Fixed points and steady solitons for the two-loop renormalization group flow

Solitons for the two-loop renormalization group flow are studied in the four-dimensional homogeneous setting, providing a classification of algebraic steady four-dimensional solitons.


Introduction
The Ricci flow (introduced by Hamilton [16] and Friedan [11]) is of fundamental importance in mathematics. It has been shown to be an effective tool for attacking important problems in geometry like Thurston's geometrization conjecture and the Poincaré conjecture. The Ricci flow also has a similar importance in physics, arising as the first-order approximation of the renormalization group flow for the non-linear sigma model of quantum field theory.
Recently, there has been interest in the second-order approximation of this renormalization group flow, or the 2-loop renormalization group flow (RG2 flow for short), which mathematically is described by where RG[g] = −2ρ − α 2Ř and α denotes a positive coupling constant. Here, ρ denotes the Ricci tensor andŘ is the symmetric (0, 2)-tensor fielď R ij = R iabc R j abc of the evolving metric g t . Independently of the physical significance of the flow (1), it is mathematically interesting as a perturbation of the Ricci flow. We refer to [7,[12][13][14] and references therein for more information on the RG2 flow.
The purpose of this paper is to study fixed points of the RG2 flow in the four-dimensional case. Genuine fixed points of the flow are provided by those manifolds where the right-hand side of (1) vanishes, i.e., ρ + α 4Ř = 0. In dimension two, this condition reduces to constant negative curvature. The analysis in dimension three has been carried out in [13], where it is shown that solutions of non-constant curvature must have Ricci curvatures Q = diag[−2/α, −2α, 0] or Q = diag[−4/α, −2α, −2/α], where Q denotes the metrically equivalent (1, 1)-tensor field associated with ρ. In the homogeneous situation, these geometries correspond to product manifolds R × N (c), where N (c) is a surface of constant curvature c, or a left-invariant metric on SU (2) with α < 0.
Tracing the tensor field RG [g], one has that if RG[g] = 0, then the coupling constant α satisfies τ + α 4 R 2 = 0, where τ is the scalar curvature and R 2 = R ijk R ijk . The previous expression does not necessarily mean that α must be constant, but it is expressible in terms of the scalar curvature and the norm of the curvature tensor, which are constant in the homogeneous case. On the other hand, the functional defined by the four-dimensional Gauss-Bonnet integrand g → M { R 2 − 4 ρ 2 + τ 2 } vol g is constant in dimension 4 and thus any compact four-dimensional manifold satisfies the curvature identity (see [4]) whereρ and R [ρ] are the symmetric (0, 2)-tensor fields given byρ ij = ρ ia ρ a j and R[ρ] ij = R iabj ρ ab . (See [10] for an extension of the previous identity to the non-compact case). If (M, g) is Einstein, then all terms in (2) vanish and it immediately follows that any Einstein four-dimensional manifold satisfies ρ + α 4Ř = 0 for α = −4τ R −2 , which shows that Einstein four-manifolds are genuine fixed points of the flow (1). In the homogeneous setting the situation is rather restrictive, and we show in Sect. 7 that any other example is a product as follows: The above result is in sharp contrast with the geometry of the Ricci flow, since genuine fixed points of the Ricci flow are Ricci-flat manifolds, which are necessarily flat in the homogeneous setting [1].
