Linear Bundle of Lie Algebras Applied to the Classiﬁcation of Real Lie Algebras

: We present a new look at the classiﬁcation of real low-dimensional Lie algebras based on the notion of a linear bundle of Lie algebras. Belonging to a suitable family of Lie bundles entails the compatibility of the Lie–Poisson structures with the dual spaces of those algebras. This gives compatibility of bi-Hamiltonian structure on the space of upper triangular matrices and with a bundle at the algebra level. We will show that all three-dimensional Lie algebras belong to two of these families and four-dimensional Lie algebras can be divided in three of these families.


Introduction
To begin, we recall the definition of a Lie bundle. Let V, W be finite dimensional vector spaces. If for any X, Y ∈ V and any S ∈ W, we can define Lie bracket on V (X, Y) → [X, Y] S , which is linear in S, then the pair (V, W) is called a linear bundle of Lie algebras or a Lie bundle, see [1][2][3]. A simple consequence of the definition is that the Lie brackets are compatible in the sense that their linear combination is again a Lie bracket [·, ·] αS+βT = α[·, ·] S + β[·, ·] T (1) where S, T ∈ W, α, β ∈ R. A classical example of such a structure is obtained by taking the space of skewsymmetric matrices n × n as V, which we denote by A(n), and the space of symmetric matrices as W, which we denote by S(n). A family of compatible brackets for this case is given by the following formula where X, Y ∈ A(n), S ∈ S(n). As a special case for S = 1, we obtain the standard matrix commutator and Lie algebra so(n). The Lie bundle (A(n), S(n)) appears in many places in the theory of integrable systems. Morosi and Pizzocchero, for example, in [4], used those brackets to construct recursion giving the Mischenko and Manakov integrals of motion for the Euler equation of an n-dimensional rigid body. In the paper [1], Bolsinov and Borisov applied those brackets to investigate the problem of multidimensional Euler and Clebsh equations. Those brackets were also used to analyze Clebsh and Neumann systems in [3]. We recommend the articles [5][6][7], where the bracket (2) or some of its modification were used to consider some integrable systems.
Let us consider a finite-dimensional manifold M. A Poisson bracket is a bilinear and skew-symmetric operation that satisfies the following conditions: for all f , g, h ∈ C ∞ (M). The first of these conditions is usually called the Jacobi identity, and the second is the Leibniz rule. A manifold M with this bracket is called a Poisson manifold (M, {·, ·}). If x = (x 1 , . . . , x n ) is a system of local coordinates on M, then the Poisson bracket can be presented in the form where π ∈ Γ 2 TM is called a Poisson tensor. The components of this tensor are given by the formula π ij (x) = −π ji (x) = {x i , x j }, and it satisfies the following system of equations, equivalent to the Jacobi identity, n ∑ s=1 ∂π ij ∂x s π sk + ∂π ki ∂x s π sj + ∂π jk ∂x s π si = 0, (6) which is equivalent to the vanishing of the Schouten-Nijenhuis bracket.
If π depends on x in a linear way, we say we have a linear Poisson structure on M.
In this paper, an important example of Poisson structure will be the so-called Lie-Poisson bracket. Let g be a real n-dimensional Lie algebra. It is a well-known fact that on the dual space g * , there exists a canonical Poisson structure. This bracket is called Lie-Poisson, and it is given by the following formula where d f (x), dg(x) ∈ (g * ) * ∼ = g. Let {e 1 , . . . , e n } be a basis of g. Using the structure constants c k ij of this Lie algebra [e i , e j ] = ∑ n k=1 c k ij e k , one can express the Poisson tensor as a linear function {x i , x j } = ∑ n k=1 c k ij x k . This gives a one-to-one correspondence between the linear Poisson structures and Lie algebras.
