Representations of nine-dimensional Levi decomposition Lie algebras

Journal of Pure and Applied Algebra 224 (2020) 1340–1363

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Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa

Representations of nine-dimensional Levi decomposition Lie algebras Sunil Khanal a , Rishi Raj Subedi b , Gerard Thompson a a

Department of Mathematics, University of Toledo, Toledo, OH 43606, USA Department of Mathematics, Florida Agricultural and Mechanical University, Tallahassee, FL 32307, USA b

a r t i c l e

i n f o

Article history: Received 21 November 2018 Received in revised form 21 June 2019 Available online 22 July 2019 Communicated by V. Suresh

a b s t r a c t We obtain a matrix representation for each of the indecomposable 9-dimensional real Lie algebras that have a non-trivial Levi decomposition. © 2019 Elsevier B.V. All rights reserved.

MSC: 17B10; 17B30 Keywords: Lie algebra Lie group Representation

1. Introduction Given a real Lie algebra L of dimension n a well known theorem due to Ado asserts that L has a faithful representation as a subalgebra of gl(p, R) for some p. In several recently published papers, one of the current authors and others have investigated the problem of finding minimal dimensional representations of indecomposable Lie algebras of dimension eight and less [2–5,7,10]. In fact minimal dimensional representations are known for all Lie algebras, indecomposable and decomposable alike, of dimension five and less [3,4]. Furthermore, minimal dimensional representations are known for six-dimensional indecomposable nilpotent Lie algebras [5,10] and also for Lie algebras of dimension five, six, seven and eight that have a non-trivial Levi decomposition [2]. In the current article we turn our attention to Lie algebras of dimension nine that have a non-trivial Levi decomposition. Such algebras were classified by Turkowski [15] and we review his

E-mail addresses: [email protected] (S. Khanal), [email protected] (R.R. Subedi), [email protected] (G. Thompson). https://doi.org/10.1016/j.jpaa.2019.07.020 0022-4049/© 2019 Elsevier B.V. All rights reserved.

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contribution in Section 3. We refer the reader also to [1], where Turkowski’s algebras were studied from the point of view of their invariants. In Section 2 we consider the problem in general of constructing Lie algebras that have a Levi decomposition. To find minimal dimension representations for the nine-dimensional indecomposable Turkowski Lie algebras is difficult in general, therefore we address the problem of finding just one faithful, linear representation for each of these 63 Lie algebras. Nonetheless, it would not be difficult to assert in many, indeed most cases, that the representations given here are minimal, since many of the underlying algebras are associated to irreducible representations. However, since we cannot be definitive in every case, we prefer to defer the issue of minimality for the present. For example, in the cases of algebras L9.4 and L9.42 given below, we have at the moment, real representations only in gl(16, R). It would be interesting to know if this dimension 16 could be decreased. To construct representations, we use various techniques and the dimension of the representation differs according to which one is employed. We use τk as the dimension of the representation that is constructed from the kth technique. So far, we have used nine different techniques labeled I, II, ... IX. The dimension of the representation of each method is given by τk for k = I, II, ...IX. For instance, in technique I, the center of the radical of the Lie algebra is trivial and we get a representation in dimension six and we write τI = 6 and so on in the other cases. The techniques are described in Section 4 and the representations are given in Sections 5 and 6. Some algebras may be done by more than one method and some are done differently from the method outlined, in those cases where we happen to be aware of certain simpler representations. 2. Constructing algebras with Levi decompositions in general Let us consider the problem of constructing a Lie algebra that has a Levi decomposition L = N  S in general. We have the following structure equations: c k k [ea , eb ] = Cab ec , [ea , ei ] = Cai ek , [ei , ej ] = Cij ek

(1)

where 1 ≤ a, b, c, d ≤ r and r + 1 ≤ i, j, k, l ≤ n and {ea } is a basis for the semi-simple subalgebra S and {ei } is a basis for the radical N . Calculation shows that the Jacobi identity is equivalent to the following conditions: j j e d c k j k C[ab Cc]e = 0, Cab Cci = Cbi Cak − Cai Cbk , k l l k l l m Cal Cij = Cai Clj − Caj Clik , C[ij Ck]l = 0.

(2)

We start with a semi-simple algebra so that the first set of conditions above is satisfied. Then the second k set say that the matrices Cal make N (merely as a vector space) into an S-module. The third set say that k the Cal are derivations of the Lie algebra N and the fourth of course that N is a Lie algebra. Therefore to find all possible Lie algebras of dimension n that have a Levi decomposition L = N  S we can proceed as follows: choose a semi-simple algebra S of dimension r. Then pick any solvable algebra N of dimension n − r and consider a representation of S in N , considered simply as a vector space of dimension n − r, should one exist. All such representations are known and are completely reducible [6] since S is semi-simple. Finally, it only remains to check that the matrices representing S act as derivations of the Lie algebra N . In the affirmative case we have our sought after Lie algebra; in the negative case there is no such algebra and we have to choose a different representation of S in N . If all such representations lead to a null result, then there can be no non-trivial Levi decomposition involving S and N .

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3. Brief description of Turkowski’s paper 3.1. In [15] Turkowski classified nine-dimensional, indecomposable Lie algebras that have a non-trivial Levi decomposition. Referring to the previous Section, Turkowski denotes by R the representation of the semi-simple factor S by automorphisms of the radical N . He argues further that S can only be one of so(3) or sl(2, R). In the case of so(3), only five representations R are possible and they are denoted in [13] as ad(so(3))⊕3D0 , R4 ⊕2D0 , R5 ⊕2D0 , R6 and 2 ad(so(3)), respectively. These representations are, respectively, the irreducible representations in gl(3, R), gl(4, R), gl(5, R), gl(6, R) supplemented by a trivial representation and the “diagonal” representation in gl(6, R). They are listed as 1, 2, 3, 4 in Table I in [15]. Here D0 denotes the trivial, one-dimensional representation, and kD0 connotes k copies of the trivial representation. In the case of sl(2, R), ten R representations are possible and are denoted in [13] as D 12 ⊕ 4D0 , D1 ⊕ 3D0 , D 32 ⊕2D0 , 2D 12 ⊕2D0 , D2 ⊕D0 , D1 ⊕D 12 ⊕D0 , D 52 , D 32 ⊕D 12 , 2D1 and 3D 12 , respectively. Here, following [13], k is either a non-negative integer or half-integer and Dk denotes the standard irreducible representation in gl(2k +1, R). They are listed as 5, 6, ..., 13, 14 in Table II in [15]. Again D0 denotes the trivial representation. Referring to equation (1), the first set of terms have been given and the second set comprise the R representation alluded to in the previous paragraph. In order to give a complete algebra, all that remains is to give the nonzero structure constants of the radical. Turkowski [15] supplies them in Table II and discerns 63 such classes of algebra. 3.2. Notation The indecomposable Lie algebras of dimension less than or equal to five and the nilpotent algebras of dimension six are denoted by Ai,j where 3 ≤ i ≤ 6, and j signifies the jth algebra in the list, following the listing in [11]. The indecomposable Lie algebras of dimension six that have a five-dimensional nilradical were classified by Mubarakzyanov [8] and are denoted by g6,i where 1 ≤ i ≤ 99; see also [12] for an updated classification. The indecomposable Lie algebras of dimension six that have a four-dimensional nilradical classified by Turkowski [14] are denoted by N6,i where 1 ≤ i ≤ 40. The Lie algebras studied by Turkowski [15], that are the principal concern of the current paper, are denoted by L9,i where 1 ≤ i ≤ 63. Finally, for abelian Lie algebras, we usually say “Abelian” rather than writing kA1 for the k-dimensional abelian algebra. 3.3. Additions In [15] two algebras were omitted: one is denoted as L∗9,7 in [1] and is a “real” form corresponding to L9,52 . The second shall be denoted here as L∗9,11 and is a semi-direct product of so(3) and R6 coming from the irreducible 6 × 6 representation of so(3). 4. Techniques We have the following techniques for finding representations of a Lie algebra L that belongs to one of Turkowski’s list of 63. 4.1. I: Radical has trivial center If the radical has trivial center then the whole Lie algebra has trivial center. Instead of taking the full adjoint representation, we take the adjoint representation restricted to the radical N .

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Theorem 4.1. Suppose that N is the radical of a Lie algebra L and that N has trivial center. Then L has a faithful representation in gl(r, R) where N is of dimension r. k k and Ei to be the matrix Cij . Then in order to have a representation Proof. Define Ea to be the matrix Cai of L we require that c k k [Ea , Eb ] = Cab Ec , [Ea , Ei ] = Cai Ek , [Ei , Ej ] = Cij Ek .