In addition to the cases above, one may consider geometrical fixed points of the RG2 flow, i.e., solutions g(t) which are fixed modulo scalings and diffeomorphisms. Given a one-parameter family ψ t of diffeomorphisms of M (with ψ 0 = Id), a solution of the form g(t) = σ(t)ψ * t g (where σ is a real-valued 4Q is a derivation of the Lie algebra (g, ·, · ) with coupling constant α if and only if D * = Q * + κα 4Q * is a derivation of the Lie algebra (g, ·, · * ) with coupling constant κα. Aimed to describe fourdimensional RG2 algebraic steady solitons, we therefore work modulo homotheties in what follows to simplify the exposition.
Let H be a Lie group with a left-invariant metric determined by an inner product on the Lie algebra (h, ·, · h ) and let G = R × H be the product Lie group with product left-invariant metric · , · g = dt ⊗ dt ⊕ · , · h . Since RG g = 0 ⊕ RG h , one has that if (h, ·, · h ) is an RG2 algebraic steady soliton, then so is (g, ·, · g ). Conversely, assume that a (complete and simply connected) Lie group G with left-invariant metric is an RG2 algebraic steady soliton. Further, assume that there exists a parallel left-invariant vector field on G. Then it splits a one-dimensional factor so that the Lie group splits isometrically (as a Riemannian manifold) G = R × N , where N is a complete and simply connected three-dimensional homogeneous manifold. Hence, N is either symmetric (in which case G is also a symmetric space) or N is isometric to a Lie group H. Correspondingly, the tensor field RG also splits as RG g = 0 ⊕ RG h and so does the corresponding (1, 1)-tensor field RG g . Hence, if G is an RG2 algebraic steady soliton, then so is H just considering the derivation determined by RG h .
Four-dimensional Lie groups are given by the product Lie groups SU (2)× R and SL(2, R) × R (where we use the non-standard notation to represent the universal covering) and the semi-direct products R E (1,1), three-dimensional Poincaré group, the Euclidean group, the Heisenberg group and the Abelian group (see the discussion in [3]). Let sl(2, R), su(2), e(1, 1), e(2), h 3 and r 3 be the Lie algebras corresponding to the three-dimensional Lie groups above. We analyze in Sect. 7 the existence of RG2 algebraic steady solitons on four-dimensional irreducible Lie groups, since otherwise it reduces to the three-dimensional case which is discussed in Sect. 2 (see also [22]) as follows. where κ > 0, κ = 1.
(2) R h 3 , for a coupling constant α = 2, given by The above result is in sharp contrast with the Ricci flow case where steady homogeneous Ricci solitons are Ricci flat and thus flat. Moreover, the Lie groups (G, ·, · ) corresponding to cases (2) and (4) are expanding (algebraic) Ricci solitons, while Lie groups corresponding to cases (1), (3) and (5) are not Ricci solitons. It follows from the analysis in Sects. 3-6 that all metrics in Theorem 1.2 represent different homothetical classes. Theorem 1.2 shows, therefore, a way in which the RG2 flow differs from the Ricci flow.
Id is a derivation and hence (G, ·, · ) is also an RG2 algebraic soliton. Therefore, there is a vector field ξ on G such that L ξ g + ρ + α 4Ř = 3( α 2 − 1) · , · for any value of the coupling constant α, thus resulting in a steady, shrinking or expanding RG2 soliton depending on the value of α, in sharp contrast to Ricci solitons.
The paper is organized as follows. We recall some known facts about RG2 algebraic steady solitons from [22] (see also [15]) in the three-dimensional case and consider also the non-unimodular setting in Sect. 2. The proof of Theorem 1.2 follows after a case by case analysis developed through Sects. 3 to 6. Finally, the proof of Theorem 1.1 and Theorem 1.2 are given in Sect. 7. In particular, we show in Sect. 7.2 that all metrics in Theorem 1.2 represent different homothetical classes.

