EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 7, No. 2, 2014, 140-155 ISSN 1307-5543 – www.ejpam.com Affine Subspaces of the Lie Algebra se(1, 1) Dennis I. Barrett, Rory Biggs, Claudiu C. Remsing∗ Department of Mathematics (Pure and Applied), Rhodes University, Grahamstown 6140, South Africa Abstract. We classify the full-rank affine subspaces (resp. parametrized affine subspaces) of the semi- Euclidean Lie algebra se(1, 1). The equivalence relations under consideration are motivated by the study of invariant control affine systems. Exhaustive lists of equivalence representatives are obtained, along with classifying conditions. 2010 Mathematics Subject Classifications: 22E60, 93B27 Key Words and Phrases: Lie algebra, affine subspace, equivalence 1. Introduction A left-invariant control affine system, evolving on a (real, finite-dimensional) Lie group, consists of a family of left-invariant vector fields and a class of “admissible controls”. The family of vector fields is affinely parametrized by the control values. A (typical) control is a piecewise continuous curve u(·) in some control set R`. Such a control system on a (matrix) Lie group G is written, in classical notation, as (cf. [11, 16]) ġ = g(A+ u1B1 + u2B2 + · · ·+ u`B`), g ∈ G, u ∈ R`. (1) Here A, B1, . . . , B` are elements of the Lie algebra g. These systems provide a fertile geometric setting for various problems in mathematical physics, mechanics, elasticity, and differential geometry [3, 8, 10]. There are two natural equivalence relations for left-invariant control affine systems, namely state space equivalence and detached feedback equivalence (cf. [9, 15]). These equivalence relations are significant in that they establish a one-to-one correspondence between the tra- jectories of equivalent systems. Two systems are state space equivalent if one can smoothly transform one system into the other, while keeping the controls fixed. For detached feed- back equivalence (a weaker equivalence relation), invariant feedback transformations of the ∗Corresponding author. Email addresses: dbarrett6@gmail.com (D. Barrett), rorybiggs@gmail.com (R. Biggs), c.c.remsing@ru.ac.za (C. Remsing) http://www.ejpam.com 140 c© 2014 EJPAM All rights reserved. D. Barrett, R. Biggs, C. Remsing / Eur. J. Pure Appl. Math, 7 (2014), 140-155 141 controls are also permitted. It turns out that these two equivalence relations can be entirely characterized at the level of Lie algebras [4]. More precisely, two systems are state space equiv- alent (resp. detached feedback equivalent) if and only if the associated parametrized affine subspaces (resp. affine subspaces) are related by a Lie algebra isomorphism. (For a system (1), the associated parametrized affine subspace is given by Π : u 7→ A+ u1B1 + · · · + u`B`, whereas the associated affine subspace is given by Γ = A+ 〈B1, . . . , B`〉.) Several classes of systems have recently been classified under these equivalence relations [1, 2, 5–7]. In this paper we classify, under the aforementioned equivalence relations, the parametrized affine subspaces (resp. affine subspaces) of the semi-Euclidean Lie algebra se(1,1). We clas- sify first the affine subspaces of se(1,1). Using these results, we then classify the parametrized affine subspaces. Both classifications are organized by distinguishing between the homogene- ity and dimension of the affine subspaces involved. Exhaustive lists of class representatives are obtained, along with associated classifying conditions. A tabulation of the main results is appended. 