(3)

However, these conditions are identical with the last three in equation (2). If the center of N is trivial then we will have a faithful representation of L. 2 In this case, τI = 6. The following list are the Lie algebras which are done by this technique: 9.1, 9.2, 9.3, 9.7, 9.17, 9.22, 9.24, 9.25, 9.26, 9.34, 9.38, 9.44, 9.46, 9.47, 9.48, 9.49, 9.50, 9.52, 9.53. 4.2. II: Abelian 5d nilradical If the radical N has a codimension one abelian nilradical then we have a representation of L in gl(6, R). The radical is from Mubar g6.i 1 ≤ i ≤ 12). Theorem 4.2. Suppose that N is the radical of a Lie algebra L and that N has an abelian nilradical of dimension r − 1 where N is of dimension r. Then N has a faithful representation in gl(r, R). k and for 1 ≤ k ≤ r − 1 define Ek to be the r × r matrix whose only Proof. Define Ea to be the matrix Cai non-zero entry is a 1 in the (k, r)th position. Then proceed as in Theorem 4.1. 2

After finding a representation of the radical it remains to add the semi-simple factor. In this case, τII = 6. Algebras done by this method are 9.5, 9.6, 9.9, 9.13, 9.14, 9.15, 9.16, 9.32, 9.33, 9.34, 9.35, 9.38, 9.43, 9.55, 9.56. 4.3. III: Radical is Abelian If the radical N is Abelian then L has a representation in gl(7, R). Theorem 4.3. Suppose that the Lie algebra g is a semi-direct product of a semi-simple subalgebra σ and an r-dimensional abelian ideal ρ in the Levi decomposition. Then g has a (faithful) representation as a subalgebra of gl(r + 1, R). Conversely, a subalgebra of gl(r + 1, R) with the upper r × r block giving a representation of a semi-simple Lie algebra and the first r entries in the last column being arbitrary and the (r + 1)th zero, gives a Levi subalgebra of gl(r + 1, R) whose radical is abelian. k = 0. To obtain Proof. We quote the structure equations in equation (1) where now we assume that Cij k the required representation consider ad(σ) restricted to ρ, in other words the matrices Cai . Define Ea to k be the same matrix as Cai but now augmented by an extra bottom row and last column of zeroes. Now define Ek to be the (r + 1) × (r + 1) matrix whose only non-zero is 1 in the (k, r + 1)th position. Then [Ea , Ei ] = Ea Ei − Ei Ea ; now Ei Ea = 0. As regards Ea Ei it is a matrix that has only non-zero entries in the k last column. That last column is Cai with a and k fixed, augmented by a zero in the (r + 1, r + 1)th position k c which means that [Ea , Ei ] = Cai Ek . Since [Ea , Eb ] = Cab Ec and [Ei , Ej ] = 0 we have a representation of g in gl(r + 1, R). 2

In this case, τIII = 7. The technique is applicable to algebras 9.10, 9.59, 9.60, 9.61, 9.63.

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4.4. IV: Entire Lie algebra has trivial center If the Lie algebra as a whole has trivial center, whereas the radical has non-trivial center, then we can take full the adjoint representation. It might be imagined that many cases could be done by using this method and that supposition is indeed correct; however, many of the Lie algebras concerned depend on parameters and it frequently happens that the full adjoint representation or its restriction to the nilradical is not faithful for special values of the parameters. In that situation one has to find a different representation for the exceptional case, whereas it is clearly preferable, if possible, to have a uniform representation that depends on the parameters. For representations constructed using method IV , we have τIV = 9. This method is used for algebras 9.11, 9.48, 9.62. It is a striking fact that only these three cases are done by method IV and that for the vast majority of the 63 algebras, the representations supplied are in dimensions much smaller than that of gl(9, R). 4.5. V: Nilradical: H ⊕ R2 We consider, more generally, a nilradical of the form H ⊕ Rn−4 . Lemma 4.1. If the nilradical is of the form H ⊕ Rn−4 then the brackets may be normalized so that only the following brackets are non-zero: n−1 j n−1 j [e1 , en ] = (a22 + a33 )e1 , [e2 , e3 ] = e1 , [e2 , en ] = j=2 a2 ej , [e3 , en ] = j=2 a3 ej and for 4 ≤ k ≤ n −  j 1 1, [ek , en ] = ak e1 + 4≤j≤n−1 ak ej . Proof. We begin by choosing a basis {e1 , e2 , ..., en } such that [e2 , e3 ] = e1 and [e1 , ej ] = 0 for 4 ≤ j ≤ n − 1 and [ei , ej ] = 0 for 4 ≤ i < j ≤ n − 1. Suppose that [e2 , en ] = αe1 + ae2 + HOT and that [e3 , en ] = βe1 + λe2 + be3 + HOT . We observe that the transformation obtained by putting en = βe2 − αe3 en has the effect of reducing α and β to zero. Next, we show that [e1 , en ] = (a + b)e1 . Indeed [e1 , en ] = [[e2 , e3 ], en ] = [e2 , [e3 , en ]] − [e3 , [e2 , en ]] = b[e2 , e3 ] − [e3 , ae2 ] = (a + b)e1 . Finally, consider the Jacobi identity [e2 , [ek , en ]] + [en , [e2 , ek ]] + [ek , [en , e2 ]] = 0 where 4 ≤ k ≤ n − 1. The second and third terms in the Jacobi identity are zero. In the [ek , en ] term only the e3 -term is of significance. As such [e2 , [ek , en ]] = a3k e1 = 0 and hence a3k = 0 for 4 ≤ k ≤ n − 1. Similarly a2k = 0 for 4 ≤ k ≤ n − 1. 2 Proposition 4.2. Let L be an n dimensional Lie algebra that has a codimension one nilradical that is of the form H ⊕ Rn−4 . Then L has a representation in gl(n, R). Proof. We may assume that the Lie algebra has only the non-zero brackets that appear in Lemma 4.1. As such the following space of matrices provides the requisite representation: ⎡

(a22 + a33 )xn ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ .. ⎢ ⎢ . ⎢ ⎢ 0 ⎢ ⎣ 0 0

x3 a22 xn a32 xn a42 xn .. .

−x2 a23 xn a33 xn a43 xn .. .

a14 xn 0 0 4 a4 x n .. .

a15 xn 0 0 4 a5 x n .. .

xn an−2 2 n−1 a 2 xn 0

an−2 xn 3 n−1 a3 x n 0

an−2 xn 4 n−1 a4 x n 0

an−2 xn 5 n−1 a5 x n 0

... ... ... ... ... ... ... ...

a1n−2 xn 0 0 4 an−2 xn .. .

a1n−1 xn 0 0 4 an−1 xn .. .

an−2 n−2 xn an−1 n−2 xn 0

an−2 n−1 xn an−1 n−1 xn 0

2x1 x2 x3 x4 .. .

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ xn−2 ⎥ ⎥ xn−1 ⎦ 0

2

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Having found a representation of the radical we examine it to see if there is room to accommodate a representation of sl(2, R). If we are able to do so then in this case, τV = 6. The following Lie algebras were done by this technique: 9.18, 9.19, 9.20, 9.21, 9.23. 4.6. VI: Nilradical: 5d-Heisenberg: Proposition 4.3. Let L be an 6 dimensional Lie algebra that has a nilradical that is H5 , that is fivedimensional Heisenberg. Then L has a representation in gl(6, R). Proof. We assume that a basis e4 , e5 , e6 , e7 , e8 , e9 is chosen so that the Lie brackets for H5 are [e4 , e6 ] = e8 , [e5 , e7 ] = e8 . We begin with e4 so as to leave room for the semi-simple part that will be spanned by e1 , e2 , e3 . As is explained in [8] and again in [12], after making a change of basis that adds suitable multiples of e4 , e5 , e6 , e7 to e9 , the most general matrix that can serve as −ad(e9 ) where e6 extends H5 to a solvable Lie algebra may be assumed to be of the form: ⎡

k ⎢0 ⎢ ⎢0 ⎢ ⎢ ⎢0 ⎢ ⎣0 0

0 a c h i 0

0 0 b e d f i k−a j −b 0 0

⎤ 0 0 f 0⎥ ⎥ g 0⎥ ⎥ ⎥. −c 0⎥ ⎥ k−d 0⎦ 0 0

(4)

As such the following space of matrices provides the requisite representation: ⎡

ks9 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎣ 0 0

−s6 as9 cs9 hs9 is9 0

−s7 bs9 ds9 is9 js9 0

s4 e s9 f s9 (k − a) s9 −bs9 0

s5 f s9 gs9 −cs9 (k − d) s9 0

⎤ 2 s8 s4 ⎥ ⎥ s5 ⎥ ⎥ ⎥. s6 ⎥ ⎥ s7 ⎦ 0

(5)

The Lie brackets are given by [e4 , e6 ] = e8 , [e4 , e9 ] = −ae4 − ce5 − he6 − ie7 , [e5 , e7 ] = e8 , [e5 , e9 ] = −be4 − de5 − ie6 − je7 ,

(6)

[e6 , e9 ] = −ee4 − f e5 + (a − k)e6 + be7 , [e7 , e9 ] = −f e4 − ge5 + ce6 + (d − k)e7 , [e8 , e9 ] = −ke8 .