We say that two Poisson tensors π 1 and π 2 are compatible if any linear combination is also a Poisson tensor. The Poisson structures π 1 and π 2 on M are compatible if and only if their Schouten-Nijenhuis bracket vanishes, which means that The manifold M equipped with two compatible Poisson structures π 1 and π 2 is called the bi-Hamiltonian manifold, and we denote it as (M, π 1 , π 2 ).
Let us consider sets of square matrices A a 1 ,...,a n−1 (n) and S a 1 ,...,a n−1 (n) = {S = (s ij ) ∈ M n (R) : s ij = a i . . . a j−1 s ji f or j > i}, where a i , i = 1, . . . , n − 1, are fixed real numbers. Elements of these sets can be represented as follows . . a n−1 x n1 x 21 0 −a 2 x 32 . . . −a 2 a 3 . . . a n−1 x n2 x 31 It was shown in [15,16] that the set A a 1 ,...,a n−1 (n) with the standard commutator is a Lie algebra. This Lie algebra depends on n − 1 parameters and is called a quasisimple orthogonal algebra. In the case, when all these parameters are nonzero, it is isomorphic to the Lie algebra so(n) or so(p, n − p). A more interesting situation is when some of the parameters are equal to zero. For instance, when a 1 = 0 and all others are nonzero, we get a contraction in the sense of Inönü-Wigner to the Euclidean algebra iso(n) or the Poincaré algebra iso(n − 1, 1). The infinite dimensional case was considered in papers [6,28].
In our article, we will only be interested in the finite-dimensional case, with the family of brackets given by where X, Y ∈ A a 1 ,...,a n−1 (n), S ∈ S a 1 ,...,a n−1 (n). We denote this Lie bundle by A a 1 ,...,a n−1 (n), S a 1 ,...,a n−1 (n) . This case was studied in detail in [29]. The dual space A a 1 ,...,a n−1 (n) * to the algebra A a 1 ,...,a n−1 (n) can be identified with the Thus, the Lie bundle A a 1 ,...,a n−1 (n), S a 1 ,...,a n−1 (n) introduces a family of Lie-Poisson brackets on the dual space L + (n): This family of brackets is indexed by S ∈ S a 1 ,...,a n−1 (n). For fixed a 1 , . . . , a n−1 the Lie bracket from Equation (12) is linear in S, and consequently, the Lie-Poisson brackets are compatible.
There are known Casimir functions for the Lie bracket (14) in the case when a 1 = . . . = a n−1 = 1 and det S = 0 given by see [2]. In the case when parameters a 1 , . . . , a n−1 are non-zero but not equal to one, this family has to be modified where δ = diag(1, a 1 , a 1 a 2 , . . . , a 1 a 2 . . . a n−1 ), see [29]. In the situation when it is allowed that some of the parameters a i are equal to zero, we get where η = diag(a 1 a 2 . . . a n−1 , a 2 . . . a n−1 , . . . , 1). However, in this case, not all of the functions have to be included. The Lie bundle A a 1 ,...,a n−1 (n), S a 1 ,...,a n−1 (n) can be generalized to the case when some parameters a i are equal to zero. Let us suppose that a k 1 = . . . = a k N = 0, where k 1 , . . . , k N is an increasing sequence, then both of the elements of A a 1 ,...,a n−1 (n) and S a 1 ,...,a n−1 (n) can be written in block form as Then nothing prevents us from replacing the almost symmetric matrix (we called it quasisimple symmetric matrix) We will denote by S a 1 ,...,a n (n) the set of matrices S of the form (18), where blocks B i in case 2 × 2 can be replaced by B i .
In this case, we will allow that matrix elements are complex numbers. It turns out that for low dimensional Lie algebras, those two families of Lie bundles, which are dependent on parameters a i , are sufficient for the classification of Lie algebras. In the next sections, it will be shown on three-and four-dimensional Lie algebras that all of these algebras belong to the Lie bundle (A a 1 ,...,a n (n), S a 1 ,...,a n (n)) or A a 1 ,...,a n (n), S a 1 ,...,a n (n) .