2. Affine Subspaces and Equivalence An `-dimensional affine subspace of a Lie algebra g is written as Γ = A+Γ0 = A+ 〈B1, . . . , B`〉 (2) where A, B1, . . . , B` ∈ g and B1, . . . , B` are linearly independent. If A ∈ Γ0, we say that Γ is homogeneous; otherwise, it is inhomogeneous. Γ is referred to as an (`, 0)-affine subspace if it is homogeneous, and as an (`, 1)-affine subspace, otherwise. We say that two affine subspaces Γ and Γ′ are L-equivalent if there exists a Lie algebra automorphismψ such thatψ·Γ = Γ′. Note that Γ = A+Γ0 and Γ′ = A′ +Γ′0 are L-equivalent if and only if there exists an automorphism ψ such that ψ · Γ0 = Γ′0 and ψ · A∈ Γ′. A related concept is that of a parametrized affine subspace, i.e., an (injective) affine g- valued map. More precisely, an `-dimensional parametrized affine subspace is a map Π : R`→ g, (u1, . . . , u`) 7→ A+ u1B1 + · · ·+ u`B` where B1, . . . , B` are linearly independent. Whenever convenient, we shall specify Π by simply writing Π : A+ u1B1 + · · · + u`B`. We say that two parametrized affine subspaces Π and Π′ are P-equivalent if there exists an automorphism ψ ∈ Aut(g) such that ψ ◦Π = Π′. Clearly Π : A+u1B1+ · · ·+u`B` is P-equivalent to Π′ : A′+u1B′1+ · · ·+u`B ′ ` if and only if there exists an automorphism ψ such that ψ · A= A′ and ψ · Bi = B′i . An affine subspace is said to have full rank if it generates the entire Lie algebra. (For control systems on Lie groups, the full-rank condition is necessary for controllability). Similarly, a parametrized affine subspace has full rank if its image has full rank. The full-rank property is invariant under both L-equivalence and P-equivalence. Throughout, we assume that all affine subspaces (resp. parametrized affine subspaces) under consideration have full rank. D. Barrett, R. Biggs, C. Remsing / Eur. J. Pure Appl. Math, 7 (2014), 140-155 142 3. Classification The (real) three-dimensional semi-Euclidean Lie algebra se(1, 1) =      0 0 0 x1 0 x3 x2 x3 0   : x1, x2, x3 ∈ R    has standard basis E1 =   0 0 0 1 0 0 0 0 0   , E2 =   0 0 0 0 0 0 1 0 0   , E3 =   0 0 0 0 0 1 0 1 0   . The commutator relations are given by [E2, E3] = −E1, [E3, E1] = E2, [E1, E2] = 0. Remark. se(1,1) is the Lie algebra of the semi-Euclidean group. This matrix Lie group is the group of motions of the Minkowski plane R1,1. The signature (−1, 1) for the Lorentz metric corresponds to the standard basis (E1, E2, E3), whereas the signature (1,−1) corresponds to the (Bianchi-Behr) basis (E1, E2,−E3) [12–14]. With respect to the standard basis (E1, E2, E3), the group of automorphisms Aut (se(1, 1)) takes the form      x y v ςy ςx w 0 0 ς   : v, w, x , y ∈ R,ς ∈ {−1, 1}, x2 6= y2    . The subsets 〈E1, E2〉 and 〈E1 + E2〉 ∪ 〈E1 − E2〉 are invariant. We now classify, under L-equivalence (resp. P-equivalence), all full-rank affine subspaces (resp. parametrized affine subspaces) of se(1, 1). We outline the approach followed in classi- fying these objects. First, we distinguish between the dimension and the homogeneity of the affine subspaces; this yields four types of affine subspaces. The invariant subsets allow us to distinguish between various (families of) equivalence classes. In each case, we simplify an arbitrary affine subspace (resp. parametrized affine subspace) by successively applying au- tomorphisms. Finally, we verify that all the candidates for class representatives are distinct and not equivalent. Families of representatives are typically parametrized by constants α > 0, β = (βi) and γ= (γi), where βi 6= 0, γi ∈ R. Remark. On se(1,1) (in