Gröbner basis
Let Q j i = ρ i g j andQ j i =Ř i g j denote the corresponding (1, 1)-tensor fields metrically equivalent to ρ andŘ, respectively. Let G be a Lie group with Lie algebra g and let D be the endomorphism of the Lie algebra determined by D = Q + α 4Q . Then, D defines an RG2 algebraic steady soliton if and only if it is a derivation (i.e., D[x, y] − [Dx, y] − [x, Dy] = 0) (see [22]). Let The components D ijk determine a system of polynomial equations {P ijk = 0} on the structure constants which is rather involved, although it can be obtained from the expressions of the Ricci tensor ρ and theŘtensor. To obtain a full classification, one needs to solve the corresponding polynomial system of equations. When the system under consideration is simple, it is an elementary problem to find all common roots, but if the number of equations and their degrees increase, it may become a quite unmanageable assignment. There exist, however, some well-known strategies to approach this kind of problem.
Given a set S of polynomials P ijk ∈ R[x 1 , . . . , x n ], an n-tuple of real numbers a = (a 1 , . . . , a n ) is a solution of the system of polynomial equations determined by S if and only if P ijk ( a) = 0 for all i, j, k. It is immediate to see that a is a solution of the polynomial system of equations determined by S if and only if it is a solution of the system determined by all the polynomials in the ideal I = P ijk generated by S: if two sets of polynomials generate the same ideal, the corresponding zero sets must be identical. Therefore, our strategy for solving the rather large polynomial systems consists of obtaining "better" polynomials that belong to the ideals generated by the corresponding polynomial systems. This is achieved by using the theory of Gröbner bases, whose construction we briefly recall below (see [8]). Let x α = x α1 1 · · · x αn n with α ∈ Z n ≥0 be a monomial in R[x 1 , . . . , x n ]. A monomial ordering is any relation on the set of monomials x α with α ∈ Z n ≥0 satisfying (1) It is a total ordering on Z n ≥0 . (2) If α > β and γ ∈ Z n ≥0 , then α + γ > β + γ.
≥0 is well ordered, so that every non-empty subset of Z n ≥0 has a smallest element with respect to the given ordering.
Establishing an ordering on Z n ≥0 will induce an ordering on the monomials. For our purposes, we will use the lexicographical order and the graded reverse lexicographical order. We say that α > lex β if in the vector α − β ∈ Z n , the leftmost non-zero entry is positive and we say that α > grevlex β if |α| > |β| or |α| = |β| and the rightmost non-zero entry of α − β ∈ Z n is negative.
The basic bricks to introduce Gröbner bases are the leading terms of the polynomials, which are defined as follows. If P = α a α x α is a polynomial in R[x 1 , . . . , x n ], any monomial ordering orders the monomials of P. The multidegree of P is the maximum α ∈ Z n ≥0 so that a α = 0, where the maximum is taken with respect to the given monomial ordering. The corresponding monomial is called the leading term, i.e., LT (P) = a α x α .
Let I ⊂ R[x 1 , . . . , x n ] be a non-zero ideal. Let LT (I) be the set of leading terms of all elements of I and let LT (I) be the ideal generated by the elements of LT (I). It is important to emphasize that if I = P ijk , then LT (I) may be strictly larger than the ideal LT (P ijk ) . A finite subset G = {g 1 , . . . , g ν } of an ideal I is said to be a Gröbner basis with respect to some monomial order if the equality above holds, i.e., LT (g 1 ), . . . , LT (g ν ) = LT (I) .
The Hilbert Basis Theorem (see, for example [8,Chapter 2]) guarantees that any non-zero ideal I ⊂ R[x 1 , . . . , x n ] has a Gröbner basis. Furthermore, any Gröbner basis for an ideal I is a basis of I (see [8] for more information). Therefore a strategy to analyze the solutions of a given system of polynomial equations consists in constructing a Gröbner basis of the ideal generated by the given polynomials and solving the polynomial equations (hopefully simpler) corresponding to the polynomials in the Gröbner basis.
Buchberger's algorithm (among others) provides a constructive algorithm to find one such basis (see, for example, [9]). We would like to emphasize that the Gröbner basis construction is very sensitive to the monomial order. For a certain ordering, simple Gröbner bases can be obtained with a reduced number of polynomials, while for other orderings both the number of polynomials and their form can be completely unmanageable. Lexicographical order is the most appropriate in most cases to get simple bases. However, it is not always possible to use such ordering by computational reasons, and other orderings must be taken into consideration. We therefore emphasize in each case the ordering under consideration.