2

In this case, τV I = 6. The algebras for which the nilradical is H5 are 9.7, 9.25, 9.26, 9.27, 9.36, 9.51, 9.52, 9.53. In each of these cases, building on the representation provided by Proposition 4.3, it is easy to add in the appropriate representation for either so(3) or sl(2, R) so as to obtain the full representation of the nine-dimensional Lie algebra. 4.7. VII: Radical direct sum pullback: There are some algebras for which one may obtain representations by a pullback method. In this case the radical has to be decomposable and we can pullback a product representation over the diagonal map. Theorem 4.4. Given representations δ1 : σ ρ1 → End(V ) and δ2 : σ ρ2 → End(W ) of Levi-decomposition algebras there is an induced representation Δ : σ (ρ1 ⊕ ρ2 ) :→ End(V ) ⊕ End(W ) defined by Δ(s, p1 , p2 ) = (δ1 (s, p1 ), δ2 (s, p2 ).

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In this case, τV II = 8 and we can apply the technique to algebras 9.37, 9.38, 9.39, 9.57, 9.58. We remark finally the many of the representations constructed in [9] for the solvable six-dimensional solvable Lie algebras with a four-dimensional nilradical, were constructed by the pullback method. 4.8. VIII: Radical has Abelian nilradical and Abelian complement In this case, τV III = 5 or 6. The proof is similar to technique II. Use restricted adjoint matrices for the complement and for the nilradical, matrices that have just a single non-zero one in the last column. Algebras done by this method are 9.8, 9.28, 9.29, 9.30, excluding those cases covered by method II. 4.9. IX: Algebras equivalent over C Some algebras are equivalent as complex Lie algebras. For example algebras 9.4, 9.42 are equivalent and both are equivalent to 9.37. As such, it is possible to find representations of 9.4 and 9.42 in gl(8, C); however, these representations may be interpreted as being real, by the device of identifying gl(8, C) as a subalgebra x y

of gl(16, R) in which a complex number x + iy is identified with the 2 × 2 matrix −y x . In this case, τIX = 16. In the list below we are content simply to give the representations of algebras 9.4 and 9.42 in gl(8, C). 4.10. Result of Humphreys We quote the following result from Humphreys [6]. Proposition 4.4. Let L be a finite dimensional complex Lie algebra acting irreducibly on the vector space V . Then L is reductive and the center is of dimension one or zero. If in addition L ⊂ sl(V ) then L is semi-simple. Notice that the condition of semi-simplicity is a conclusion not an assumption. In practice we apply this result to semi-simple algebras to deduce that for a particular representation there is only a trivial one-dimensional extension to a Levi decomposition or more generally to an irreducible block in a reducible representation. Unfortunately Humphreys’ Proposition only applies to complex Lie algebras; nonetheless as an experiential result it is nearly always true for real algebras. A useful counterexample to keep in mind in this regard is given by the isomorphism so(4) ≈ so(3) ⊕ so(3) as it is also for the “real” version of Schur’s Lemma. 5. Representations with semi-simple factor so(3) In this section we provide representations for each of the 11 nine-dimensional Lie algebras L9.1 , L9.2 , ...L9.11 that have a non-trivial Levi-decomposition where the semi-simple factor is so(3) with Lie brackets [e1 , e2 ] = e3 , [e2 , e3 ] = e1 , [e3 , e1 ] = e2 . The subsections are separated according to the four different representations of so(3) that occur and the Lie brackets are supplied. In [15], these representations, as well as the ten for sl(2, R), were listed in Table I. Each of the algebras L9.1, L9.2 , ...L9.11 has its own subsubsection in which the brackets for the radical are given. Following [15], such radicals are denoted generically by N . We give in each case the radical following the notation described in subsection 3.2. Many such radicals involve parameters which are allowed to assume certain special values. The matrix that defines the Lie algebra representation is denoted generically by S.

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5.1. so(3): representation ad(so(3)) ⊕ 3D0 [e1 , e5 ] = e6 , [e2 , e4 ] = −e6 , [e3 , e4 ] = e5 , [e1 , e6 ] = −e5 , [e2 , e6 ] = e4 , [e3 , e5 ] = −e4 . 5.1.1. Lp,q 9.1 N = g6.1pq=0 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = e6 , [e7 , e9 ] = pe7 , [e8 , e9 ] = qe8 . ⎡

−s9 ⎢ s ⎢ 3 ⎢ −s ⎢ 2 S=⎢ ⎢ 0 ⎢ ⎣ 0 0

−s3 −s9 s1 0 0 0

s2 −s1 −s9 0 0 0

0 0 0 −ps9 0 0

0 0 0 0 −qs9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

5.1.2. Lp9.2 N = g6.2, p=0 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = e6 , [e7 , e9 ] = pe7 , [e8 , e9 ] = e7 + pe8 . ⎡

−s9 ⎢ s ⎢ 3 ⎢ −s ⎢ 2 S=⎢ ⎢ 0 ⎢ ⎣ 0 0

−s3 −s9 s1 0 0 0

s2 −s1 −s9 0 0 0

0 0 0 −ps9 0 0

0 0 0 −s9 −ps9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

5.1.3. Lp,q 9.3 N = g6.8,p=0,q≥0 , [e4 , e9 ] = pe4 , [e5 , e9 ] = pe5 , [e6 , e9 ] = pe6 , [e7 , e9 ] = qe7 − e8 , [e8 , e9 ] = e7 + qe8 . ⎡

−ps9 ⎢ s ⎢ 3 ⎢ −s ⎢ 2 S=⎢ ⎢ 0 ⎢ ⎣ 0 0

−s3 −ps9 s1 0 0 0

s2 −s1 −ps9 0 0 0

0 0 0 −qs9 s9 0

0 0 0 −s9 −qs9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

5.2. Semi-simple so(3): representation R4 ⊕ 2D0 [e1 , e4 ] = 12 e7 , [e1 , e4 ] = 12 e5 , [e3 , e4 ] = 12 e6 , [e1 , e5 ] = 12 e6 , [e2 , e5 ] = − 12 e4 , [e3 , e5 ] 1 − 2 e5 , [e2 , e6 ] = 12 e7 , [e3 , e6 ] = − 12 e4 , [e1 , e7 ] = − 12 e4 , [e2 , e7 ] = − 12 e6 , [e3 , e7 ] = 12 e5 .

= − 12 e7 , [e1 , e6 ] =

5.2.1. L9.4 N = A6.5 , [e4 , e5 ] = −e9 , [e4 , e7 ] = e8 , [e5 , e6 ] = −e8 , [e6 , e7 ] = e9 . ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ S=⎢ ⎢ ⎢ ⎢ ⎣

is1 2

1 2 (s2 +is3 ) is 1 (−s +is ) − 21 2 3 2 2(is4 −is5 −s6 +s7 ) (is +is −s −s ) √ − 4 25√2 6 7 2 2

0 0

0 0

0 0 0 0 0

(is4 +is5 −s6 −s7 ) √ 2 2 (is4 −is5 −s6 +s7 ) √ 2 2 1 (−is 8 +s9 ) 2

0 0

0

0

0

0

0 0

0 0

0 0

0 0

0

0

0

0

0 0 is1 2

0 0 1 2 (s2 +is3 )

1 2 (−s2 +is3 ) (is6 +is7 −s4 −s5 ) √ 2 2

is − 21 (is6 −is7 −s4 +s5 ) √ 2 2

0

0

0

0

0

0

0 0 0 0 (is −is −s +s ) 0 − 6 27√2 4 5 0 0 0

(is6 +is7 −s4 −s5 ) √ 2 2 − 12 (is8 +s9 )

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦

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5.2.2. Lp9.5 N = g6.1 p=0 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = e6 , [e7 , e9 ] = e7 , [e8 , e9 ] = pe8 . ⎡

−s9

− s22 −s9

⎢ s2 ⎢ 2 ⎢ s3 ⎢ S = ⎢ s21 ⎢ 2 ⎢ ⎣ 0 0

s1 2 − s23

0 0

− s23 − s21 −s9

− s21

s2 2

−s9 0 0

s3 2 − s22

0 0

0 0 0 0 −ps9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

5.2.3. Lp,q 9.6 g6.11 q=0 , [e4 , e9 ] = pe4 − e6 , [e5 , e9 ] = pe5 − e7 , [e6 , e9 ] = e4 + pe6 , [e7 , e9 ] = e5 + pe7 , [e8 , e9 ] = qe8 . ⎡ ⎢ ⎢ ⎢ ⎢ S=⎢ ⎢ ⎢ ⎣

