Minimal Representations of Lie Algebras With Non-Trivial Levi Decomposition
Ryad Ghanam, Manoj Lamichhane, Gerard Thompson
TL;DR
This paper constructs minimal dimension matrix representations for certain low-dimensional Lie algebras with non-trivial Levi decompositions, using subspace techniques related to semi-simple Lie algebra representations.
Contribution
It provides explicit minimal matrix representations for specific low-dimensional Lie algebras with Levi decomposition, advancing understanding of their structure.
Findings
Minimal representations for 5-8 dimensional Lie algebras obtained
Uses subspace techniques related to semi-simple Lie algebra representations
Provides explicit construction methods for these representations
Abstract
We obtain minimal dimension matrix representations for each of the Lie algebras of dimension five, six, seven, and eight obtained by Turkowski that have a non-trivial Levi decomposition. The Key technique involves using subspace associated to a particular representation of semi-simple Lie algebra to help in the construction of the radical in the putative Levi decomposition.
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Minimal representations of Lie algebras with non-trivial Levi decomposition
Abstract.
We obtain minimal dimension matrix representations for each of the Lie algebras of dimensions five, six, seven and eight obtained by Turkowski that have a non-trivial Levi decomposition. The key technique involves using the invariant subspaces associated to a particular representation of a semi-simple Lie algebra to help the in the construction of the radical in the putative Levi decomposition.
**Ryad Ghanam1, Manoj Lamichhane2 and Gerard Thompson3
**
1 Department of Mathematics, Virginia Commonwealth University in Qatar
PO Box 8095, Doha, Qatar
2 Department of Mathematics, Western Kentucky University
Bowling Green, KY 42101
3 Department of Mathematics, University of Toledo,
Toledo, OH 43606, U.S.A.
Key words: semi-simple Lie algebra, minimal representation, radical, Levi decomposition, invariant subspaces.
AMS Mathematics Subject Classification (2010): 22E60 17B05, 17B30, 20G05, 22E25.
1. Introduction
Given a real Lie algebra of dimension a well known theorem due to Ado asserts that has a faithful representation as a subalgebra of \mathfrak{g}\mathfrak{l}(p,\mbox{\mathbb{R}}) for some . In several recently published papers the authors and others have investigated the problem of finding minimal dimensional representations of indecomposable dimensional Lie algebras of dimension five and less [TW, KB, GT1, GT2]. In two recent papers [KB, GT1], the problem of finding such a minimal representation is considered for the four-dimensional Lie algebras. In [KB], the main technique is somewhat indirect and depends on a construction known as a left symmetric structure. In [GT1, GT2], the minimal representations have been calculated directly without the need for considering left symmetric structures.
Burde [Bur] has defined an invariant of a Lie algebra to be the dimension of its minimal faithful representation. So far minimal representations have been found for all the indecomposable and decomposable algebras of dimension five and less [GT1, GT2]. In this paper we investigate the issue of minimal representations for Lie algebras that have a non-trivial Levi decomposition. Such algebras have been classified in dimensions five to eight by Turkowski [Turk]. Of course it is interesting to ascertain the value of from a theoretical point of view. However, an important practical reason is that calculations involving symbolic programs such as Maple and Mathematica use up lots of memory when storing matrices; accordingly, calculations are likely to be faster if one can represent matrix Lie algebras using matrices of a small size. Besides the value of having explicit representations of low-dimensional Lie algebras they also add to the growing body of results that seek to provide alternatives to Ado’s Theorem for the construction of representations [Bur].
In this paper we shall use the classification of the low-dimensional indecomposable Lie algebras found in [PSWZ] which is in turn taken from [Mub1]. Such algebras are denoted as where denotes the dimension of the algebra and the th one in the list. In dimension , the algebra is the unique non-abelian algebra. The same notation is adopted in [Mub2], which concerns a class of six-dimensional algebras, and is used in [Turk], which is the most important reference for the current paper. We shall also use to denote the irreducible representation of and for the irreducible representation of order of \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}) where is either an integer of half-integer. Furthermore, denotes copies of and the trivial one-dimensional representation.