The unimodular case
Three-dimensional RG2 algebraic steady solitons have been classified by Wears in the unimodular case [22] (see also [15]). Following Milnor [20], all Remark 2.2. Metrics corresponding to case (1.a) are algebraic Ricci solitons for λ = −2 (i.e., Q + 2 Id is a derivation), while metrics corresponding to case (1.b) are not. Moreover, the Heisenberg Lie group is an algebraic Ricci soliton for λ = − 3 2 , while the special unitary group does not admit any non-Einstein Ricci soliton.

The non-unimodular case
In addition to the previous RG2 algebraic steady solitons, there are some non-unimodular ones, which can be described as follows:

Lemma 2.3. Let G be a three-dimensional non-unimodular Lie group. Then G is a non-locally symmetric RG2 algebraic steady soliton if and only if it is homothetic to a left-invariant metric determined by the Lie algebra
where {e 1 , e 2 , e 3 } is an orthonormal basis and one of the following holds: , for a coupling constant α = 1 Proof. Following Milnor [20], any non-locally symmetric left-invariant metric on a non-unimodular Lie group is determined by Lie brackets where {e 1 , e 2 , e 3 } is an orthonormal basis and η ≥ 0, ξ > 0, excluding the case η = 0, ξ = 1. A straightforward calculation shows that D = Q + α 4Q is a derivation of the Lie algebra if and only if the following polynomials vanish identically: Computing a Gröbner basis G of the ideal generated by the polynomials D ijk ∈ R[ξ, η, α] above with respect to the lexicographical order, one gets that such a basis G = {g k } consists of seven polynomials, among which one has the polynomials g 1 Since the polynomials g k also belong to the ideal generated by the D ijk ∈ R[ξ, η, α], any solution of the system of equations {D ijk = 0} must also be a solution of the equations {g k = 0}. Hence, g 1 leads to the following cases: α = 2, η = 0 and ξ 2 = 4(α − 1) 2 . Setting α = 2, since ξ > 0 one easily gets that D = Q + α 4Q is never a derivation of the Lie algebra. Assuming η = 0, one has that D = Q + α 4Q is a derivation if and only if ((ξ 2 + 6)ξ 2 + 1)α − 2(ξ 2 + 1) = 0, which corresponds to Assertion (1).
Now, a standard calculation shows that left-invariant metrics in Assertion (1) corresponding to different values of the parameter ξ are never homothetical, since τ = −2(ξ 2 + 3) and R 2 = 4(3ξ 4 + 10ξ 2 + 3). The same result holds for metrics in Assertion (2), where τ = −4 η 2 + 2 − η η 2 + 1 and R 2 = 16(5η 2 + 4) 2η 2 + 1 ± 2η η 2 + 1 . 3. The direct products SL(2, R) × R and SU (2) × R Let g = g 3 × R be a direct extension of the unimodular Lie algebra g 3 = sl(2, R) or g 3 = su (2). Let ·, · be an inner product on g and let ·, · 3 denote its restriction to g 3 . Following the work of Milnor [20], there exists an where λ 1 , λ 2 , λ 3 ∈ R and λ 1 λ 2 λ 3 = 0. Moreover, the associated Lie group corresponds to SU (2) (resp., . , e 4 } is an orthonormal basis with brackets given by Now, a straightforward calculation shows that the components ρ ij of the Ricci tensor are where the coefficients C ij are polynomials on the structure constants given by  where the coefficients R ij are polynomials on the structure constants given by Vol. 25 (2023) Fixed points and steady solitons Page 11 of 29 42 Since λ 1 λ 2 λ 3 = 0, assume λ 1 = 1 and so we just work with the homothetic metric determined byẽ i = 1 λ1 e i . The expressions of the Ricci tensor and theŘ-tensor imply that D = Q + α 4Q is a derivation of the Lie algebra if and only if the system of polynomial equations {P ijk = 0} holds true, where P ijk are polynomials associated with the coefficients D ijk (which we omit for the sake of brevity). We consider separately the cases corresponding to different possibilities (up to rotation) for the constants k 1 , k 2 and k 3 as follows.

k 1 = k 2 = 0
Simplifying the polynomials {P ijk } when possible as in the previous cases and computing a Gröbner basis G 3 of the ideal generated by {P ijk } ⊂ R[k 3 , K, α, λ 2 , λ 3 ] with respect to the graded reverse lexicographical order, one gets that the polynomial g 31 = k 3 3 (λ 2 − 1) 2 K 2 belongs to G 3 . Hence, either k 3 = 0 or λ 2 = 1 and, in both cases, e 4 determines a parallel leftinvariant vector field. Now, a direct calculation shows that, in this case, any non-symmetric RG2 algebraic steady soliton is determined by Lemma 2.1-(3), obtaining the case given in Lemma 3.1. Finally, the tensor field RG[g] vanishes, which finishes the proof.

The semi-direct products R E(1, 1) and R E(2)
Let g 3 be either the Poincaré algebra e(1, 1) or the Euclidean algebra e(2) and let g = R g 3 be a semi-direct extension. Let ·, · be an inner product on g and ·, · 3 its restriction to g 3 . Following the work of Milnor [20], there exists an orthonormal basis {v 1 where λ 1 , λ 2 ∈ R and λ 1 λ 2 = 0. Moreover, g 3 = e(2) (resp., g 3 = e(1, 1)) if λ 1 λ 2 > 0 (resp., λ 1 λ 2 < 0). The algebra of derivations of g 3 is given by To simplify the notation, set A = a λ1 +k 3 , C = c−k 2 λ 1 and D = d+k 1 λ 2 . Now, a straightforward calculation shows that the components ρ ij of the Ricci tensor become 2K 2 ρ 11 = C 11 , 2K 2 ρ 12 = C 12 , 2K 2 ρ 13 = C 13 , 2Kρ 14 where the coefficients C ij are polynomials on the structure constants given by Recall that any Einstein metric is a genuine fixed point of the RG2 flow. Moreover, the product manifold R × E(1, 1) is an RG2 algebraic steady soliton just considering the RG2 algebraic steady solitons in Lemma 2.1-(1). Henceforth, we focus on the irreducible non-Einstein case. Proof. Let ·, · be a left-invariant metric as described in (6). A standard calculation shows that the components of theŘ-tensor are given by

