂3, ∂4; 2) ∂1, x1∂1 + x2∂2, x 2 1 ∂1 + 2x1x2∂2 + x2∂3, x2∂1 + 2x2x3∂2 + (x 2 3 + x4)∂3; 3) ∂1, x1∂1 + x2∂2, x 2 1 ∂1 + 2x1x2∂2 + x2∂3, x2∂1 + 2x2x3∂2 + (x 2 3 + c)∂3, c ∈ {−1; 0; 1}; 4) ∂1, x1∂1 + x2∂2, (x 2 1 + x 2 )∂1 + 2x1x2∂2, ∂3; 5) ∂1, x1∂1 + x2∂2, (x 2 1 − x 2 )∂1 + 2x1x2∂2, ∂3; 6) ∂1, x1∂1 + x2∂2, x 2 1 ∂1 + 2x1x2∂2, ∂3; 7) ∂1, x1∂1 + x2∂2, x 2 1 ∂1 + 2x1x2∂2, x2x3∂2; 8) ∂1, x1∂1 + x2∂2, x 2 1 ∂1 + 2x1x2∂2, x2∂2; 9) ∂1, x1∂1, x 2 1 ∂1, ∂2. Proof. The automorphism group of the algebra sl(2,R)⊕A1 is a direct product of the automorphism groups of sl(2,R) and A1. We extend the realizations of sl(2,R) to realizations of sl(2,R) ⊕ A1 with the operator e4, beginning from the most general form e4 = η (x)∂a. Realizations of real semisimple low-dimensional Lie algebras Consider the realization R(sl(2,R), 1). The general form of the operator e4 which commutates with the basis elements of R(sl(2,R), 1) is as follows: e4 = ξ 1x2∂1 + (2ξ 1x3 + ξ 2)x2∂2 + (ξ 1x23 + ξ 2x3 + ξ 3)∂3 + ξ ∂j , where ξ are arbitrary functions of x̂. The form of operators e1, e2 and e3 is preserved by the transformation: x̃1 = x1+ f1x2 1− fx3 , x̃2 = f2x2 (1− fx3) , x̃3 = f2x3 1− fx3 +f, x̃j = f , where f are arbitrary functions of x̂. After action of this transformation the operator e4 turns into the operator ẽ4 of the same form with following functions ξ̃a: ξ̃ = 1 f (ξ + ξf + ξ(f) + ξf j ), ξ̃ 2 = ξ + 2ξf − 2ξ̃(f) + ξ f j f , ξ̃ = ξf − ξ̃(f) − ξ̃f + ξf + ξf j , ξ̃ = ξf j k . Here and below subscripts mean differentiation with respect to the corresponding variables xa. There are two possible cases. 1) ∃j: ξ 6= 0. Then the operator e4 can be transformed to the form ẽ4 = ∂4 and we obtain the realization R(sl(2,R)⊕A1, 1). 2) ξ̃ = 0. The expression I = (ξ) − 4ξξ is an invariant of the above transformations of ξ. Therefore, we can make ξ̃1 = 1, ξ̃2 = 0, ξ̃3 = I. If I = const then we obtain the realization R(sl(2,R)⊕A1, 3), otherwise we can choose new variable x̃4 = I and obtain the realization R(sl(2,R)⊕A1, 2). We omit calculations on the realizations R(sl(2,R)⊕A1, 4–9), because they are simpler than the adduced ones and are made in the same way, starting from three other realizations of the algebra sl(2,R). Inequivalence of the obtained realizations can be easily proved by means of technics proposed in [9]. Theorem 3. There are only two inequivalent realizations of the algebra so(3): 1) − sinx1 tanx2∂1 − cos x1∂2, − cos x1 tan x2∂1 + sinx1∂2, ∂1; 2) − sinx1 tanx2∂1 − cos x1∂2 + sinx1 sec x2∂3, − cosx1 tanx2∂1 + sinx1∂2 + cosx1 sec x2∂3, ∂1. Remark 2. The realizations R(so(3), 1) and R(so(3), 2) are well-known. At the best of our knowledge, completeness of the list of these realizations was first proved in [8]. We do not assert that the adduced forms of realizations are optimal for all applications and the classification from Theorem 3 is canonical. Realizations of real semisimple low-dimensional Lie algebras Consider the realization R(so(3), 1) of rank 2 in more details. It acts transitively on the manifold S. With the stereographic projection tan x1 = t/x, cotan x2 = √ x + t it can be reduced to the well known realization on the plane [10]: (1 + t2)∂t + xt∂x, x∂t − t∂x, −xt∂t − (1 + x2)∂x If dimension of the x-space is not smaller than 3, the variables x1, x2 and the implicit variable x3 in R(so(3), 1) can be interpreted as the angles and the radius of the spherical coordinates (imbedding S in R). Then in the corresponding Cartesian coordinates R(so(3), 1) has the well-known form: x2∂3 − x3∂2, x3∂1 − x1∂3, x1∂2 − x2∂1, which is generated by the standard representation of SO(3) in R. Theorem 4. A list of inequivalent