The outline of this paper is as follows. In Section 2 we consider the construction of abstract Lie algebras that have a Levi decomposition. In Section 3 we present several results about representations that will needed be in the sequel. In Section 4 we carry out the construction of the minimal representations proper. A key element here is the use of the invariant subspaces that come from the representation of the semi-simple factor, in all cases here either or \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}) in a given representation, that acts by commutator on \mathfrak{g}\mathfrak{l}(N,\mbox{\mathbb{R}}) for various small values of , to construct a representation of the radical ideal of the algebra. It is argued in every case that the representations thus constructed are of minimal dimension. Finally, in Section 5 we give a list of minimal dimension representations for each of the algebras in Turkowski’s list [Turk]. Occasionally we give more than one such representation, sometimes depending on parameters and if it seems to be of particular interest although we do not consider the difficult issue of the inequivalence of different representations. Finally, the reader may object that we have not supplied details of the Turkowski Lie algebras but to do so would involve repeating large Sections of [Turk]. In any case the Lie brackets can be readily computed from the representations given in Section 5.
2. Constructing Levi algebras in general
Let us consider the problem of constructing a Lie algebra that has a Levi decomposition in general. We have the following structure equations:
[TABLE]
where and and is a basis for the semi-simple subalgebra and is a basis for the radical . Calculation shows that the Jacobi identity is equivalent to the following conditions:
[TABLE]
We interpret these conditions as follows: we start with a semi-simple algebra so that the first set of conditions above is satisfied. Then the second set say that the matrices make (merely as a vector space) into a -module. The third set say that the are derviations of the Lie algebra and the fourth of course that is a Lie algebra. Therefore to find all possible Lie algebras of dimension that have a Levi decomposition we can proceed as follows: choose a semi-simple algebra of dimension . Then pick any solvable algebra of dimension and consider a representation of in considered simply as the vector space \mbox{\mathbb{R}}^{n-r}, all of which are known and are completely reducible [Hum] since is semi-simple. Of course it is a difficult question as to whether a representation of a particular semi-simple algebra exists in a certain dimension: see [Hum] for more details. Finally, it only remains to check that the matrices representing act as derivations of the Lie algebra . 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 in . If all such representations lead to a null result then there can be no non-trivial Levi decomposition involving and .
3. Representation results
We quote the following result from Humphreys [Hum].
Proposition 3.1**.**
Let be a finite dimensional complex Lie algebra acting irreducibly on the vector space . Then is reductive and the center is of dimension one or zero. If in addition then 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.
The following two results lie at opposite extremes.
Theorem 3.1**.**
Suppose that the Lie algebra is a semi-direct product of a semi-simple subalgebra and an -dimensional abelian ideal in the Levi decomposition. Then has a (faithful) representation as a subalgebra of .
Conversely, a subalgebra of with the upper block giving a representation of semi-simple Lie algebra and the first entries in the last column being arbitrary and the th zero, gives a Levi subalgebra of whose radical is abelian.
Proof.
We quote the structure equations in eqn 2.1 where now we assume that . To obtain the required representation consider ad restricted to , in other words the matrices . Define to be the same matrix as but now augmented by an extra bottom row and last column of zeroes. Now define to be the matrix whose only non-zero is in the th position. Then ; now . As regards it is a matrix that has only non-zero entries in the last column. That last column is with and fixed, augmented by a zero in the th position which means that . Since and we have a representation of in . ∎
Theorem 3.2**.**
*Suppose that is the radical of a Lie algebra and that has trivial center. Then has a faithful representation of in where is of dimension . *
Proof.
Define to be the matrix and to be the matrix . Then in order to have a representation of we require that
[TABLE]
However, these conditions are identical with the last three in eqn. 2.2. If the center of is trivial then we will have a faithful representation of . ∎
The previous Theorem admits the following variation, having been already noted in the solvable case [GT2].
Theorem 3.3**.**
*Suppose that is the radical of a Lie algebra and that has an abelian nilradical of dimension where is of dimension . Then has a faithful representation in . *
Proof.
Define to be the matrix and to be the matrix . For define to be the matrix whose only non-zero entry is a in the th position. Then proceed as in Theorem 3.2. ∎
4. Minimal Representations
4.1. Algebras of dimension
4.1.1. Algebras of dimension
The first algebra that has a non-trivial Levi decomposition is denoted either by in [PSWZ] or in [Turk]. We obtain a representation for it in \mathfrak{g}\mathfrak{l}(3,\mbox{\mathbb{R}}) by using Theorem 3.1.