Lemma 4.1. Let G be a semi-direct product R E(1, 1) or R E(2). Then, G admits a non-Einstein irreducible RG2 algebraic steady soliton if and only if it is homothetic to the Lie group R E(1, 1) determined by
Since λ 1 λ 2 = 0, we work with a homothetic basisẽ i = 1 λ1 e i so that we may assume λ 1 = 1. The expressions of the Ricci tensor and theŘtensor imply that D = Q + α 4Q is a derivation of the Lie algebra if and only if the system of polynomial equations {P ijk = 0} holds true, where P ijk ∈ R[A, b, λ 2 , C, D, K, α] are the polynomials associated with the coefficients D ijk (which we omit for the sake of brevity). We compute a Gröbner basis G of the ideal I = P ijk with respect to the graded reverse lexicographical order and a detailed analysis of that basis shows that the polynomials belong to G. Thus, C = D = 0 and 4Ab 2 (λ 2 − 1)K 4 = 0; so we have three different possibilities corresponding to b = 0, A = 0 or λ 2 = 1. We consider the three situations separately.

b = 0
Constructing a Gröbner basis G 1 of the ideal G ∪ {b} ⊂ R[A, b, λ 2 , K, α] with respect to the lexicographical order, one gets that the polynomial belongs to G 1 . This shows that λ 2 must take one of the different values λ 2 = 1, 3 . If λ 2 = 1, then the metric is Einstein. We analyze the other three cases separately.

λ
, and a straightforward calculation shows that, in this case, D is a derivation of the Lie algebra. Moreover, setting γ = − A K , one has the Lie algebra structure
Moreover, a direct calculation shows that these metrics are irreducible. Furthermore, the metric is a Ricci soliton if and only Q + 2 Id is a derivation, which may occur if and only if κ(κ 2 − 1) = 0. Hence, it is a Ricci soliton if and only if it is Einstein. We conclude that these metrics correspond to the ones given in Lemma 4.1.

λ 2 = 1 and bA = 0
In this case, the manifold is symmetric and isometric to a product R × N (c), where N (c) is a space of constant negative curvature, which finishes the proof.

The semi-direct product R H 3
Let g = R h 3 be a semi-direct product of R with the Heisenberg algebra h 3 . Let ·, · be an inner product on g and let {v 1 , v 2 , v 3 } be an orthonormal basis of h 3 so that The algebra of derivations of h 3 with respect to a rotated basis that we also denote by {v 1 , v 2 , v 3 } is given by (see [5]) Let {v 1 , v 2 , v 3 , v 4 } be a basis of g where ad(e 4 ) is determined by a derivation as above. After normalization, as in the previous sections, there is an orthonormal basis {e 1 , e 2 , e 3 , e 4 } of (g, ·, · ) where the non-zero Lie brackets are given as follows: Vol. 25 (2023) Fixed points and steady solitons Page 17 of 29 42 We use the notation F = f − k 1 γ and H = h + k 2 γ. Then the non-zero components of the Ricci tensor are given by where the coefficients C ij are determined by the structure constants as follows: In addition to Einstein metrics and symmetric products, R × H 3 is an RG2 algebraic steady soliton considering the RG2 algebraic steady solitons in Lemma 2.1-(2). Henceforth, we focus on the irreducible non-Einstein case. Proof. Let ·, · be a left-invariant metric on R H 3 determined by the Lie algebra inner product (7). A straightforward calculation shows that the components of theŘ-tensor are given by where the coefficients R ij are polynomials on the structure constants given by Note that since γ = 0, one may work with a homothetic basisẽ i = 1 γ e i , so that we may assume γ = 1. It follows from the expressions obtained for the Ricci tensor and for theŘ-tensor that D = Q + α 4Q is a derivation of the Lie algebra if and only if the system of polynomial equations {P ijk = 0} holds true, where P ijk ∈ R[a, c, d, H, F, K, α] are the polynomials associated with the coefficients D ijk (which we omit for the sake of brevity). We construct a Gröbner basis G of the ideal generated by the polynomials {P ijk } with respect to the lexicographical order and we get that the polynomial g 1 = d 4 F HK 2 is in the basis. Therefore, we have three possibilities which we analyze separately.