realizations of the algebra so(3) ⊕ A1 in vector fields on a space of an arbitrary (finite) number of variables is exhausted by the following ones: 1) − sinx1 tanx2∂1 − cos x1∂2, − cos x1 tan x2∂1 + sinx1∂2, ∂1, ∂3; 2) − sinx1 tanx2∂1 − cos x1∂2 + sinx1 sec x2∂3, − cos x1 tan x2∂1 + sinx1∂2 + cos x1 sec x2∂3, ∂1, ∂3; 3) − sinx1 tanx2∂1 − cos x1∂2 + sinx1 sec x2∂3, − cos x1 tan x2∂1 + sinx1∂2 + cos x1 sec x2∂3, ∂1, x4∂3; 4) − sinx1 tanx2∂1 − cos x1∂2 + sinx1 sec x2∂3, − cos x1 tan x2∂1 + sinx1∂2 + cos x1 sec x2∂3, ∂1, ∂4. Proof. The automorphism group of so(3) ⊕ A1 is the direct product of the automorphism groups of so(3) and A1. To classify realizations of so(3) ⊕A1, we start from the realizations R(so(3), 1) and R(so(3), 2). For convenience we rewrite them as a realization parameterized with α ∈ {0; 1}: e1 = − sinx1 tanx2∂1 − cos x1∂2 + α sinx1 sec x2∂3, e2 = − cosx1 tanx2∂1 + sinx1∂2 + α cos x1 secx2∂3, e3 = ∂1 The values α = 0 and α = 1 correspond to the realizations R(so(3), 1) and R(so(3), 2). We take the operator e4 in the most general form e4 = ξ (x)∂a and obtain the equations for ξ(x) from condition of vanishing commutators of e4 with the other basis elements: ξ 1 = 0, ξ 2 2 = 0, ξ j 2 = 0, (1a) Realizations of real semisimple low-dimensional Lie algebras αξ 3 − ξ cos x2 = 0, αξ 3 cos x2 + ξ = 0, ξ 2 cos x2 + αξ = 0, (1b) ξ 2 − ξ tanx2 = 0, αξ 3 − αξ tan x2 = 0, αξ 3 = 0. (1c) It follows from (1a) that ξ = ξ(x̌) and ξ = ξ(x̌). In the case α = 0 we obtain from (1b) that ξ = 0, ξ = 0 and ξ = ξ(x̌). Then e4 has the form e4 = ξ (x̌)∂p, where p = 3, n, and one of the coefficients ξ does not vanish. Using allowable transformations of variables x̃1 = x1, x̃2 = x2, x̃p = f (x̌), we can can make ξ = 1 and ξ = 0. (”Allowable” means that such transformations preserve the form of e1, e2 and e3.) As a result, we obtain realization R(so(3)⊕A1, 1). Consider the case α = 1. The general solution of system (1a)–(1c) is: ξ = φ sinx3 + φ 2 cosx3 cos x2 , ξ = φ sinx3 − φ cos x3, ξ3 = φ 3 − (φ sinx3 + φ cos x3) tan x2, ξ = φ , where φ are arbitrary functions of x̂. Therefore, the operator e4 can be presented in the form: e4 = φ 1e′1 + φ 2e′2 + φ 3e′3 + φ ∂j , where operators e 1 –e 3 are obtained from e1–e3 with transposition of the variables x1 and x3. The next step is to simplify the operator e4. Since in this case the direct method of finding allowable transformations of variables is too cumbersome and complicated, we use the infinitesimal approach. An one-parametric group of local transformations in the space of variables x preserves the form of operators e1–e3 if its infinitesimal generator Q commutes with these operators. Therefore, Q has the same form as e4: Q = ρ1e′1 + ρ 2e′2 + ρ 3e′3 + ρ ∂j , where ρ are arbitrary functions of x̂. There are two possible cases: ξ = 0 or ∃j: ξ 6= 0. In any case the operator e4 can be transformed by means of allowable transformations x̃1 = x1, x̃2 = x2, x̃3 = x3, x̃j = f (x̂) to the form: e4 = φ 1e′1 + φ 2e′2 + φ 3e′3 + β∂4, β ∈ {0, 1}. Below we use only transformations preserving x̂ and, therefore, assume ρ = 0. Introducing the vector notations φ̄ = (φ, φ, φ), ρ̄ = (ρ, ρ, ρ), ρ̄4 = (ρ 4 , ρ 4 , ρ 4 ), and ē = (e 1 , e 2 , e 3 ), we can present the commutator [e4, Q] as follows: [e4, Q] = (ρ̄× φ̄− βρ̄4) · ē′, Realizations of real semisimple low-dimensional Lie algebras where ”×” and ”·” denote the vector and scalar products. The finite transformations ̃̄ φ = γ̄(ε, φ̄, x̂) generated by Q are found by integrating the Lie equations: dγ̄ dε = ρ̄× γ̄ − βρ̄4, γ̄|ε=0 = φ̄, (2) where ε is a group parameter and x̂ are assumed constants. Therefore, γ̄ = OJ(ε)Oφ̄− βO ∫ ε