−ps9 s3 2

s2 2

− s22 −ps9 s1 2

+ s9

− s23

s1 2

0 0

− s23 − s9 − s21 −ps9 s2 2

+ s9 0 0

0 0

− s21 s3 2 − s9 − s22 −ps9 0 0

0 0 0 0 −qs9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

5.2.4. Lp9.7 g6.92∗ , [e4 , e6 ] = e8 , [e4 , e9 ] = pe4 + e6 , [e5 , e7 ] = e8 , [e5 , e9 ] = pe5 − e7 , [e6 , e9 ] = pe6 − e4 , [e7 , e9 ] = pe7 − e5 , [e8 , e9 ] = 2pe8 . ⎡

−ps9

⎢ s2 ⎢ 2 ⎢ s3 − s ⎢ 2 9 S = ⎢ s1 ⎢ 2 ⎢ ⎣ −s6 0

− s22 −ps9

− s23 + s9 − s21 −ps9

s1 2

− s23 − s9 −s7 0

s2 2

s4 0

− s21 s3 2 + s9 − s22 −ps9 s5 0

0 0 0 0 −2ps9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

5.2.5. L9.7∗ N = g6.82 , [e4 , e6 ] = e8 , [e4 , e9 ] = e4 , [e5 , e7 ] = e8 , [e5 , e9 ] = e5 , [e6 , e9 ] = e6 , [e7 , e9 ] = e7 , [e8 , e9 ] = 2e8 . ⎡

−2 s9 ⎢ 0 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

−s6 −s9 s2 2 s3 2 s1 2

0

−s7 − s22 −s9 s1 2 − s23

0

s4 − s23 − s21 −s9

s5 − s21

s2 2

−s9 0

0

s3 2 − s22

⎤ 2 s8 s4 ⎥ ⎥ s5 ⎥ ⎥ ⎥. s6 ⎥ ⎥ s7 ⎦ 0

5.2.6. L9.8 N = N6.18α=0,β=1,γ=1 , [e4 , e8 ] = e4 , [e4 , e9 ] = e6 , [e5 , e8 ] = e5 , [e5 , e9 ] = e7 , [e6 , e8 ] = e6 , [e6 , e9 ] = −e4 , [e7 , e8 ] = e7 , [e7 , e9 ] = −e5 .

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⎡ ⎢ ⎢ ⎢ S=⎢ ⎢ ⎣

− s22 s8

s8 s3 2

s2 2

− s9

s1 2

0

− s23 + s9 − s21 s8

s1 2

− s23 − s9 0

s2 2

0

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⎤ s4 s5 ⎥ ⎥ ⎥ s6 ⎥ . ⎥ s7 ⎦ 0

− s21 s3 2 + s9 − s22 s8 0

5.3. Semi-simple so(3): representation R5 ⊕ D0 [e1 , e4 ] = 12 e7 , [e2 , e4 ] = 12 e6 , [e3 , e4 ] = 2e5 , [e1 , e5 ] = − 12 e6 , [e2 , e5 ] = 12 e7 , [e3 , e5 ] = −2e4 , [e1 , e6 ] = 2e5 , [e2 , e6 ] = −2e4 , [e3 , e6 ] = e7 , [e1 , e7 ] = −2e4 , [e2 , e7 ] = −e8 , [e1 , e8 ] = 3e6 , [e2 , e8 ] = 3e7 . 5.3.1. L9.9 N = g6.1 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e5 , e8 ] = e5 , [e6 , e9 ] = e6 , [e7 , e9 ] = e7 , [e8 , e9 ] = e8 . ⎡

−s9 ⎢ 2s ⎢ 3 ⎢ s2 ⎢ S = ⎢ s21 ⎢ 2 ⎢ ⎣ 0 0

−2 s3 −s9 − s21 s2 2

0 0

−2 s2 2 s1 −s9 s3 −s1 0

−2 s1 −2 s2 −s3 −s9 −s2 0

0 0 3 s1 3 s2 −s9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

5.4. Semi-simple so(3): representation 2 ad (so(3) [e1 , e5 ] = e6 , [e2 , e4 ] = −e6 , [e3 , e4 ] = e5 , [e1 , e6 ] = −e5 , [e2 , e6 ] = e4 , [e3 , e5 ] = −e4 , [e1 , e8 ] = e9 , [e2 , e7 ] = −e9 , [e3 , e7 ] = e8 , [e1 , e9 ] = −e8 , [e2 , e9 ] = e7 , [e3 , e8 ] = −e7 . 5.4.1. L9.10 N = Abelian. ⎡

0 ⎢ s3 ⎢ ⎢ −s ⎢ 2 ⎢ S=⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0

−s3 0 s1 0 0 0 0

s2 −s1 0 0 0 0 0

0 0 0 0 s3 −s2 0

0 0 0 −s3 0 s1 0

0 0 0 s2 −s1 0 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥ s7 ⎥ . ⎥ s8 ⎥ ⎥ s9 ⎦ 0

5.4.2. L9.11 N = A6.3 , [e4 , e5 ] = e9 , [e4 , e6 ] = −e8 , [e5 , e6 ] = e7 . ⎡

0 ⎢ s ⎢ 3 ⎢ ⎢ −s2 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ s6 ⎢ −s ⎢ 5 ⎢ ⎢ 0 ⎢ ⎣ s9 −s8

−s3 0 s1 −s6 0 s4 −s9 0 s7

s2 −s1 0 s5 −s4 0 s8 −s7 0

0 0 0 0 s3 −s2 0 s6 −s5

0 0 0 −s3 0 s1 −s6 0 s4

0 0 0 s2 −s1 0 s5 −s4 0

0 0 0 0 0 0 0 s3 −s2

0 0 0 0 0 0 −s3 0 s1

⎤ 0 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥. 0 ⎥ ⎥ ⎥ s2 ⎥ ⎥ −s1 ⎦ 0

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5.5. Semi-simple so(3): representation R6 [e1 , e6 ] = −e8 , [e1 , e7 ] = −e9 , [e1 , e8 ] = e6 , [e1 , e9 ] = e7 , [e2 , e4 ] = −e8 , [e2 , e5 ] = −e9 , [e2 , e8 ] = e4 , [e2 , e9 ] = e5 , [e3 , e4 ] = −e6 , [e3 , e5 ] = −e7 , [e3 , e6 ] = e4 , [e3 , e7 ] = e5 . 5.5.1. L9.11∗ N =Abelian. ⎡

0 ⎢ 0 ⎢ ⎢ −s ⎢ 3 ⎢ S=⎢ 0 ⎢ ⎢ −s2 ⎢ ⎣ 0 0

0 0 0 −s3 0 −s2 0

s3 0 0 0 −s1 0 0

0 s3 0 0 0 −s1 0

s2 0 s1 0 0 0 0

0 s2 0 s1 0 0 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥ s7 ⎥ . ⎥ s8 ⎥ ⎥ s9 ⎦ 0

6. Representations with semi-simple factor sl(2, R) In this section we provide representations for each of the 52 nine-dimensional Lie algebras L9.12 , L9.13 , ...L9.63 that have a non-trivial Levi-decomposition where the sl(2, R) semi-simple factor is sl(2, R) with Lie brackets [e1 , e2 ] = 2e2 , [e1 , e3 ] = −2e3 , [e2 , e3 ] = e1 . The subsections are separated according to the different representations of sl(2, R) that occur and the Lie brackets are supplied. Finally each of the algebras L9.12 , L9.13 , ...L9.63 has its own subsubsection in which the brackets for the radical are given. The notation here is taken from [15]. 6.1. sl(2, R): representation D 12 ⊕ 4D0 [e1 , e4 ] = e4 , [e2 , e5 ] = e4 , [e3 , e4 ] = e5 , [e1 , e5 ] = −e5 . 6.1.1. L9.12 N = A6.12 , [e4 , e5 ] = e6 , [e7 , e9 ] = e6 , [e8 , e9 ] = e7 . ⎡

