E extracta mathematicae Vol. 33, Núm. 2, 219 – 227 (2018) Minimal Matrix Representations of Decomposable Lie Algebras of Dimension Less Than or Equal to Five Ryad Ghanam, Manoj Lamichhane, Gerard Thompson Department of Mathematics, Virginia Commonwealth University in Qatar, PO Box 8095, Doha, Qatar, raghanam@vcu.edu Department of Mathematics, University of Wisonsin at Waukesha, Waukesha, WI 53188, U.S.A., manoj.lamichhane@uwc.edu Department of Mathematics, University of Toledo, Toledo, OH 43606, U.S.A., gerard.thompson@utoledo.edu Presented by Consuelo Mart́ınez Received October 27, 2017 Abstract: We obtain minimal dimension matrix representations for each decomposable five- dimensional Lie algebra over R and justify in each case that they are minimal. Key words: Lie algebra, Lie group, minimal representation. AMS Subject Class. (2000): 17Bxx, 53Cxx. 1. Introduction Given a real Lie algebra g of dimension n a well known theorem due to Ado asserts that g has a faithful representation as a subalgebra of gl(p, R) for some p. In several recently published papers the authors and others have investigated the problem of finding minimal dimensional representations of indecomposable Lie algebras of dimension five and less [7, 5, 2, 3]. In particu- lar, in [3] minimal representations have been found for all the indecomposable algebras of dimension five and less. There remains the question of finding minimal dimensional representations of the decomposable Lie algebras of di- mension five and less and that is the goal of this paper. It should be regarded as a sequel to [3] and as a result, we now have minimal dimensional represen- tations of all Lie algebras of dimension up to and including five. Burde [1] has defined an invariant µ(g) of a Lie algebra g to be the dimen- sion of its minimal faithful representation. In case an algebra g is decompos- able, that is, g = g1 ⊕g2 and one has representations for g1 and g2, then there is an obvious way to construct a representation for g, that is, by taking a block diagonal representation, so that µ(g1 ⊕g2) ≤ µ(g1)+µ(g2). However, it an in- teresting phenomenon that in some cases there are representations of smaller 219 220 r. ghanam, m. lamichhane, g. thompson size than this diagonal representation, so that µ(g1 ⊕ g2) < µ(g1) + µ(g2) is possible. A particular circumstance in which this latter inequality holds is when g1 = R, as an abelian one-dimensional Lie algebra and g2 is indecompos- able: then µ(g1 ⊕g2) = µ(g2) and the representation for g1 ⊕g2 is obtained by adding multiples of the identity to the minimal dimensional representation for g2. The principal concern of the present paper is to furnish examples where µ(g1 ⊕g2) < µ(g1)+µ(g2) but that are not obtained by the trivial mechanism of adding multiples of the identity. In this paper we shall use the classification of the low-dimensional indecom- posable Lie algebras found in [6] which is in turn taken from [4]. Such algebras are denoted as Am.n where m denotes the dimension of the algebra and n the nth one in the list. We shall need indecomposable algebras of dimension from 2 to 4. In dimension 2, the algebra A2.1 is the unique non-abelian algebra. In dimension 3 there are two simple algebras A3.8 = sl(2, R) and A3.9 = so(3) and the remaining seven are solvable. In dimension 4 there are no simple or semi-simple algebras and all 12 classes of algebra are solvable. An outline of this paper is as follows. In Section 2 we list all the decom- posable Lie algebras of dimension ≤ 5. In Section 3 we consider classes of algebras given in Section 2, for example abelian Lie algebras. To show that µ has a certain value we argue in every case that µ has a certain lower bound; then all that is required is to exhibit a particular representation for which that value of µ is attained. In Section 4 we consider algebras of the form A2.1⊕ A3.n, which need slightly lengthier arguments. Finally, in Section 5 we give a list of representations with at least one for each decomposable Lie algebra where the value of µ is attained. Occasionally we give more than one such representation if it seems to be of particular interest although we do not consider the difficult issue of the equivalence of different representations. 