fact, on any three-dimensional Lie algebra), the full-rank condition for an affine subspace (2) can be characterized as follows. No (1, 0)-affine subspace has full rank. A (1,1)-affine subspace has full rank if and only if A, B1 and [A, B1] are linearly independent, whereas a (2,0)-affine subspace has full rank if and only if B1, B2 and [B1, B2] are linearly independent. Also, it is clear that any (2,1)-affine subspace or (3, 0)-affine subspace has full rank. D. Barrett, R. Biggs, C. Remsing / Eur. J. Pure Appl. Math, 7 (2014), 140-155 143 3.1. Affine Subspaces We begin by classifying the affine subspaces of se(1,1). Such a classification has been obtained elsewhere [7]. However, for the sake of completeness, we include full proofs here. We denote by E∗3 the corresponding element of the dual basis. Theorem 1. Any (1, 1)-affine subspace Γ = A+Γ0 is L-equivalent to exactly one of the following affine subspaces ¨ Γ(1,1) 1 = E1 + 〈E3〉 E∗3(Γ 0) 6= {0} Γ(1,1) 2,α = αE3 + 〈E1〉 E∗3(Γ 0) = {0}. Here α > 0, with different values of the parameter yielding distinct (non-equivalent) class repre- sentatives. Proof. Suppose that E∗3(Γ 0) 6= {0}. Then Γ = a1E1 + a2E2 + 〈b1E1 + b2E2 + E3〉 and    a1 a2 1−a2 2 − a2 a2 1−a2 2 0 − a2 a2 1−a2 2 a1 a2 1−a2 2 0 0 0 1      1 0 −b1 0 1 −b2 0 0 1   · Γ = E1 + 〈E3〉= Γ (1,1) 1 ; as Γ has full rank, we have a2 1 6= a2 2. Thus Γ is L-equivalent to Γ(1,1) 1 . Suppose E∗3(Γ 0) = {0}. Then Γ = a1E1 + a2E2 + a3E3 + 〈b1E1 + b2E2〉 and    b1 b2 1−b2 2 − b2 b2 1−b2 2 0 − sgn(a3)b2 b2 1−b2 2 sgn(a3)b1 b2 1−b2 2 0 0 0 sgn(a3)      1 0 − a1 a3 0 1 − a2 a3 0 0 1   · Γ = |a3|E3 + 〈E1〉= Γ (1,1) 2,α whereα= |a3|> 0. Due to the full-rank assumption, we have b2 1 6= b2 2. Hence Γ isL-equivalent to Γ(1,1) 2,α . As 〈E1, E2〉 is an invariant subspace, Γ(1,1) 1 cannot be L-equivalent to Γ(1,1) 2,α . It is easy to show that Γ(1,1) 2,α and Γ(1,1) 2,α′ are L-equivalent only if α= α′. If Γ = 〈A, B〉 is a (2, 0)-affine subspace, then A+ 〈B〉 is a (1,1)-affine subspace and hence is L-equivalent to either Γ(1,1) 1 or Γ(1,1) 2,α . Thus Γ is L-equivalent to Γ(1,1) 1 � or Γ(1,1) 2,α � . Accordingly, we get the following classification of (2, 0)-affine subspaces. Corollary 1. Any (2,0)-affine subspace is L-equivalent to Γ(2,0) = 〈E1, E3〉. Theorem 2. Any (2, 1)-affine subspace Γ = A+Γ0 is L-equivalent to exactly one of the following affine subspaces      Γ(2,1) 1 = E2 + 〈E1, E3〉 E∗3(Γ 0) 6= {0}, E1 + E2 /∈ Γ0 and E1 − E2 /∈ Γ0 Γ(2,1) 2 = E1 + 〈E1 + E2, E3〉 E∗3(Γ 0) 6= {0}, E1 + E2 ∈ Γ0 or E1 − E2 ∈ Γ0 Γ(2,1) 3,α = αE3 + 〈E1, E2〉 E∗3(Γ 0) = {0}. Here α > 0, with different values of the parameter yielding distinct (non-equivalent) class repre- sentatives. D. Barrett, R. Biggs, C. Remsing / Eur. J. Pure Appl. Math, 7 (2014), 140-155 144 Proof. Suppose that E∗3(Γ 0) 6= {0}, E1 + E2 /∈ Γ0 and E1 − E2 /∈ Γ0. Then Γ = a1E1 + a2E2 + 〈b1E1 + b2E2, c1E1 + c2E2 + E3〉 with b2 1 6= b2 2. Hence Γ′ =   1 0 −c1 0 1 −c2 0 0 1   · Γ = a1E1 + a2E2 + 〈b1E1 + b2E2, E3〉 where b1a2 − a1 b2 6= 0 (as Γ is inhomogeneous). Consequently    b2 1−b2 2 b1a2−a1 b2 0 0 0 b2 1−b2 2 b1a2−a1 b2 0 0 0 1       b1 b2 1−b2 2 − b2 b2 1−b2 2 0 − b2 b2 1−b2 2 b1 b2 1−b2 2 0 0 0 1    · Γ′ = a1 b2 − a2 b2 b1a2 − a1 b2 E1 + E2 + ® b2 1 − b2 2 b1a2 − a1 b2 E1, E3 ¸ = E2 + 〈E1, E3〉= Γ (2,1) 1 . Thus Γ is L-equivalent to Γ(2,1) 1 . On the other hand, suppose that E∗3(Γ 0) 6= {0} and E1 ± E2 ∈ Γ0. Then Γ = a1E1 + a2E2 + 〈E1 ± E2, b1E1 + b2E2 + E3〉 and Γ′ =   1 0 −b1 0 1 −b2 0 0 1   · Γ = a1E1 + a2E2 + 〈E1 ± E2, E3〉 where a1 ∓ a2 6= 0 (as Γ is