0 s9 ⎢0 0 ⎢ ⎢0 0 ⎢ S=⎢ ⎢0 0 ⎢ ⎣0 0 0 0

0 s9 0 0 0 0

s5 0 0 s1 s3 0

−s4 0 0 s2 −s1 0

⎤ −2s6 2s7 ⎥ ⎥ −2s8 ⎥ ⎥ ⎥. s4 ⎥ ⎥ s5 ⎦ 0

6.1.2. Lpqr 9.13 N = g6.1pqr=0 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = pe6 , [e7 , e9 ] = qe7 , [e8 , e9 ] = re8 . ⎡

s1 − s9 ⎢ s 3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − s9 0 0 0 0

0 0 −ps9 0 0 0

0 0 0 −qs9 0 0

0 0 0 0 −rs9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

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6.1.3. Lp,q 9.14 N = g6.2p=0 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = pe6 , [e7 , e9 ] = qe7 , [e8 , e9 ] = e7 + qe8 . ⎡

s1 − s9 ⎢ s 3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − s9 0 0 0 0

0 0 −ps9 0 0 0

0 0 0 −qs9 0 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

0 0 0 −s9 −qs9 0

6.1.4. Lp9.15 N = g6.3 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = pe6 , [e7 , e9 ] = pe7 + e6 , [e8 , e9 ] = e7 + pe8 . ⎡

s1 − s9 ⎢ s 3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − s9 0 0 0 0

0 0 −ps9 0 0 0

0 0 −s9 −ps9 0 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

0 0 0 −s9 −ps9 0

6.1.5. Lp,q,r 9.16 N = g6.8 pq=0,r≥0 , [e4 , e9 ] = pe4 , [e5 , e9 ] = pe5 , [e6 , e9 ] = qe6 , [e7 , e9 ] = re7 − e8 , [e8 , e9 ] = e7 + re8 . ⎡

−ps9 + s1 ⎢ s3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −ps9 − s1 0 0 0 0

0 0 −qs9 0 0 0

0 0 0 −rs9 s9 0

0 0 0 −s9 −rs9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

6.1.6. Lp,q 9.17 N = g6.13 pq=0 , [e4 , e5 ] = e6 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = 2e6 , [e7 , e9 ] = pe7 , [e8 , e9 ] = qe8 . ⎡

s1 − s9 ⎢ s 3 ⎢ ⎢ −s ⎢ 5 S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − s9 s4 0 0 0

0 0 −2 s9 0 0 0

0 0 0 −ps9 0 0

0 0 0 0 −qs9 0

⎤ s4 s5 ⎥ ⎥ 2 s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

6.1.7. Lp,q 9.18 N = g6.13 p2 +q2 =0 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = (p + q)e6 , [e7 , e8 ] = e6 , [e7 , e9 ] = pe7 , [e8 , e9 ] = qe8 . ⎡

s1 − s9 ⎢ s 3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − s9 0 0 0 0

0 0 − (p + q) s9 0 0 0

0 0 s8 −ps9 0 0

0 0 −s7 0 −qs9 0

⎤ s4 s5 ⎥ ⎥ −2s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

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6.1.8. Lp9.19 N = g6.14 , [e4 , e5 ] = e6 , [e4 , e9 ] = pe4 , [e5 , e9 ] = pe5 , [e6 , e9 ] = 2pe6 , [e7 , e9 ] = e7 , [e8 , e9 ] = e6 + 2pe8 . ⎡

−2 ps9 ⎢ 0 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

−s4 s2 −ps9 − s1 0 0 0

s5 −ps9 + s1 s3 0 0 0

0 0 0 −s9 0 0

−s9 0 0 0 −2 ps9 0

⎤ −2 s6 s4 ⎥ ⎥ s5 ⎥ ⎥ ⎥. s7 ⎥ ⎥ −2s8 ⎦ 0

6.1.9. Lp9.20 N = g6.21 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = 2pe6 , [e7 , e8 ] = e6 , [e7 , e9 ] = pe7 , [e8 , e9 ] = e7 + pe8 . ⎡

s1 − s9 ⎢ s 3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − s9 0 0 0 0

0 0 −2p s9 0 0 0

0 0 −s8 −ps9 0 0

0 0 −s7 s9 −ps9 0

⎤ s4 s5 ⎥ ⎥ −2s6 ⎥ ⎥ ⎥. −s7 ⎥ ⎥ s8 ⎦ 0

6.1.10. Lp9.21 N = g6.25 , [e4 , e5 ] = e6 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = 2e6 , [e7 , e9 ] = pe7 , [e8 , e9 ] = e7 + pe8 . ⎡

−2 s9 ⎢ 0 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s5 s1 − s9 s3 0 0 0

−s4 s2 −s1 − s9 0 0 0

0 0 0 −ps9 0 0

0 0 0 −s9 −ps9 0

⎤ −2s6 s4 ⎥ ⎥ s5 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

6.1.11. L9.22 N = g6.26 , [e4 , e5 ] = e6 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = 2e6 , [e7 , e9 ] = e6 + 2e7 . ⎡

s1 − s9 ⎢ s 3 ⎢ ⎢ −s ⎢ 5 S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − s9 s4 0 0 0

0 0 −2 s9 0 0 0

0 0 −s9 −2 s9 0 0

0 0 0 −s9 −2 s9 0

⎤ s4 ⎥ s5 ⎥ 2 s6 + s7 ⎥ ⎥ ⎥. 2 s7 + s8 ⎥ ⎥ 2 s8 ⎦ 0

6.1.12. Lp,q 9.23 N = g6.32p=0 , [e4 , e5 ] = e6 , [e4 , e9 ] = pe4 , [e5 , e9 ] = pe5 , [e6 , e9 ] = 2qe6 , [e7 , e8 ] = e6 , [e7 , e9 ] = qe7 − e8 . ⎡

s1 − ps9 ⎢ s3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − ps9 0 0 0 0

0 0 0 0 0 s8 0 0 0 s9 0 0

0 0 −s7 −s9 0 0

⎤ s4 s5 ⎥ ⎥ −2s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

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6.1.13. Lp,q 9.24 N = g6.35,p=0,q≥0 , [e4 , e5 ] = e6 , [e4 , e9 ] = pe4 , [e5 , e9 ] = pe5 , [e6 , e9 ] = 2pe6 , [e7 , e9 ] = qe7 − e8 , [e8 , e9 ] = e7 + e8 . ⎡

s1 − ps9 ⎢ s3 ⎢ ⎢ −s ⎢ 5 S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − ps9 s4 0 0 0

0 0 −2ps9 0 0 0

0 0 0 −qs9 s9 0

⎤ s4 s5 ⎥ ⎥ 2 s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

0 0 0 −s9 −qs9 0

6.1.14. Lp9.25 N = g6.82|p|≤1 , [e4 , e5 ] = e6 , [e4 , e9 ] = pe4 , [e5 , e9 ] = pe5 , [e6 , e9 ] = 2pe6 , [e7 , e9 ] = e7 , [e8 , e9 ] = (2p − 1)e8 . ⎡

s1 − ps9 ⎢ s3 ⎢ ⎢ s5 ⎢ ⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − ps9 −s4 0 0 0

0 0 −2ps9 0 0 0

0 0 s8 −s9 0 0

0 0 −s7 0 − (2 p − 1) s9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

6.1.15. L9.26 N = g6.85,=±1 , [e4 , e5 ] = e6 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = 2e6 , [e7 , e8 ] = e6 , [e7 , e9 ] = e7 , [e8 , e9 ] = e7 + e8 . ⎡

−2 s9 ⎢ 0 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s5 −s9 0 0 0 0

−s7 0 −s9 0 −s9 0

s4 0 0 −s9 0 0

s8 0 0 0 −s9 0

⎤ −2 s6

s4 ⎥ ⎥ s8 ⎥ ⎥ ⎥. −s5 ⎥ ⎥ s7 ⎦ 0

6.1.16. Lp9.27 N = g6.89 , [e4 , e5 ] = e6 , [e4 , e9 ] = pe4 , [e5 , e9 ] = pe5 , [e6 , e9 ] = 2pe6 , [e7 , e8 ] = e6 , [e7 , e9 ] = pe7 − e8 , [e8 , e9 ] = e7 + pe8 . ⎡

−2 ps9 ⎢ 0 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s5 s1 − ps9 s3 0 0 0

−s4 s2 −s1 − ps9 0 0 0

s8 0 0 −ps9 s9 0

−s7 0 0 −s9 −ps9 0

⎤ −2s6 s4 ⎥ ⎥ s5 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

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6.1.17. Lp,q 9.28pq=0 pqpq N = N6.1pq =0 , [e4 , e8 ] = pe4 , [e4 , e9 ] = qe4 , [e5 , e8 ] = pe5 , [e5 , e9 ] = qe5 , [e6 , e8 ] = e6 , [e7 , e9 ] = e7 . ⎡

s1 − ps8 − qs9 ⎢ s3 ⎢ ⎢ S=⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − ps8 − qs9 0 0 0

0 0 −s8 0 0

0 0 0 −s9 0

⎤ s4 s5 ⎥ ⎥ ⎥ s6 ⎥ . ⎥ s7 ⎦ 0

6.1.18. Lp9.29 1pp N = N6.2 , [e4 , e8 ] = e4 , [e4 , e9 ] = pe4 , [e5 , e8 ] = e5 , [e5 , e9 ] = pe5 , [e6 , e9 ] = e6 , [e7 , e8 ] = e6 , [e7 , e9 ] = e7 . ⎡

s1 − s8 − ps9 ⎢ s3 ⎢ ⎢ S=⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − s8 − ps9 0 0 0