2. Classification of decomposable Lie algebras The first step is to determine, as abstract Lie algebras, all the decompos- able Lie algebras of dimension ≤ 5. Here is a summary of all such algebras. • dimension 2: abelian R2. • dimension 3: abelian R3: nonabelian R ⊕ A2.1. • dimension 4: abelian R4: nonabelian: R2 ⊕ A2.1; A2.1 ⊕ A2.1;R ⊕ A3.n. • dimension 5: abelian R5: nonabelian R3 ⊕ A2.1; R ⊕ A2.1 ⊕ A2.1; R2⊕ A3.n; A2.1⊕ A3.n; R ⊕ A4.n representations of decomposable lie algebras 221 3. Minimal dimension matrix representations Now we shall consider each of the algebras above with regard to finding minimal dimension matrix representations. 3.1. Abelian subalgebras. First of all for an abelian Lie algebra of dimension n we have µ = ⌈2 √ n − 1⌉ [1]. For n = 2, 3, 4, 5 we obtain µ(Rn) = 2, 3, 4, 4. Representations will be given in Section 5. 3.2. Algebras that have R as a single summand. In each of the cases R⊕A2.1, R⊕A3.1, . . . , R⊕A3.7, R⊕A4.1, . . . , R⊕A4.12 the second factor is solvable. Lie’s Theorem guarantees an upper triangular representation for the adjoint representation, at least when we complexify the algebra. In fact for all these algebras representations have been found which are almost upper triangular and of minimal dimension [2, 3]. By “almost upper triangular” we mean that the complexified algebra is upper triangular. As such, we can simply add multiples of the identity so as to accommodate the extra factor of R. In the case of R ⊕ A2.1, equivalently we are simply looking at the space of 2 × 2 upper triangular matrices. In the case of R ⊕ A3.8 we have R ⊕ sl(2, R), which is isomorphic to gl(2, R) and therefore µ = 2. In the case of R ⊕ A3.9 we have R ⊕ so(3) and again we can simply add multiples of the identity to the standard three-dimensional representation of so(3). See Section 5 for such representations. 3.3. Dimension four non-abelian. Consulting the list in Section 2, the first classes of algebra for which we have not yet supplied a representation are R2 ⊕ A2.1 and A2.1 ⊕ A2.1. In the first case there is an abelian three- dimensional subalgebra and consequently µ ≥ 3; in fact µ = 3 and we give two representations in Section 5. For A2.1 ⊕ A2.1 clearly µ > 2 because the only four-dimensional subalgebra with µ = 2 is gl(2, R); however, we can find representations in gl(3, R). 3.4. Dimension five non-abelian. 3.4.1. R3 ⊕A2.1. The algebra has a four dimensional abelian subalgebra and as such µ ≥ 4. In fact µ = 4. 3.4.2. R ⊕ A2.1 ⊕ A2.1. For this algebra µ = 3. 222 r. ghanam, m. lamichhane, g. thompson 3.4.3. R2 ⊕ A3.n. For 1 ≤ n ≤ 7 each of the algebras has a four dimen- sional abelian subalgebra and as such µ ≥ 4. In fact in each of these cases we can simply take a block diagonal representation of each three dimensional representation R ⊕ A3.n with R giving µ = 4. For R2 ⊕ A3.8 we can again take a block diagonal representation of R ⊕ A3.8 ≈ gl(2, R) with R giving µ = 3. Finally for R2 ⊕A3.9 the fact that µ ≥ 4 follows from an application of Schur’s Lemma. Again we can take a block diagonal representation. 