inhomogeneous). Therefore    a1 a2 1−a2 2 − a2 a2 1−a2 2 0 ∓ a2 a2 1−a2 2 ± a1 a2 1−a2 2 0 0 0 ±1    · Γ′ =E1 + ® a1 ∓ a2 a2 1 − a2 2 (E1 + E2),±E3 ¸ =E1 + 〈E1 + E2, E3〉= Γ (2,1) 2 . Thus Γ is L-equivalent to Γ(2,1) 2 . Lastly, suppose that E∗3(Γ 0) = {0}. Then Γ = a3E3 + 〈E1, E2〉 and   1 0 0 0 sgn(a3) 0 0 0 sgn(a3)   · Γ = |a3|E3 + 〈E1, sgn(a3)E2〉= Γ (2,1) 3,α D. Barrett, R. Biggs, C. Remsing / Eur. J. Pure Appl. Math, 7 (2014), 140-155 145 where α= |a3|> 0. Hence Γ is L-equivalent to Γ(2,1) 3,α . As 〈E1, E2〉 and 〈E1 + E2〉 ∪ 〈E1 − E2〉 are invariant subsets, no two of Γ(2,1) 1 , Γ(2,1) 2 and Γ(2,1) 3,α are L-equivalent. It is a simple matter to show that Γ(2,1) 3,α is L-equivalent to Γ(2,1) 3,α′ only if α= α′. Remark. There is only one (3,0)-affine subspace, namely se(1, 1) itself. 3.2. Parametrized Affine Subspaces When convenient, a parametrized affine subspace specified by Π : 3 ∑ i=1 ai Ei + u1 3 ∑ i=1 bi Ei + u2 3 ∑ i=1 ci Ei + u3 3 ∑ i=1 di Ei will be represented (in matrix form) as   a1 b1 c1 d1 a2 b2 c2 d2 a3 b3 c3 d3   . Since any automorphism ψ is identified with its matrix, the composition ψ ◦ Π becomes a matrix multiplication. We begin by classifying the parametrized (1, 1)-affine subspaces. Theorem 3. Let Π be a parametrization of a (1, 1)-affine subspace Γ. (i) If Γ isL-equivalent to Γ(1,1) 1 , thenΠ isP-equivalent to exactly one of the following parametrized affine subspaces Π(1,1) 1,α,γ : E1 + γ1E3 + u(αE3). (ii) If Γ isL-equivalent to Γ(1,1) 2,α , thenΠ isP-equivalent to exactly one of the following parametrized affine subspaces Π(1,1) 2,α : αE3 + uE1. Hereα > 0 and γ1 ∈ R, with different values of these parameters yielding distinct (non-equivalent) class representatives. Proof. Let Π : ∑3 i=1 ai Ei+u ∑3 i=1 bi Ei . By Theorem 1, Γ is L-equivalent to Γ(1,1) 1 = E1+〈E3〉 or Γ(1,1) 2,α = αE3 + 〈E1〉. (i) Suppose that Γ is L-equivalent to Γ(1,1) 1 . Then b3 6= 0 and    1 0 − b1 b3 0 sgn(b3) − sgn(b3)b2 b3 0 0 sgn(b3)      a1 b1 a2 b2 a3 b3  =   a′1 0 a′2 0 a′3 |b3|   D. Barrett, R. Biggs, C. Remsing / Eur. J. Pure Appl. Math, 7 (2014), 140-155 146 (for some a′1, a′2, a′3 ∈ R). Since Γ is L-equivalent to Γ(1,1) 1 , we have a′1 6= 0, a′2 6= 0 and (a′1) 2 6= (a′2) 2. Accordingly,     a′1 (a′1) 2−(a′2)2 − a′2 (a′1) 2−(a′2)2 0 − a′2 (a′1) 2−(a′2)2 a′1 (a′1) 2−(a′2)2 0 0 0 1       a′1 0 a′2 0 a′3 |b3|  =   1 0 0 0 γ1 α   where α= |b3|> 0 and γ1 = a′3 ∈ R. Thus Π is P-equivalent to Π(1,1) 1,α,γ. (ii) Suppose that Γ is L-equivalent to Γ(1,1) 2,α . Then b3 = 0, a3 6= 0, b2 1 6= b2 2 and   1 0 − a1 a3 0 sgn(a3) − sgn(a3)a2 a3 0 0 sgn(a3)     a1 b1 a2 b2 a3 0  =   0 b1 0 sgn(a3)b2 |a3| 0   . Furthermore,    b1 b2 1−b2 2 − sgn(a3)b2 b2 1−b2 2 0 − sgn(a3)b2 b2 1−b2 2 b1 b2 1−b2 2 0 0 0 1      0 b1 0 sgn(a3)b2 |a3| 0  =   0 1 0 0 α 0   where α= |a3|> 0. Thus Π is P-equivalent to Π(1,1) 2,α . By Theorem 1, Γ(1,1) 1 and Γ(1,1) 2,α are not L-equivalent. Hence Π(1,1) 1,α,γ is not P-equivalent to Π(1,1) 2,α . Suppose there exists ψ ∈ Aut (se(1, 1)) such that ψ ◦Π(1,1) 1,α,γ = Π (1,1) 1,α′,γ′ . Then   x + vγ1 vα ςy +wγ1 wα ςγ1 ςα  =   1 0 0 0 γ′1 ςα′   (for some v, w ∈ R, x2 6= y2 and ς ∈ {−1,1}) which implies that α = α′, ς = 1 and γ = γ′. If ψ ◦Π(1,1) 2,α = Π (1,1) 2,α′ for some automorphism ψ, then   vα x wα ςy ςα 0  =   0 1 0 0 α′ 0   and so α= α′. If Π : A + u1B + u2C is a parametrized (2,0)-affine subspace, then u 7→ B + uC is a parametrized (1,1)-affine subspace, and hence is P-equivalent to Π(1,1) 1,α,γ or Π(1,1) 2,α . We use this fact to arrive at the following classification of the parametrized (2,0)-affine