4.1.2. Algebras of dimension
In dimension six there are four algebras that have a non-trivial Levi decomposition denoted by in [Turk]. According to [TW] only one of them, , has a representation in \mathfrak{g}\mathfrak{l}(3,\mbox{\mathbb{R}}) which is supplied by Theorem 3.2. Furthermore, Theorem 3.1 provides representations of minimal dimension for and in \mathfrak{g}\mathfrak{l}(4,\mbox{\mathbb{R}}). Furthermore has a representation in \mathfrak{g}\mathfrak{l}(4,\mbox{\mathbb{R}}) which is also of minimal dimension.
4.1.3. Algebras of dimension
According to [TW] the only seven-dimensional algebra that has a representation in \mathfrak{g}\mathfrak{l}(3,\mbox{\mathbb{R}}) is decomposable. There are seven algebras that have a non-trivial Levi decomposition denoted by in [Turk]. Theorem 3.2 provides representations for in \mathfrak{g}\mathfrak{l}(4,\mbox{\mathbb{R}}) whereas Theorem 3.1 yields representations for and in \mathfrak{g}\mathfrak{l}(5,\mbox{\mathbb{R}}). In fact also has a representation in \mathfrak{g}\mathfrak{l}(4,\mbox{\mathbb{R}}).
It remains to show that the representations for coming from Theorem (3.1) are minimal. We have said already that they cannot be subalgebras of \mathfrak{g}\mathfrak{l}(3,\mbox{\mathbb{R}}). If they had representations in \mathfrak{g}\mathfrak{l}(4,\mbox{\mathbb{R}}) there would be a basis in which the radical \mbox{\mathbb{R}}^{4} would correspond to the upper right block according to a Theorem of Schur-Jacobson [Jac]. As such the form of the representation would have to be . The occurrence of the lower left zero block is a consequence of the fact that the radical is an ideal. In the case of there is then no room to accommodate a representation of ; for we could only take either the standard representation of \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}) or the “diagonal” representation of \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}). In the former case we obtain algebra and in the latter a decomposable subalgebra but certainly not .
5. Algebras of dimension
Now we consider Turkowski’s algebras of dimension . At the outset it is clear that for such algebras because the only eight-dimensional subalgebra of \mathfrak{g}\mathfrak{l}(3,\mbox{\mathbb{R}}) is \mathfrak{s}\mathfrak{l}(3,\mbox{\mathbb{R}}), which is simple [TW]. We shall find the algebras for which .
5.1. Algebras of dimension for which
According to Proposition 3.1 we cannot have the irreducible representation of \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}) or . Suppose next that we had the irreducible representation of \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}) or . A complement has to have matrices of the form . However, the complement will be solvable only if either or in which case it gives an eight-dimensional decomposable algebra. So the only possibilities, up to isomorphism, for the simple factor is either of the representations or .
In the first case the invariant subspaces may be described as where u,v,x,y\in\mbox{\mathbb{R}}^{2} and each of them engenders a two-dimensional invariant subspace, together with a kernel . The radical cannot consist entirely of the kernel or else we will obtain a decomposable algebra, so one of must occur. Suppose that occurs. We will suppose also that occurs. We will need to add in a one-dimensional subspace of the kernel. It can be one of four kinds:
[TABLE]
It is not our concern here to look at all possible subalgebras of \mathfrak{s}\mathfrak{l}(4,\mbox{\mathbb{R}}) (see [GT3]); however, if we put in the first matrix in 5.1, we obtain precisely . In the second case we obtain , in the third and in the fourth .
Now suppose that but that . Then another way to obtain an algebra is to take and independent of . We will need to have in the kernel. We obtain thereby a radical that is isomorphic to the five-dimensional Heisenberg algebra: it is in [Turk]. Other choices of invariant subspaces will lead to different representations but not to different underlying algebras.
Finally we come back to the other representation of \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}). This time the invariant subspaces may be described as , where each of is a copy of \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}) together with the kernel . To avoid having a decomposable algebra we include and we need to take a two-dimensional kernel. Again to avoid decomposability, we shall have to take, up to equivalence, and . We obtain .