d = 0
Constructing a Gröbner basis G 1 of the ideal generated by G ∪ {d} ⊂ R[a, c, d, H, F, K, α] with respect to the lexicographical order, one has that the polynomials g 11 = aHK 4 and g 12 = aF K 4 are in G 1 . Thus, a = 0 or F = H = 0, a = 0. Therefore, in this case, any RG2 algebraic steady soliton is reducible and one easily checks that it is obtained as a product extension of Lemma 2.1-(2).

F = H = 0 and a = 0.
Since 4K 4 D 131 = a 3 cα, we get c = 0 and thus 4K 5 D 343 = a 3 (4a 2 α + (α − 8)K 2 ), which shows that α = 8K 2 4a 2 +K 2 . Now, a straightforward calculation shows that D = Q + α 4Q is a derivation of the Lie algebra if and only if a = ε Moreover, a direct calculation shows that this metric is never Einstein and that it is irreducible. Furthermore, a straightforward calculation shows that Q + 3 2 Id is a derivation of the Lie algebra and thus an algebraic Ricci soliton. Thus, taking ε = −1, the above left-invariant metric determines an RG2 algebraic steady soliton which corresponds to Assertion (1) with κ = 0.

F = 0, dH = 0
Construct a Gröbner basis G 3 of the ideal generated by G ∪ {F } ⊂ R[a, c, d, H, F, K, α] with respect to the lexicographical order. Since the polynomial g 31 = dH(12 H 2 + 7K 2 )K 4 belongs to G 3 , it follows that no RG2 algebraic steady solitons may exist in this setting, finishing the proof.

The semi-direct product R R 3
Let r 3 be the Abelian algebra. The corresponding algebra of derivations is gl(3, R). For any D ∈ gl(3, R), decomposing it into its symmetric and skewsymmetric part, one has (see [5]) Now, the non-zero components of the Ricci tensor are K 2 ρ 11 = C 11 , K 2 ρ 12 = C 12 , K 2 ρ 13 = C 13 , K 2 ρ 22 = C 22 , K 2 ρ 23 = C 23 , K 2 ρ 33 = C 33 , K 2 ρ 44 = C 44 , where the coefficients C ij are given in terms of the structure constants as follows: In addition to Einstein metrics and symmetric products, R R 3 is an RG2 algebraic steady soliton just considering the RG2 algebraic steady solitons in Lemma 2.3. Henceforth, we focus on the irreducible non-Einstein case. third-order Riemannian scalar curvature invariantŘ = R ijk R k pq R pq ij and setting p = −f − 1, one has that τ −3Ř = f (f + 1) f (f + 1) f 2 + f + 9 + 3 + 1 from where it follows that two different left-invariant metrics in Assertion (1) with p = −f − 1 are never homothetic since 0 < f ≤ 1.
Proof. The non-zero components of theŘ-tensor are given by  Let ·, · be a left-invariant metric on R R 3 determined by the Lie algebra inner product (8). We consider the diagonal matrix diag[a, f, p] in the decomposition of elements of der r 3 and we analyze separately the cases of the determinant being null and non-null.

af p = 0
In this case, at least one of a, f and p must be zero. Thus, without loss of generality, we may assume a = 0. Moreover, one may work with a homothetic basisẽ i = Ke i so that we may assume K = 1. A key observation in this case is that if b = c = 0, then e 1 determines a parallel left-invariant vector field. Hence, if b = c = 0 and G admits an RG2 algebraic steady soliton, then G splits as a product R × H, where H corresponds to the non-unimodular Lie group determined by the Lie algebra h = span{e 2 , e 3 , e 4 } with Otherwise, the expressions obtained for the Ricci tensor and for thě R-tensor imply that D = Q + α 4Q is a derivation of the Lie algebra if and