0 0 −s9 0 0

0 0 −s8 −s9 0

⎤ s4 s5 ⎥ ⎥ ⎥ s6 ⎥ . ⎥ s7 ⎦ 0

6.1.19. Lp,q 9.30 pqpq N = N6.13p 2 +q 2 =0 [e4 , e8 ] = qe4 , [e4 , e9 ] = pe4 , [e5 , e8 ] = qe5 , [e5 , e9 ] = pe5 , [e6 , e8 ] = e6 , [e6 , e9 ] = −e7 , [e7 , e8 ] = e7 , [e7 , e9 ] = e6 . ⎡

s1 − qs8 − ps9 ⎢ s3 ⎢ ⎢ S=⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − qs8 − ps9 0 0 0

⎤ s4 s5 ⎥ ⎥ ⎥ s6 ⎥ . ⎥ s7 ⎦ 0

0 0 −s8 s9 0

0 0 −s9 −s8 0

0 0 0 −s8 0 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ −s9 ⎦ 0

6.1.20. L9.31 1,0 N = N6.20 , [e4 , e8 ] = e4 , [e5 , e8 ] = e5 , [e6 , e9 ] = e6 , [e8 , e9 ] = e7 . ⎡

s1 − s8 ⎢ s 3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − s8 0 0 0 0

0 0 −s9 0 0 0

0 0 0 0 0 0

6.2. sl(2, R): representation D1 ⊕ 3D0 [e1 , e4 ] = 2e4 , [e2 , e5 ] = 2e4 , [e3 , e4 ] = e5 , [e1 , e6 ] = −2e6 , [e2 , e6 ] = e5 , [e3 , e5 ] = 2e6

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6.2.1. Lp,q 9.32 N = g6.1pq=0 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = e6 , [e7 , e9 ] = pe7 , [e8 , e9 ] = qe8 . ⎡

2 s1 − s9 ⎢ s3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

2 s2 −s9 2 s3 0 0 0

0 s2 −2 s1 − s9 0 0 0

0 0 0 −ps9 0 0

0 0 0 0 −qs9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

6.2.2. Lp9.33 N = g6.2 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = e6 , [e7 , e9 ] = pe7 , [e8 , e9 ] = e7 + pe8 . ⎡

2 s1 − s9 ⎢ s3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

2 s2 −s9 2 s3 0 0 0

0 s2 −2 s1 − s9 0 0 0

0 0 0 −ps9 0 0

0 0 0 −s9 −ps9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

6.2.3. Lp,q 9.34 N = g6.8p=0,q≥0 , [e4 , e9 ] = pe4 , [e5 , e9 ] = pe5 , [e6 , e9 ] = pe6 , [e7 , e9 ] = qe7 − e9 , [e8 , e9 ] = e7 + qe8 . ⎡

2s1 − ps9 ⎢ s3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

2 s2 −ps9 2 s3 0 0 0

0 s2 −2s1 − ps9 0 0 0

0 0 0 −qs9 s9 0

0 0 0 −s9 −qs9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

6.3. sl(2, R): representation D 32 ⊕ 2D0 [e1 , e4 ] = 3e4 , [e2 , e5 ] = 3e4 , [e3 , e4 ] = e5 , [e1 , e5 ] = e5 , [e2 , e6 ] = 2e5 , [e3 , e5 ] = 2e6 , [e1 , e6 ] = −e6 , [e2 , e7 ] = e6 , [e3 , e6 ] = 3e7 , [e1 , e7 ] = −3e7 . 6.3.1. Lp9.35 N = g6.1p=0 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = e6 , [e7 , e9 ] = e7 , [e8 , e9 ] = pe8 . ⎡

3 s1 − s9 ⎢ s3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

3 s2 s1 − s9 2 s3 0 0 0

0 2 s2 −s1 − s9 3 s3 0 0

0 0 s2 −3 s1 − s9 0 0

0 0 0 0 −ps9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

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6.3.2. L9.36 N = g6.82 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = e6 , [e7 , e9 ] = e7 , [e8 , e9 ] = 2e8 , [e4 , e7 ] = e8 , [e5 , e6 ] = −3e8 . ⎡

3 s1 − s9 ⎢ s3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ −s7 0

3 s2 s1 − s9 2 s3 0 3 s6 0

0 2 s2 −s1 − s9 3 s3 −3 s5 0

0 0 s2 −3 s1 − s9 s4 0

0 0 0 0 −2 s9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ 2 s8 ⎦ 0

6.4. sl(2, R): representation 2D 12 ⊕ 2D0 [e1 , e4 ] = e4 , [e2 , e5 ] = e4 , [e3 , e4 ] = e5 , [e1 , e5 ] = −e5 , [e1 , e6 ] = e6 , [e2 , e7 ] = e6 , [e3 , e6 ] = e7 , [e1 , e7 ] = −e7 . 6.4.1. L9.37 N = 2A3.1 , [e4 , e5 ] = e8 , [e6 , e7 ] = e9 . ⎡

s1 ⎢ ⎢ s3 ⎢ ⎢ s5 ⎢ ⎢ 0 S=⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ ⎣ 0 0

s2 −s1 −s4 0 0 0 0 0

0 s4 0 s5 0 −2s8 0 0 0 0 0 0 0 0 0 0

0 0 0 0 s1 s3 s7 0

0 0 0 0 s2 −s1 −s6 0

0 0 0 0 0 0 0 0

⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥. s6 ⎥ ⎥ s7 ⎥ ⎥ ⎥ −2s9 ⎦ 0

6.4.2. L9.38 N = 2A3.3 , [e4 , e8 ] = e4 , [e5 , e8 ] = e5 , [e6 , e9 ] = e6 , [e7 , e9 ] = e7 . ⎡

s1 − s8 ⎢ s 3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − s8 0 0 0 0

s4 s5 0 0 0 0

0 0 0 s1 − s9 s3 0

0 0 0 s2 −s1 − s9 0

6.4.3. L9.39 N = A3.1 ⊕ A3.3 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e7 ] = e8 . ⎡

s1 ⎢ s3 ⎢ ⎢ 0 ⎢ ⎢ S=⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 0 0 0 0 0

s4 s5 s9 0 0 0 0

0 0 0 s1 s3 s7 0

0 0 0 s2 −s1 −s6 0

0 0 0 0 0 0 0

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ s6 ⎥ . ⎥ s7 ⎥ ⎥ −2 s8 ⎦ 0

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥. s6 ⎥ ⎥ s7 ⎦ 0

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6.4.4. L9.40 N = A6.3 , [e6 , e7 ] = e8 , [e6 , e9 ] = e4 , [e7 , e9 ] = e5 . ⎡

0 ⎢0 ⎢ ⎢0 ⎢ S=⎢ ⎢0 ⎢ ⎣0 0

0 s1 s3 0 0 0

0 s2 −s1 0 0 0

s7 −s9 0 s1 s3 0

−s6 0 −s9 s2 −s1 0

s6 s2 −s1 0 0

s8 s4 s5 0 0

⎤ −2 s8 s4 ⎥ ⎥ s5 ⎥ ⎥ ⎥. s6 ⎥ ⎥ s7 ⎦ 0

6.4.5. L9.41 N = A6.4 , [e4 , e7 ] = e8 , [e5 , e6 ] = −e8 , [e6 , e7 ] = e9 . ⎡

0 ⎢0 ⎢ ⎢ S = ⎢0 ⎢ ⎣0 0

−s7 s1 s3 0 0

⎤ 2s9 s6 ⎥ ⎥ ⎥ s7 ⎥ . ⎥ 0 ⎦ 0

6.4.6. L9.42 N = A6.5 , [e4 , e5 ] = −e9 , [e4 , e7 ] = e8 , [e5 , e6 ] = −e8 , [e6 , e7 ] = e9 . ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ S=⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