subspaces. These representatives parametrize Γ(2,0). D. Barrett, R. Biggs, C. Remsing / Eur. J. Pure Appl. Math, 7 (2014), 140-155 147 Corollary 2. Let Π : ∑3 i=1 ai Ei + u1 ∑3 i=1 bi Ei + u2 ∑3 i=1 ci Ei be a parametrized (2, 0)-affine subspace. Π is P-equivalent to exactly one of the following parametrized affine subspaces ( Π(2,0) 1,α,γ : γ1E1 + γ2E3 + u1(E1 + γ3E3) + u2(αE3) c3 6= 0 Π(2,0) 2,α,γ : γ1E1 + γ2E3 + u1(αE3) + u2E1 c3 = 0. Here α > 0 and γ1,γ2,γ3 ∈ R, with different values of these parameters yielding distinct (non- equivalent) class representatives. We now proceed to the classification of the parametrized (2, 1)-affine subspaces. Lemma 1. Let X = ∑3 i=1 x i Ei , Y = ∑3 i=1 yi Ei and Z = z3E3 be linearly independent elements of se(1,1) and let σ ∈ {−1, 1}. (i) If y2 1 6= y2 2 , then there exists ψ ∈ Aut (se(1, 1)) such that ψ · X =   x ′1 σx ′2 σx3   , ψ · Y =   1 0 σy3   , ψ · Z =   0 0 σz3   for some x ′1, x ′2 ∈ R. (ii) If y2 1 = y2 2 and x2 1 6= x2 2 , then there exists % ∈ {−1,1} and ψ ∈ Aut (se(1, 1)) such that ψ · X =   x ′1 0 σx3   , ψ · Y =   1 %σ σy3   , ψ · Z =   0 0 σz3   for some x ′1 ∈ R. (iii) If y2 1 = y2 2 and x2 1 = x2 2 , then there exists % ∈ {−1,1} and ψ ∈ Aut (se(1, 1)) such that ψ · X =   1 −σ σ%x3   , ψ · Y =   1 σ σ% y3   , ψ · Z =   0 0 σ%z3   . Proof. (i) Suppose that y2 1 6= y2 2 . Then    y1 y2 1−y2 2 − y2 y2 1−y2 2 0 − σy2 y2 1−y2 2 σy1 y2 1−y2 2 0 0 0 σ      x1 y1 0 x2 y2 0 x3 y3 z3  =   x ′1 1 0 σx ′2 0 0 σx3 σy3 σz3   . (ii) Suppose that y2 1 = y2 2 and x2 1 6= x2 2 . Let y0 = y1 6= 0. Then y2 = ±y0 and    x1 x2 1−x2 2 − x2 x2 1−x2 2 0 ∓ x2 x2 1−x2 2 ± x1 x2 1−x2 2 0 0 0 ±1      x1 y0 0 x2 ±y0 0 x3 y3 z3  =   1 y ′0 0 0 y ′0 0 ±x3 ±y3 ±z3   . D. Barrett, R. Biggs, C. Remsing / Eur. J. Pure Appl. Math, 7 (2014), 140-155 148 Furthermore,    1 y ′0 0 0 0 ± σy ′0 0 0 0 ±σ      1 y ′0 0 0 y ′0 0 ±x3 ±y3 ±z3  =   x ′′1 1 0 0 %σ 0 σx3 σy3 σz3   where % = ±1. (The composition of these two automorphisms yields ψ.) (iii) Suppose that y2 1 = y2 2 and x2 1 = x2 2 . Let y0 = y1 6= 0 and x0 = x1 6= 0. Then (since X and Y are linearly independent) y2 = ±y0 and x2 = ∓x0. We have   1 y0 0 0 0 ± 1 y0 0 0 0 ±1     x0 y0 0 ∓x0 ±y0 0 x3 y3 z3  =   x ′0 1 0 −x ′0 1 0 ±x3 ±y3 ±z3   . Moreover,     x ′0+1 2x ′0 x ′0−1 2x ′0 0 σ(x ′0−1) 2x ′0 σ(x ′0+1) 2x ′0 0 0 0 σ       x ′0 1 0 −x ′0 1 0 ±x3 ±y3 ±z3  =   1 1 0 −σ σ 0 σ%x3 σ% y3 σ%z3   where % = ±1. Theorem 4. Let Π : ∑3 i=1 ai Ei + u1 ∑3 i=1 bi Ei + u2 ∑3 i=1 ci Ei be a parametrization of a (2, 1)- affine subspace Γ. (i) If Γ isL-equivalent to Γ(2,1) 1 , thenΠ isP-equivalent to exactly one of the following parametrized affine subspaces ( Π(2,1) 1,α,β ,γ : γ1E1 + β1E2 + γ2E3 + u1(E1 + γ3E3) + u2(αE3) c3 6= 0 Π(2,1) 2,α,β ,γ : γ1E1 + β1E2 + γ2E3 + u1(αE3) + u2E1 c3 = 0. (ii) If Γ isL-equivalent to Γ(2,1) 2 , thenΠ isP-equivalent to exactly one of the following parametrized affine subspaces            Π(2,1) 3,β ,γ : β1E1 + γ1E3 + u1(E1 + E2 + γ2E3) + u2(β2E3) c3 6= 0, � a1c3−a3c1 c3 �2 6= � a2c3−a3c2 c3 �2 Π(2,1) 4,β ,γ : E1 − E2 + γ1E3 + u1(E1 + E2 + γ2E3) + u2(β1E3) c3 6= 0, � a1c3−a3c1 c3 �2 = � a2c3−a3c2 c3 �2 Π(2,1) 5,β ,γ : β1E1 + γ1E3 + u1(β2E3) + u2(E1 + E2) c3 = 0, � a1 b3−a3 b1 b3 �2 6= � a2 b3−a3 b2 b3 �2 Π(2,1) 6,β ,γ : E1 − E2 + γ1E3 + u1(β1E3) + u2(E1 + E2) c3 = 0, � a1 b3−a3 b1 b3 �2 = � a2 b3−a3 b2 b3 �2 . (iii) If Γ isL-equivalent to Γ(2,1) 3,α , thenΠ isP-equivalent to exactly one of the following parametrized affine subspaces      Π(2,1) 7,α,β ,γ : β1E3 + u1(γ1E1 +αE2) + u2E1 c2 1 6= c2 2 Π(2,1) 8,β : β1E3 + u1(β2E1) + u2(E1 + E2) c2 1 = c2 2 , b2 1 6= b2 2 Π(2,1) 9,β : β1E3 + u1(E1 − E2) + u2(E1 + E2) c2 1 = c2 2 , b2 1 = b2 2. D. Barrett, R. Biggs, C. Remsing / Eur. J. Pure Appl. Math, 7 (2014), 140-155 149 Here α > 0, βi 6= 0 and γi ∈ R, with different values of these parameters yielding distinct (non- equivalent) class representatives. Proof. By Theorem 2, we have that Γ is L-equivalent to Γ(2,1) 1 = E2 + 〈E1, E3〉, Γ(2,1) 2 = E1 + 〈E1 + E2, E3〉 or Γ(2,1) 3,α = αE3 + 〈E1, E2〉. (i) Assume Γ is L-equivalent to Γ(2,1) 1 . The affine subspace ∑3 i=1 bi Ei+ ∑3 i=1 ci Ei � has full rank (as 〈E1, E3〉 has full rank) and so the parametrized affine subspace u 7→ 3 ∑ i=1 bi Ei + u 3 ∑ i=1 ci Ei is P-equivalent to Π(1,1) 1,α,γ or Π(1,1) 2,α , by Theorem 3. It follows that Π is P-equivalent to Π(2,1) 1,α,β ,γ when c3 6= 0 and Π is P-equivalent to Π(2,1) 2,α,β ,γ when c3 = 0. As Π is inhomogeneous, β1 6= 0. (ii) Assume Γ is L-equivalent to Γ(2,1) 2 (in this case b3 6= 0 or c3 6= 0). Suppose that c3 6= 0. Then   1 0 − c1 c3 0 1 − c2 c3 0 0 1     a1 b1 c1 a2 b2 c2 a3 b3 c3  =   a′1 b′1 0 a′2 b′2 0 a3 b3 c3   where a′1 = a1c3−a3c1 c3 , a′2 = a2c3−a3c2 c3 and b′1, b′2 ∈ R. As Γ is L-equivalent to Γ(2,1) 2 , we have (b′1) 2 = (b′2) 2. Suppose (a′1) 2 6= (a′2) 2. By the lemma (with X = a′1E1 + a′2E2 + a3E3, Y = b′1E1 + b′2E2 + b3E3 and Z = c3E3) there exists ψ ∈ Aut (se(1, 1)) such that ψ ·   a′1 b′1 0 a′2 b′2 0 a3 b3 c3  =   β1 1 0 0 1 0 γ1 γ2 β2   for some β1,β2 6= 0 and γ1,γ2 ∈ R. Therefore Π is P-equivalent to Π(2,1) 3,β ,γ. On the other hand, suppose that (a′1) 2 = (a′2) 2. By the lemma (with X , Y and Z as before) there exists ψ ∈ Aut (se(1,1)) such that ψ ·   a′1 b′1 0 a′2 b′2 0 a3 b3 c3  =   1 1 0 −1 1 0 γ1 γ2 β1   for some β1 6= 0 and γ1,γ2 ∈ R. Hence Π is P-equivalent to Π(2,1) 4,β ,γ. Suppose that c3 = 0. Then b3 6= 0 and    1 0 − b1 b3 0 1 − b2 b3 0 0 1      a1 b1 c1 a2 b2 c2 a3 b3 0  =   a′1 0 c′1 a′2 0 c′2 a3 b3 0   D. Barrett, R. Biggs, C. Remsing / Eur. J. Pure Appl. Math, 7 (2014), 140-155 150 where a′1 = a1 b3−b1a3 b3 , a′2 = a2 b3−b2a3 b3 and c′1, c′2 ∈ R. Since Γ is L-equivalent to Γ(2,1) 2 , we have (c′1) 2 = (c′2) 2. Suppose that (a′1) 2 6= (a′2) 2. By the lemma (with X = a′1E1 + a′2E2 + a3E3, Y = c′1E1 + c′2E2 and Z = b3E3), there exists ψ ∈ Aut (se(1,1)) such that ψ ·   a′1 0 c′1 a′2 0 c′2 a3 b3 0  =   β1 0 1 0 0 1 γ1 β2 0   for some β1,β2 6= 0 and γ1 ∈ R. Thus Π is P-equivalent to Π(2,1) 5,β ,γ. On the other hand, suppose that (a′1) 2 = (a′2) 2. By the lemma (with X , Y and Z as before), there exists ψ ∈ Aut (se(1, 1)) such that ψ ·   a′1 0 c′1 a′2 0 c′2 a3 b3 0  =   1 0 1 −1 0 1 γ1 β1 0   for some β1 6= 0 and γ1 ∈ R. Hence Π is P-equivalent to Π(2,1) 6,β ,γ. (iii) Assume Γ is L-equivalent to Γ(2,1) 3,α (in this case, b3 = c3 = 0 and a3 6= 0). We have   1 0 − a1 a3 0 1 − a2 a3 0 0 1     a1 b1 c1 a2 b2 c2 a3 0 0  =   0 b1 c1 0 b2 c2 a3 0 0   . Suppose that c2 1 6= c2 2 . By the lemma (with X = b1E1 + b2E2, Y = c1E1 + c2E2 and Z = a3E3), there exists ψ ∈ Aut (se(1, 1)) such that ψ ·   0 b1 c1 0 b2 c2 a3 0 0  =   0 γ1 1 0 α 0 β1 0 0   for some α > 0, β1 6= 0 and γ1 ∈ R. Therefore Π is P-equivalent to Π(2,1) 7,α,β ,γ. Suppose that c2 1 = c2 2 and b2 1 6= b2 2. By the lemma (with X = b1E1 + b2E2, Y = c1E1 + c2E2 and Z = a3E3), there exists ψ ∈ Aut (se(1,1)) such that ψ ·   0 b1 c1 0 b2 c2 a3 0 0  =   0 β2 1 0 0 1 β1 0 0   for some β1,β2 6= 0. Thus Π is P-equivalent to Π(2,1) 8,β . Suppose that c2 1 = c2 2 and b2 1 = b2 2. By the lemma (with X = b1E1 + b2E2, Y = c1E1 + c2E2 and Z = a3E3), there exists ψ ∈ Aut (se(1,1)) such that ψ ·   0 b1 c1 0 b2 c2 a3 0 0  =   0 1 1 0 −1 1 β1 0 0   D. Barrett, R. Biggs, C. Remsing / Eur. J. Pure Appl. Math, 7 (2014), 140-155 151 for some β1 6= 0. Hence Π is P-equivalent to Π(2,1) 9,β . Clearly, parametrized affine subspaces corresponding to different (2, 1)-affine subspace class representatives (Γ(2,1) 1 , Γ(2,1) 2 and Γ(2,1) 3,α ) cannot be P-equivalent. No two families in case (i), case (ii) and case (iii) are P-equivalent, as the subsets 〈E1, E2〉 and 〈E1 + E2〉 ∪ 〈E1 − E2〉 are invariant. For each family, it is straightforward to verify that two representatives are P- equivalent only if their parameters are equal. Suppose Π : A+ u1B + u2C + u3D is a parametrization of a (3,0)-affine subspace Γ. We shall denote by bΠ the parametrization bΠ : B + u1C + u2D of the associated affine subspace bΓ = B+ 〈C , D〉. As bΠ is a parametrized (2, 1)-affine subspace, it is P-equivalent to exactly one of the representatives listed in Theorem 4. Accordingly, we get the following classification of parametrized (3,0)-affine subspaces (we again use Theorem 2, the classification of (2,1)-affine subspaces, to organize the results). Corollary 3. LetΠ : ∑3 i=1 ai Ei+u1 ∑3 i=1 bi Ei+u2 ∑3 i=1 ci Ei+u3 ∑3 i=1 di Ei be a parametrization of a (3, 0)-affine subspace Γ. (i) If bΓ isL-equivalent to Γ(2,1) 1 , thenΠ isP-equivalent to exactly one of the following parametrized subspaces ( Π(3,0) 1,α,β ,γ : ∑3 i=1 γi Ei + u1(γ4E1 + β1E2 + γ5E3) + u2(E1 + γ6E2) + u3(αE3) d3 6= 0 Π(3,0) 2,α,β ,γ : ∑3 i=1 γi Ei + u1(γ4E1 + β1E2 + γ5E3) + u2(αE3) + u3E1 d3 = 0. (ii) If bΓ isL-equivalent to Γ(2,1) 2 , thenΠ isP-equivalent to exactly one of the following parametrized subspaces                              Π(3,0) 3,β ,γ : ∑3 i=1 γi Ei + u1(β1E1 + γ4E3) + u2(E1 + E2 + γ5E3) + u3(β2E3) d3 6= 0, � b1d3−b3d1 d3 �2 6= � b2d3−b3d2 d3 �2 Π(3,0) 4,β ,γ : ∑3 i=1 γi Ei + u1(E1 − E2 + γ4E3) + u2(E1 + E2 + γ5E3) + u3(β1E3) d3 6= 0, � b1d3−b3d1 d3 �2 = � b2d3−b3d2 d3 �2 Π(3,0) 5,β ,γ : ∑3 i=1 γi Ei + u1(β1E1 + γ4E3) + u2(β2E3) + u3(E1 + E2) d3 = 0, � b1c3−b3c1 c3 �2 6= � b2c3−b3c2 c3 �2 Π(3,0) 6,β ,γ : ∑3 i=1 γi Ei + u1(E1 − E2 + γ4E3) + u2(β1E3) + u3(E1 + E2) d3 = 0, � b1c3−b3c1 c3 �2 = � b2c3−b3c2 c3 �2 . (iii) If bΓ isL-equivalent to Γ(2,1) 3,α , thenΠ isP-equivalent to exactly one of the following parametrized subspaces      Π(3,0) 7,α,β ,γ : ∑3 i=1 γi Ei + u1(β1E3) + u2(γ4E1 +αE2) + u3E1 d2 1 6= d2 2 Π(3,0) 8,β ,γ : ∑3 i=1 γi Ei + u1(β1E3) + u2(β2E1) + u3(E1 + E2) d2 1 = d2 2 , c2 1 6= c2 2 Π(3,0) 9,β ,γ : ∑3 i=1 γi Ei + u1(β1E3) + u2(E1 − E2) + u3(E1 + E2) d2 1 = d2 2 , c2 1 = c2 2 . REFERENCES 152 Here α > 0, βi 6= 0 and γi ∈ R, with different values of these parameters yielding distinct (non- equivalent) class representatives. 4. Final Remark In this paper we classified, under L-equivalence (resp. P-equivalence), the affine sub- spaces (resp. parametrized affine subspaces) of the semi-Euclidean Lie algebra se(1,1). This can be interpreted as a classification, under detached feedback equivalence (resp. state space equivalence), of left-invariant control affine systems on the semi-Euclidean group SE(1,1) (i.e., the connected matrix Lie group with Lie algebra se(1,1)). For instance, by corollary 1, any two-input homogeneous system (1) on SE(1, 1) is detached feedback equivalent to the system ġ = g(u1E1 + u2E3). Likewise, by Corollary 2, any homogeneous two-input system on SE(1, 1) is state space equivalent to exactly one of the systems ġ = g � γ1E1 + γ2E3 + u1(E1 + γ3E3) + u2(αE3) � ġ = g � γ1E1 + γ2E3 + u1(αE3) + u2E1 � . References [1] R Adams, R Biggs, and C Remsing. Equivalence of control systems on the Euclidean group SE(2). Control and Cybernetics, 41(3):513–524, 2012. [2] R Adams, R Biggs, and C Remsing. Control systems on the orthogonal group SO(4). Communications in Mathematics, 21(2):107–128, 2013. [3] A Agrachev and Y Sachkov. Control Theory from the Geometric Viewpoint. Springer-Verlag, Berlin, 2004. [4] R Biggs and C Remsing. A category of control systems. Analele Ştiinţifice ale Universită̧tii “Ovidius” Constanţa. Seria Matematică, 20(1):355–368, 2012. [5] R Biggs and C Remsing. Control affine systems on semisimple three-dimensional Lie groups. Analele Ştiinţifice ale Universită̧tii “Al. I. Cuza” din Iaşi. Serie Nouă. Matematică, 59(2):399–414, 2013. [6] R Biggs and C Remsing. Control affine systems on solvable three-dimensional Lie groups, I. Archivum Mathematicum, 49(3):187–197, 2013. [7] R Biggs and C Remsing. Control affine systems on solvable three-dimensional Lie groups, II. Note di Matematica, 33(2):19–31, 2013. [8] A Bloch. Nonholonomic Mechanics and Control. Springer-Verlag, New York, 2003. [9] B Jakubczyk. Equivalence and invariants of nonlinear control systems. In H Sussmann, editor, Nonlinear Controllability and Optimal Control. Marcel Dekker, New York, 1990. REFERENCES 153 [10] V Jurdjevic. Geometric Control Theory. Cambridge University Press, Cambridge, 1997. [11] V Jurdjevic and H Sussmann. Control systems on Lie groups. Journal of Differential Equations, 12(2):313–329, 1972. [12] A Krasiński, C Behr, E Schücking, F Estabrook, H Wahlquist, G Ellis, R Jantzen, and W Kundt. The Bianchi classification in the Schücking-Behr approach. General Relativity and Gravitation, 35(3):475–489, 2003. [13] M MacCallum. On the classification of the real four-dimensional Lie algebras. In A Har- vey, editor, On Einstein’s Path: Essays in Honour of E. Schücking. Springer-Verlag, New York, 1999. [14] J Ratcliffe. Foundations of Hyperbolic Manifolds. Springer-Verlag, New York, second edi- tion, 2006. [15] W Respondek and I Tall. Feedback equivalence of nonlinear control systems: a survey on formal approach. In J-P Barbot and W Perruquetti, editors, Chaos in Automatic Control. Control Eng. (Taylor & Francis), CRC Press, 2006. [16] Y Sachkov. Control theory on Lie groups. Journal of Mathematical Sciences, 156(3):381– 439, 2009. REFERENCES 154 Appendix Table 1: Classification of affine subspaces and parametrized affine subspaces Type Affine subspaces Parametrized affine subspaces (1, 1) E1 + 〈E3〉   1 0 0 0 γ1 α   αE3 + 〈E1〉   0 1 0 0 α 0   (2, 0) 〈E1, E3〉   γ1 1 0 0 0 0 γ2 γ3 α     γ1 0 1 0 0 0 γ2 α 0   (2, 1) E2 + 〈E1, E3〉   γ1 1 0 β1 0 0 γ2 γ3 α     γ1 0 1 β1 0 0 γ2 α 0   E1 + 〈E1 + E2, E3〉   β1 1 0 0 1 0 γ1 γ2 β2     1 1 0 −1 1 0 γ1 γ2 β1     β1 0 1 0 0 1 γ1 β2 0     1 0 1 −1 0 1 γ1 β1 0   αE3 + 〈E1, E2〉   0 γ1 1 0 α 0 β1 0 0     0 β2 1 0 0 1 β1 0 0     0 1 1 0 −1 1 β1 0 0   α > 0, βi 6= 0, γi ∈ R REFERENCES 155 Table 2: Classification of parametrized (3,0)-affine subspaces Π : A+ u1B + u2C + u3D Classifying conditions Parametrized affine subspaces E∗3(〈C , D〉) 6= {0} E1 + E2, E1 − E2 /∈ 〈C , D〉   γ1 γ4 1 0 γ2 β1 0 0 γ3 γ5 γ6 α     γ1 γ4 0 1 γ2 β1 0 0 γ3 γ5 α 0   E∗3(〈C , D〉) 6= {0} E1 ± E2 ∈ 〈C , D〉   γ1 β1 1 0 γ2 0 1 0 γ3 γ4 γ5 β2     γ1 1 1 0 γ2 −1 1 0 γ3 γ4 γ5 β1     γ1 β1 0 1 γ2 0 0 1 γ3 γ4 β2 0     γ1 1 0 1 γ2 −1 0 1 γ3 γ4 β1 0   E∗3(〈C , D〉) = {0}   γ1 0 γ4 1 γ2 0 α 0 γ3 β1 0 0     γ1 0 β2 1 γ2 0 0 1 γ3 β1 0 0     γ1 0 1 1 γ2 0 −1 1 γ3 β1 0 0   α > 0, βi 6= 0, γi ∈ R