We know now that for all the remaining algebras . Next we observe that Theorem 3.2 is applicable to the following algebras: and for them we have . There remain nine cases which are not yet covered: . We note that Theorem 3.1 implies for that . In fact for we have . The cases of and will be discussed below. As regards it depends on a parameter ; for the radical has trivial center but for there is a one-dimensional center. However, in the case we can find a representation by appealing to Theorem 3.3. This representation, unlike the adjoint representation of , extends uniformly for all values of . Hence for we have . The case will be discussed below. We shall also consider where the radical is the five-dimensional Heisenberg algebra.
5.2. and
As stated above, Theorem 3.1 furnishes representations for and in dimension . The question is whether representations exist in lower dimensions. We know that the only possibility is a representation in dimension . We examine first. We know that has a unique irreducible representation in every dimension starting at . To find a representation in \mathfrak{g}\mathfrak{l}(5,\mbox{\mathbb{R}}) we must consider the irreducible representations in dimensions and . The case is impossible according to Proposition 3.1. For the case we argue as follows. Without loss of generality the representation for may be assumed to be of the form where is the standard irreducible representation of . Now consider the action of this representation of on \mathfrak{g}\mathfrak{l}(5,\mbox{\mathbb{R}}). We write such a matrix in \mathfrak{g}\mathfrak{l}(5,\mbox{\mathbb{R}}) as where u,v,x,y\in\mbox{\mathbb{R}}^{3} and a,b,c,d,\alpha\in\mbox{\mathbb{R}}. Now engender invariant subspaces and so too do the one-dimensional subspaces corresponding to ; the latter five span the kernel of the representation. In order to have a -dimensional radical we must choose one of and a -dimensional subspace of the kernel. However, we will never obtain as the representation of on the radical, the irreducible representation of . The argument for is similar if even easier.
The preceding argument lends itself to immediate generalization as follows. Whenever a irreducible representation of a semi-simple algebra is augmented by adding a certain number of zero rows and columns and acts by commutator on \mathfrak{g}\mathfrak{l}(N,\mbox{\mathbb{R}}) for some , the corresponding invariant subspaces give a number of copies of the associated standard or definition representation, that is, a matrix acting by matrix multiplication on a vector, together with the kernel and a complement to the irreducible representation inside of \mathfrak{g}\mathfrak{l}(k,\mbox{\mathbb{R}}). In particular, it is impossible to obtain an irreducible representation unless or from the complementary subspace; of course for this subspace is not available as it would change the given representation into \mathfrak{g}\mathfrak{l}(N,\mbox{\mathbb{R}}).
Now we examine and apply the discussion of the last paragraph. The representation of \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}) on the radical is supposed to give . As such we can exclude the irreducible representation of sizes and . The representation is also not allowed because of Proposition 3.1. There remain two more representations to consider:
[TABLE]
In the first case the invariant subspaces can be described as:
[TABLE]
[TABLE]
We see that none of these invariant subspaces are five-dimensional and hence we cannot obtain the representation .
In the second case the invariant subspaces are given by:
[TABLE]
where are each copies of \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}),u,v,x,y\in\mbox{\mathbb{R}}^{3} and a,b,c,d,\alpha\in\mbox{\mathbb{R}}. Again there is no possibility of obtaining the representation .
5.3. Radical -dimensional Heisenberg
Proposition 5.1**.**
Of the algebras only has a faithful representation in \mathfrak{g}\mathfrak{l}(4,\mbox{\mathbb{R}}).
Proof.
Each of the three algebras has a one-dimensional center spanned by . In each case also spans the center of the Heisenberg algebra of dimension five which is the radical in these cases. In any matrix representation of Heisenberg, a non-zero center must be represented by a nilpotent matrix. Up to isomorphism, in \mathfrak{g}\mathfrak{l}(4,\mbox{\mathbb{R}}) there are precisely three non-zero nilpotent matrices of rank three, two and one:
[TABLE]
The centralizer of a matrix of type (i) is of the form , which is four-dimensional abelian. As regards a matrix of type (ii), its centralizer is of the form . The latter space of matrices comprise an eight-dimensional decomposable Levi algebra, \mbox{\mathbb{R}}^{2}\oplus L_{6.4} in fact. The conclusion is that if any of the algebras has a faithful representation in \mathfrak{g}\mathfrak{l}(4,\mbox{\mathbb{R}}), a non-zero element in its center must be a multiple of a matrix of type (iii) in the appropriate basis.