0

0

0

0

0

0

0

0

0

s4 +is6 2 s5 +is7 2 s9 −is8 2

0

0

0

0

0

0

0

0

0

0 4 − s6 +is 2 5 − s7 +is 2 8 − s9 +is 2

s1

s2

0

s3

−s1

0

s5 +is7 2

6 − s4 +is 2

0

0

0

0

0

0

s1

s2

0

0

0

0

0

0

0

0

−s1 s6 +is4 2

0

0

s3 s7 +is5 − 2

0

0

0

0

0

0

0

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0

6.4.7. Lp,q 9.43 N = g6.1q=0,|p|≤1| , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = pe6 , [e7 , e9 ] = pe7 , [e8 , e9 ] = qe8 . ⎡

s1 − s9 ⎢ s 3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − s9 0 0 0 0

0 0 s1 − ps9 s3 0 0

0 0 s2 −s1 −ps9 0 0

0 0 0 0 −qs9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

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6.4.8. Lp9.44 N = g6.6 , [e4 , e9 ] = pe4 , [e5 , e9 ] = pe5 , [e6 , e9 ] = e4 + pe6 , [e7 , e9 ] = e5 + pe7 , [e8 , e9 ] = e8 . ⎡

s1 −ps9 ⎢ s 3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

−s9 0 s1 − ps9 s3 0 0

s2 −s1 − ps9 0 0 0 0

0 −s9 s2 −s1 − ps9 0 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

0 0 0 0 −s9 0

6.4.9. Lp,q 9.45 N = g6.11q=0 , [e4 , e9 ] = pe4 − e6 , [e5 , e9 ] = pe5 − e7 , [e6 , e9 ] = e4 + pe6 , [e7 , e9 ] = e5 + pe7 , [e8 , e9 ] = qe8 . ⎡

s1 − ps9 ⎢ s3 ⎢ ⎢ s9 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − ps9 0 s9 0 0

−s9 0 s1 − ps9 s3 0 0

0 −s9 s2 s1 − ps9 0 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

0 0 0 0 −qs9 0

6.4.10. Lp9.46 N = g6.13 , [e4 , e5 ] = e8 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = pe6 , [e7 , e9 ] = pe7 , [e8 , e9 ] = 2e8 . ⎡

s1 − s9 ⎢ s 3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ −s5 0

s2 −s1 − s9 0 0 s4 0

0 0 s1 − ps9 s3 0 0

0 0 s2 −s1 − ps9 0 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ 2 s8 ⎦ 0

0 0 0 0 −2 s9 0

6.4.11. L9.47 N = g6.15 , [e6 , e7 ] = e8 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = e4 + e6 , [e7 , e9 ] = e5 + e7 , [e8 , e9 ] = 2e8 . ⎡

s1 − s9 ⎢ s 3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − s9 0 0 0 0

s9 0 s1 − s9 s3 −s7 0

0 −s9 s2 −s1 − s9 s6 0

0 0 0 0 −2 s9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ 2 s8 ⎦ 0

6.4.12. L9.48 N = g6.53 , [e6 , e8 ] = e4 , [e6 , e9 ] = e6 , [e7 , e8 ] = e5 , [e7 , e9 ] = e7 , [e8 , e9 ] = 2e8 . ⎡

s1 ⎢s ⎢ 3 S=⎢ ⎣ 0 0

s2 −s1 0 0

s6 s7 s9 0

⎤ s4 s5 ⎥ ⎥ ⎥. s8 ⎦ 0

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6.4.13. Lp9.49 N = g6.54 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e8 ] = e4 [e6 , e9 ] = (1 − p)e6 , [e7 , e8 ] = e5 , [e7 , e9 ] = (1 − p)e7 , [e8 , e9 ] = pe8 . ⎡

s1 − s9 ⎢ s ⎢ 3 S=⎢ ⎣ 0 0

s2 −s1 − s9 0 0

s6 s7 −ps9 0

⎤ s4 s5 ⎥ ⎥ ⎥. s8 ⎦ 0

6.4.14. L9.50 N = g6.76 , [e4 , e9 ] = 3e4 , [e5 , e9 ] = 3e5 , [e6 , e7 ] = e8 , [e6 , e8 ] = e4 , [e6 , e9 ] = e6 , [e7 , e8 ] = e5 , [e7 , e9 ] = e7 , [e8 , e9 ] = 2e8 . ⎡

s1 − 3 s9 ⎢ s3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 − 3 s9 0 0 0 0

−s8 0 s1 − s9 s3 −s7 0

0 −s8 s2 −s1 − s9 s6 0

s6 s7 0 0 −2s9 0

⎤ 3 s4 3 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ 2 s8 ⎦ 0

6.4.15. Lp9.51 N = g6.82|p|≤1 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e4 , e7 ] = e8 , [e5 , e6 ] = −e8 , [e6 , e9 ] = pe6 , [e7 , e8 ] = e5 , [e7 , e9 ] = pe7 , [e8 , e9 ] = (1 + p)e8 . ⎡

− (p + 1) s9 ⎢ 0 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

s7 s1 − s9 s3 0 0 0

−s6 s2 −s1 − s9 0 0 0

s5 0 0 s1 − ps9 s3 0

−s4 0 0 s2 −s1 − ps9 0

⎤ −2s8 s4 ⎥ ⎥ s5 ⎥ ⎥ ⎥. s6 ⎥ ⎥ s7 ⎦ 0

6.4.16. L9.52 N = g6.82 , [e4 , e5 ] = e8 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e7 ] = e8 , [e6 , e9 ] = e6 , [e7 , e9 ] = e7 , [e8 , e9 ] = 2e8 . ⎡

2s9 ⎢ 0 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

−s5 s1 + s9 s3 0 0 0

s4 s2 s9 − s1 0 0 0

−s7 0 0 s1 + s9 s3 0

s6 0 0 s2 s9 − s1 0

⎤ −s8 s4 ⎥ ⎥ s5 ⎥ ⎥ ⎥. s6 ⎥ ⎥ s7 ⎦ 0

6.4.17. Lp9.53 N = g6.92p≥0 , [e4 , e5 ] = e8 , [e4 , e9 ] = pe4 − e6 , [e5 , e9 ] = pe5 − e7 , [e6 , e7 ] = e8 , [e6 , e9 ] = e4 + pe6 , [e7 , e9 ] = e5 + pe7 , [e8 , e9 ] = 2pe8 .

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⎡

−2 ps9 ⎢ 0 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

−s4 s2 −s1 − ps9 0 s9 0

s5 s1 − ps9 s3 s9 0 0

s7 −s9 0 s1 − ps9 s3 0

−s6 0 −s9 s2 −s1 − ps9 0

⎤ −2s8 s4 ⎥ ⎥ s5 ⎥ ⎥ ⎥. s6 ⎥ ⎥ s7 ⎦ 0

6.4.18. L9.54 N = N6.18 , [e4 , e8 ] = e4 , [e4 , e9 ] = e6 , [e5 , e8 ] = e5 , [e5 , e9 ] = e7 , [e6 , e8 ] = e6 [e6 , e9 ] = −e4 , [e7 , e8 ] = e7 , [e7 , e9 ] = −e5 . ⎡

s1 − s8 ⎢ s 3 ⎢ ⎢ S = ⎢ −s9 ⎢ ⎣ 0 0

s2 −s1 − s8 0 −s9 0

s9 0 s1 − s8 s3 0

0 s9 s2 −s1 − s8 0

⎤ s4 s5 ⎥ ⎥ ⎥ s6 ⎥ . ⎥ s7 ⎦ 0

6.5. sl(2, R): representation D2 ⊕ D0 [e1 , e4 ] = 4e4 , [e1 , e5 ] = 2e5 , [e1 , e7 ] = −2e7 , [e1 , e8 ] = −4e8 , [e2 , e5 ] = 4e4 , [e2 , e6 ] = 3e5 , [e2 , e7 ] = 2e6 , [e2 , e8 ] = e7 , [e3 , e4 ] = e5 , [e3 , e5 ] = 2e6 , [e3 , e6 ] = 3e7 , [e3 , e7 ] = 4e8 . 6.5.1. L9.55 N = g6.1 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = e6 , [e7 , e9 ] = e7 , [e8 , e9 ] = e8 . ⎡

4 s1 − s9 ⎢ s3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

4 s2 2 s1 − s9 2 s3 0 0 0

0 3 s2 −s9 3 s3 0 0

0 0 2 s2 −2 s1 − s9 4 s3 0

0 0 0 s2 −4 s1 − s9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

6.6. sl(2, R): representation D1 ⊕ D 12 ⊕ D0 [e1 , e4 ] = 2e4 , [e2 , e5 ] = 2e4 , [e3 , e4 ] = e5 , [e1 , e6 ] = −2e6 , [e2 , e6 ] = e5 , [e3 , e5 ] = 2e6 , [e1 , e7 ] = e7 , [e2 , e8 ] = e7 , [e3 , e7 ] = e8 , [e1 , e8 ] = −e8 . 6.6.1. Lp9.56 N = g6.1p=0 ; A1,1 4,5 ⊕ 2A1 p=0 , [e4 , e9 ] = e4 , [e5 , e9 ] = e5 , [e6 , e9 ] = e6 , [e7 , e9 ] = pe7 , [e8 , e9 ] = pe8 . ⎡

2 s1 − s9 ⎢ s3 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ ⎣ 0 0

2 s2 −s9 2 s3 0 0 0

0 s2 −2 s1 − s9 0 0 0

0 0 0 s1 − ps9 s3 0

0 0 0 s2 −s1 − ps9 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥. s7 ⎥ ⎥ s8 ⎦ 0

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6.6.2. L9.57 N = 3A1 ⊕ A3.3 , [e7 , e9 ] = e7 , [e8 , e9 ] = e8 .