4. Algebras A2.1 ⊕ A3.n It remains to discuss the cases A2.1 ⊕A3.n where 1 ≤ n ≤ 9. For 1 ≤ n ≤ 7 these algebras have a three dimensional abelian subalgebra and as such µ ≥ 3. On the other hand for each A3.n we have µ ≤ 3. Hence for A2.1 ⊕ A3.n we must have µ ≤ 5 by taking a block diagonal representation. Lemma 4.1. Over R every 3×3 matrix is equivalent under change of basis to one of the following: a)  α 0 00 β 0 0 0 γ   , b)  λ 1 00 λ 0 0 0 µ   , c)   α 1 0−1 α 0 0 0 β   , d)  λ 1 00 λ 1 0 0 λ   . Lemma 4.2. Any abelian three-dimensional subalgebra of gl(3, R) con- tains a multiple of the identity. Proof. We refer to Lemma 4.1. In case d) the space of matrices that commutes with the given matrix is of the form [ a b c 0 a b 0 0 a ] . Likewise in case c) the space of matrices that commutes with the given matrix is of the form[ a b 0 −b a 0 0 0 c ] and hence the result follows in these two cases. In case b) if λ ̸= µ the centralizer consists of matrices of the form [ a b c 0 a 0 0 0 c ] so if there is a an abelian three-dimensional algebra it will contain I. In case b) if λ = µ the centralizer consists of matrices of the form [ a b c 0 a 0 0 d e ] . Now we may assume that e = a or else we would be in a different subcase. Given two such commuting matrices by subtracting multiples of the original matrix we may assume that b = 0 and so a linear a combination of them will be a multiple of I. Finally, in case a) we can assume that all three matrices in the algebra have real eigenvalues of algebraic multiplicity one; otherwise the algebra would fall into case b), c) or d). Thus all three matrices are diagonalizable and representations of decomposable lie algebras 223 since they commute, simultaneously diagonalizable and linearly independent. Hence again the algebra will contain I. Corollary 4.3. A subalgebra of gl(3, R) cannot have a subalgebra iso- morphic to A2.1 ⊕ A3.n where 1 ≤ n ≤ 7. Corollary 4.4. A subalgebra of gl(3, R) that contains an abelian three- dimensional subalgebra is decomposable as an abstract Lie algebra. In view of Corollary 4.3 it follows for A2.1 ⊕ A3.n, where 1 ≤ n ≤ 7, that µ ≥ 4. In fact µ = 4. It remains to discuss A2.1⊕A3.8 and A2.1⊕A3.9. In the first case we cannot have µ = 3. If we could then the representation for A3.8 = sl(2, R) would be either the 3×3 irreducible representation, which is excluded because the only matrices that commute with it are multiples of the identity and so A2.1 could not be accommodated; or, the 2 × 2 irreducible representation augmented by an extra column of rows and zeros. However, in this latter case the centralizer of sl(2, R) is two-dimensional abelian and again there would not be room for A2.1. The block diagonal representation of A2.1 and A3.8 is four-dimensional and so µ = 4. In the case of A2.1 ⊕ A3.9 again µ ̸= 3 because of Schur’s Lemma. We can apply Schur’s Lemma again to show that µ ̸= 4. In fact one needs to exercise some caution because of the same reason that the 4×4 standard representation of so(4) ≈ so(3) ⊕ so(3); as such one can find alternative 4 × 4 representations for R2 ⊕ A3.9 but not for A2.1 ⊕ A3.9. On the other hand the block diagonal representation of A2.1 and A3.9 is five-dimensional and so µ = 5. 5. The representations 5.1. Dimension three. 5.1.1. R3.  x 0 00 y 0 0 0 z    x y z0 x 0 0 0 x   . 5.1.2. R ⊕ A2.1. [ x z 0 y ] . 224 r. ghanam, m. lamichhane, g. thompson 5.2. Dimension four. 5.2.1. R4.   w 0 0 0 0 x 0 0 0 0 y 0 0 0 0 z     0 0 w x 0 0 y z 0 0 0 0 0 0 0 0     w 0 x z 0 w 0 y 0 0 w 0 0 0 0 w   . 5.2.2. R2 ⊕ A2.1 : [e1, e2] = e2. x w 00 y 0 0 0 z    αx + z y w0 (α + 1)x + z 0 0 0 αx + z   (α ∈ R) . 5.2.3. A2.1 ⊕ A2.1 : [e1, e2] = e2, [e3, e4] = e4. αx + λz y w0 (α + 1)x + λz 0 0 0 αx + µz   (α, λ, µ ∈ R, λ2 + µ2 ̸= 0) . 5.2.4. R ⊕ A3.k (1 ≤ k ≤ 9). Lie brackets are the same as for A3.k (1 ≤ k ≤ 9): use the representations given in [3] and add multiples of the identity. 5.3. Dimension five. 5.3.1. R5.   q 0 w x 0 q y z 0 0 q 0 0 0 0 q   . 5.3.2. R3 ⊕ A2.1 : [e1, e2] = e2.  x y 0 0 0 z 0 0 0 0 w 0 0 0 0 q   . representations of decomposable lie algebras 225 5.3.3. R ⊕ A2.1 ⊕ A2.1 : [e2, e3] = e2, [e4, e5] = e4.  x q w 0 y 0 0 0 z   . 5.3.4. R ⊕ A4.8,4.9b(−1