The centralizer of a matrix of type (iii) is of the form . This latter space of matrices comprise a decomposable ten-dimensional Levi algebra. In fact it is isomorphic to \mbox{\mathbb{R}}\oplus(\mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{6.82\delta=0a=b=1}) [Mub2]. Since the derived algebra of Heisenberg is spanned by the matrix of type (iii), the only way to obtain it as a subalgebra is in the obvious way, that is, by taking and : it gives the algebra . ∎
Now we shall investigate whether algebras that have a -dimensional Heisenberg radical can be represented in \mathfrak{g}\mathfrak{l}(2,\mbox{\mathbb{R}}). Again the central element must be represented by a nilpotent matrix . The rank of cannot be otherwise its centralizer is abelian; nor can it be or else the centralizer is solvable. Suppose then that the rank of is and we may assume that . Its centralizer may be written as where are , x,y\in\mbox{\mathbb{R}}^{2} and \alpha\in\mbox{\mathbb{R}}. We deduce first of all that this case cannot correspond to since we have to have a representation of .
Now we may assume that is the standard representation of \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}) together with a multiple of the identity. In fact we have the “” representation of the form and a putative representation of -dimensional Heisenberg in the form . Now in the case of algebra the kernel of the representation is “”, that is, three-dimensional; however, this circumstance could arise only from having and both non-zero and independent in which case we would obtain a decomposable algebra. Hence is excluded. In the cases and the kernel is one-dimensional and is spanned by . As such we may assume that . Now consider invariant subspaces. Both and give invariant subspaces and so does the trace-free part of . However, the only way to have an invariant subspace of dimension five, since the element must be included, is to take as multiples of the identity, which, however does not produce a subalgebra. The conclusion is that if there is a representation of an algebra that has the -dimensional Heisenberg as its radical, then the element must have rank one.
Now we assume that has rank one and is of the form . Its centralizer may be written as where are , x,y\in\mbox{\mathbb{R}}^{2} and \alpha\in\mbox{\mathbb{R}}. In order to obtain we would have to take the three-dimensional representation of in the upper left block. However, in that case because and span invariant subspaces the only possibility is to obtain the seven-dimensional Heisenberg algebra.
Considering the cases we see that the only possibility for the block is to have either the three-dimensional irreducible representation of or two-dimensional irreducible representation augmented by an extra row and column of zeros of \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}). In the former case we will obtain again the seven-dimensional Heisenberg algebra. In the latter case we have to consider the algebras that we are trying to represent: . For and the kernel of the representation is one-dimensional, which leads to the self-same representation that we found above for but now augmented by an extra middle row and column of zeroes. On the other hand for the representation is given by “” whereas and will each give unless they are correlated, that is, which gives . Thus in the cases and in fact for these three algebras .
5.4.
Now let us look at . First of all note that unlike the algebras just discussed, where the radical is five-dimensional Heisenberg, although the radical is again nilpotent, , the algebra as a whole has trivial center and so the adjoint representation is faithful. Thus . Moreover, it was shown in [GT2] that for alone. Thus for . In fact in Section 5 the reader will see a representation. Thus all that remains is to show that for .
As in the preceding subsection we can argue that must be nilpotent. Again we only have to consider the case where has rank one or two.
5.4.1. of rank
If it has rank two, as above, its centralizer may be written as where are , x,y\in\mbox{\mathbb{R}}^{2} and \alpha\in\mbox{\mathbb{R}}. Although is not a central element, its centralizer is spanned by . Hence the simple factor is of the form where correspond to , respectively, and . Imposing the conditions and implies that must be of the form and likewise from and that .
Next we impose the condition which implies that
[TABLE]
However, now we find that and implies that and are zero. Hence no such representation can exist.
5.4.2. of rank
As above we assume that has rank one and is of the form . Its centralizer is . There are two possibilities: either the upper left is the irreducible representation of \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}}) or it is the representation augmented by an column of rows and zeros, which we take as the third row and third column. Consider the former case first so that
[TABLE]
However, the condition implies that . Hence there is no representation in this case.