⎡

2 s1 ⎢ s3 ⎢ ⎢ 0 ⎢ ⎢ S=⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0

2 s2 0 2 s3 0 0 0 0

0 s2 −2 s1 0 0 0 0

s4 s5 s6 0 0 0 0

0 0 0 0 s1 s3 0

0 0 0 0 s2 −s1 0

⎤ 0 0 ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥. ⎥ s7 ⎥ ⎥ s8 ⎦ s9

6.6.3. L9.58 N = 3A1 ⊕ A3.1 , [e7 , e8 ] = e9 .

⎡

s1 ⎢ ⎢ s3 ⎢ ⎢ s8 ⎢ ⎢ 0 S=⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ ⎣ 0 0

s2 −s1 −s7 0 0 0 0 0

0 s7 0 s8 0 s9 0 0 0 0 0 0 0 0 0 0

0 0 0 0 2 s1 s3 0 0

0 0 0 0 2 s2 0 2 s3 0

0 0 0 0 0 s2 −2 s1 0

⎤ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥. s4 ⎥ ⎥ s5 ⎥ ⎥ ⎥ s6 ⎦ 0

6.7. sl(2, R): representation D 52

[e1 , e4 ] = 5e4 , [e2 , e5 ] = 5e4 , [e3 , e4 ] = e5 , [e1 , e5 ] = 3e5 , [e2 , e6 ] = 4e5 , [e3 , e5 ] = 2e6 , [e1 , e6 ] = e6 , [e2 , e7 ] = 3e6 , [e3 , e6 ] = 3e7 , [e1 , e7 ] = −e7 , [e2 , e8 ] = 2e7 , [e3 , e7 ] = 4e8 , [e1 , e8 ] = −3e8 , [e2 , e9 ] = e8 , [e3 , e8 ] = e9 , [e1 , e9 ] = −e9 .

6.7.1. L9.59 N = Abelian.

⎡

5 s1 ⎢ s3 ⎢ ⎢ 0 ⎢ ⎢ S=⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0

5 s2 3 s1 2 s3 0 0 0 0

0 4 s2 s1 3 s3 0 0 0

0 0 3 s2 −s1 4 s3 0 0

0 0 0 2 s2 −3 s1 5 s3 0

0 0 0 0 s2 −5 s1 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥ s7 ⎥ . ⎥ s8 ⎥ ⎥ s9 ⎦ 0

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6.8. sl(2, R): representation D 32 ⊕ D 12 6.8.1. L9.60 N = Abelian. ⎡

3 s2 s1 2 s3 0 0 0 0

3 s1 ⎢ s3 ⎢ ⎢ 0 ⎢ ⎢ S=⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0

0 2 s2 −s1 3 s3 0 0 0

0 0 s2 −3 s1 0 0 0

0 0 0 0 s1 s3 0

0 0 0 0 s2 −s1 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥ s7 ⎥ . ⎥ s8 ⎥ ⎥ s9 ⎦ 0

6.9. sl(2, R): representation 2D1 [e1 , e4 ] = 2e4 , [e2 , e5 ] = 2e4 , [e3 , e4 ] = e5 , [e1 , e6 ] = −2e6 , [e2 , e6 ] = e5 , [e3 , e5 ] = 2e6 , [e1 , e7 ] = 2e7 , [e2 , e8 ] = 2e7 , [e3 , e7 ] = e7 , [e1 , e9 ] = −2e9 , [e2 , e9 ] = e8 , [e3 , e8 ] = 2e9 . 6.9.1. L9.61 N = Abelian. ⎡

2 s1 ⎢ s ⎢ 3 ⎢ S=⎢ 0 ⎢ ⎣ 0 0

2 s2 0 2 s3 0 0

0 s2 −2 s1 0 0

s4 s5 s6 0 0

⎤ s7 s8 ⎥ ⎥ ⎥ s9 ⎥ . ⎥ 0 ⎦ 0

6.9.2. L9.62 N = A6,3 , [e4 , e5 ] = 2e7 , [e4 , e6 ] = e8 , [e5 , e9 ] = e8 . [e4 , e5 ] = e7 , [e4 , e6 ] = e8 , [e5 , e6 ] = 2e9 . ⎡

2 s1 ⎢2s ⎢ 3 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ S=⎢ ⎢ 0 ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 0 s3 0 0 0 0 0 0

0 2 s2 −2 s1 0 0 0 0 0 0

2 s5 2 s6 0 2 s1 2 s3 0 0 0 0

−s4 0 s6 s2 0 s3 0 0 0

0 −2 s4 −2 s5 0 2 s2 −2 s1 0 0 0

−2 s8 −2 s9 0 2 s5 2 s6 0 2 s1 2 s3 0

s7 0 −s9 −s4 0 s6 s2 0 s3

⎤ 0 2 s7 ⎥ ⎥ ⎥ 2 s8 ⎥ ⎥ 0 ⎥ ⎥ −2 s4 ⎥ ⎥. −2 s5 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 2 s2 ⎦ −2 s1

6.10. sl(2, R): representation 3D 12 [e1 , e4 ] = e4 , [e1 , e5 ] = −e5 , [e1 , e6 ] = e6 , [e1 , e7 ] = −e7 , [e1 , e8 ] = e8 , [e1 , e9 ] = −e9 , [e2 , e5 ] = e4 , [e2 , e7 ] = e6 , [e2 , e8 ] = e9 , [e3 , e4 ] = e5 , [e3 , e6 ] = e7 , [e3 , e8 ] = e9 .

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6.10.1. L9.63 N = Abelian. ⎡

s1 ⎢ s3 ⎢ ⎢ 0 ⎢ ⎢ S=⎢ 0 ⎢ ⎢ 0 ⎢ ⎣ 0 0

s2 −s1 0 0 0 0 0

0 0 s1 s3 0 0 0

0 0 s2 −s1 0 0 0

0 0 0 0 s1 s3 0

0 0 0 0 s2 −s1 0

⎤ s4 s5 ⎥ ⎥ s6 ⎥ ⎥ ⎥ s7 ⎥ . ⎥ s8 ⎥ ⎥ s9 ⎦ 0

References [1] R. Campoamor-Stursberg, Structural Data and Invariants of Nine Dimensional Real Lie Algebras with Non-Trivial Levi Decomposition, Nova Science Publishers Inc., New York, 2009. [2] R. Ghanam, M. Lamichhane, G. Thompson, Minimal representations of Lie algebras with non-trivial Levi decomposition, Arab. J. Math. 6 (4) (December 2017) 281–296. [3] R. Ghanam, M. Lamichhane, G. Thompson, Minimal dimension representations of decomposable Lie algebras, Extr. Math. 33 (2) (2018) 219–227. [4] R. Ghanam, G. Thompson, Minimal matrix representations of five-dimensional Lie algebras, Extr. Math. 30 (1) (2015) 95–133. [5] R. Ghanam, G. Thompson, Minimal matrix representations for six-dimensional nilpotent Lie algebras, Math. Æterna 8 (3) (2018) 113–138. [6] J. Humphreys, Lie Algebras and Their Representations, Springer, 1997. [7] Y. Kang, C. Bai, Refinement of Ado’s theorem in low dimensions and applications in affine geometry, Commun. Algebra 36 (1) (2008) 82–93. [8] G.M. Mubarakzyanov, Classification of Solvable Lie Algebras in dimension six with one non-nilpotent basis element, Izv. Vysš. Učebn. Zaved., Mat. 4 (35) (1963) 104–116. [9] M. Rawashdeh, G. Thompson, The inverse problem for the canonical connection of six-dimensional codimension two nilradical Lie Algebras, J. Math. Phys. 47 (2006) 112901. [10] N. Rojas, Minimal representations for 6-dimensional nilpotent Lie algebras, J. Algebra Appl. 15 (10) (2016) 1650191. [11] L. Snobl, P. Winternitz, Classification and Identification of Lie algebras, American Mathematical Society CRM Monograph Series, vol. 33, 2014. [12] A. Shabanskaya, G. Thompson, Six-dimensional Lie algebras with a five-dimensional nilradical, J. Lie Theory 23 (2) (2013) 313–355. [13] P. Turkowski, Low-dimensional real Lie algebras, J. Math. Phys. 29 (10) (1988) 2139–2144. [14] P. Turkowski, Solvable Lie Algebras of dimension six, J. Math. Phys. 31 (6) (1990) 1344–1350. [15] P. Turkowski, Structure of real Lie algebras, Linear Algebra Appl. 171 (1992) 197–212.