Now we assume that
[TABLE]
Then implies that and we find that and are identically satisfied. Furthermore we find that . Now if then and and is identically satisfied. For we obtain . Assume first that then . At this point we obtain a Lie algebra whose radical is and not . If instead we assume then we have . Again we obtain a Lie algebra whose radical is and not .
6. Minimal Representations of Turkowski’s algebras
6.1. Dimension 5
6.1.1. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes\mbox{\mathbb{R}}^{2}:L_{5.1}
[TABLE]
6.2. Dimension 6
6.2.1. \mathfrak{s}\mathfrak{o}(3)\rtimes\mbox{\mathbb{R}}^{3}:L_{6.1}
[TABLE]
6.2.2. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{3.1}:L_{6.2}
[TABLE]
6.2.3. (\lambda\neq\mu)\,\mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{3.3}:L_{6.3}
[TABLE]
6.2.4. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes\mbox{\mathbb{R}}^{3}:L_{6.4}
[TABLE]
6.3. Dimension 7
6.3.1.
[TABLE]
6.3.2. \mathfrak{s}\mathfrak{o}(3)\rtimes\mbox{\mathbb{R}}^{4}:L_{7.2}
[TABLE]
6.3.3. (\lambda\neq\mu,p=\frac{1}{\lambda-\mu})\mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{4.5a=1}=L_{7.3p}
[TABLE]
6.3.4. (\lambda\neq\mu)\mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{4.9b=1}=L_{7.4}
[TABLE]
6.3.5. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{4.5a=b=1}:L_{7.5}
[TABLE]
6.3.6. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes\mbox{\mathbb{R}}^{4}:L_{7.6}
[TABLE]
6.3.7. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes\mbox{\mathbb{R}}^{4}:L_{7.7}
[TABLE]
6.4. Dimension 8
6.4.1.
[TABLE]
6.4.2.
[TABLE]
6.4.3.
[TABLE]
6.4.4.
[TABLE]
6.4.5. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes\mbox{\mathbb{R}}^{5}:L_{8.5}
[TABLE]
6.4.6. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{5.4}:L_{8.6}
[TABLE]
6.4.7. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{5.7pq\neq 0}:L_{8.7}
[TABLE]
6.4.8. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes(A_{5.8p=1},A_{5.9\frac{1}{p}\frac{1}{p}p\neq 0}):L_{8.8}
[TABLE]
6.4.9. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes(A_{5.13a=1q\neq 0}):L_{8.9}
[TABLE]
6.4.10. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes(A_{5.19a=2\,p\neq 0}):L_{8.10}
[TABLE]
6.4.11. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes(A_{5.20a=2\,p\neq 0}):L_{8.11}
[TABLE]
6.4.12. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{5.7b=c}:L_{8.12p=1}
[TABLE]
6.4.13. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{5.7b=c}:L_{8.12p}
[TABLE]
6.4.14. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{5.4}:L_{8.13\epsilon=1}
[TABLE]
6.4.15. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{5.4}:L_{8.13\epsilon=-1}
[TABLE]
To obtain the form used by Turkowski [Turk] use:
[TABLE]
6.4.16. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{5.1}=L_{8.14}
[TABLE]
6.4.17. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{5.3}=L_{8.15}
[TABLE]
6.4.18. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{5.15a=1}:L_{8.16}
[TABLE]
6.4.19. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{5.7b=c}:L_{8.17}
[TABLE]
6.4.20. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes A_{5.17p=q=s=1}:p=\mu-\lambda\,L_{8.18}
[TABLE]
6.4.21. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes\mbox{\mathbb{R}}^{5}:L_{8.19}
[TABLE]
6.4.22. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes\mbox{\mathbb{R}}^{5}:L_{8.20}
[TABLE]
6.4.23. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes\mbox{\mathbb{R}}^{5}:L_{8.21}
[TABLE]
6.4.24. \mathfrak{s}\mathfrak{l}(2,\mbox{\mathbb{R}})\rtimes\mbox{\mathbb{R}}^{5}:L_{8.22}
[TABLE]
7. Acknowledgment
The authors thank the Qatar Foundation and Virginia Commonwealth University in Qatar for funding this project.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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