Three-Dimensional Lie Algebras and Lie Bundles
First Lie bundle. Let us consider the Lie algebra so(3) with the standard commutator. Note that A 1,1 (3) as a set coincides with it but carries another Lie bracket where X 1 , Y 1 ∈ A 1,1 (3) and S 1 ∈ S 1,1 (3). We assume the forms of X 1 and S 1 as follows From the definition in Equation (14), we get the Lie-Poisson bracket on the dual space L + (3) to the Lie algebra A 1,1 (3) In our calculation, we took x ∈ L + (3) of the form Thus, the matrix of the Poisson tensor assumes the following form

Second Lie bundle.
We can also get another Lie-Poisson bracket by taking X 2 , Y 2 ∈ A 1,0 (3) and S 2 ∈ S 1,0 (3) as follows Using a similar calculation as in Equation (21), we get the next Lie-Poisson bracket Thus, the matrix of the Poisson tensor in this case is given by Table 1 describes which Lie-Poisson structures that are related to the three-dimensional Lie algebras g ij , according to the classification taken from [30], can be obtained from those two brackets. An isomorphism is given by mapping (x 1 , x 2 , x 3 ) → (e 1 , e 2 , e 3 ). It is easy to see that some of them can be obtained from both Equations (19) and (23). Furthermore, these brackets split all of these three-dimensional Lie algebras into two families-algebras in the same family are all compatible with each other but not with those from the other one. Table 1. Three-dimensional Lie algebras with their Poisson structure (22) or (26) and corresponding matrices S 1 , S 2 from Equations (19) and (23).
Obviously, we would get the same result by analyzing the compatibility of Lie-Poisson structures related, respectively, to those three-dimensional Lie algebras, see Table 2.
Belonging to the appropriate family of a Lie bundle also requires that Casimirs for the Poisson bracket (respectively, invariants for Lie algebras) are generally described by one function. For the first family of Lie algebras, after direct calculation from Equation (15), we get = s 11 x 2 1 + s 22 x 2 2 + s 33 x 2 3 + 2s 21 x 1 x 2 + 2s 31 x 1 x 3 + 2s 32 x 2 x 3 . Table 3 displays the results in some specific situations. For the second family, the Casimir functions can be obtained by solving the linear first order partial differential equation

Four-Dimensional Lie Algebras and Lie Bundles
Let us consider all real Lie algebras of dimension equal to four. A complete list of these algebras is given, for example, by Mubarakzyanov [22]. There are twelve real algebras of dimension four, and five of them depend on parameters. Our list of algebras gathered in Table 4 is based on the article [30].
First Lie bundle.
The first family can be obtain by taking the bracket where X 1 , Y 1 ∈ A 1,0,1 (4) and S 1 ∈ S 1,0,1 (4) are matrices of the form After direct calculation from Equation (14), we get the following formula where x ∈ L + (4) is an upper triangular matrix given by As we can see, we obtained six-dimensional cases. To get our four-dimensional families, we limit ourselves to variables (x 2 , x 3 , y 1 , y 3 ) by putting x 1 = y 2 = 0 and s 11 = s 31 = s 41 = 0. This means restricting to the subspaces V ⊂ A 1,0,1 (4) and S ⊂ S 1,0,1 (4), whose elements are given by We get the Lie-Poisson structure related to the Lie algebra g 4,1 by putting s 44 = s 32 = −1 and the other parameters equal to zero in matrix S 1 . To get the Lie-Poisson structure associated with the Lie algebra g 4,3 in matrix S 1 , we put s 32 = s 43 = −1 and all others parameters equal to zero. To get the Lie-Poisson structure associated with Lie algebra g 4,12 , we put s 33 = s 44 = −1, s 12 = 1 and other parameters equal to zero in matrix S 1 . In these cases, an isomorphism is given by the mapping (x 2 , y 1 , x 3 , y 3 ) → (e 1 , e 2 , e 3 , e 4 ).
In Table 6, there are Lie-Poisson structures related to four-dimensional Lie algebras g 4,i from the first family that can be obtained from bracket (31) by choosing a suitable matrix S 1 and putting x 1 = y 2 = 0.

Conclusions
We proposed a new approach to the classification of low-dimensional Lie algebras using concepts well known in the theory of integrable systems, such as the bi-Hamiltonian structure. In the case of three-Dimensional Lie algebras, we were able to describe them using two families of linear bundles of Lie algebras. In the case of the four-dimensional Lie algebras, we described them in terms of three such families. In the future, we plan to use these tools to study higher dimensional Lie algebras.
The bi-Hamilton property is very useful in the study of integrable systems. Let us consider the well-known Euler equations for n = 3: The Euler top describes the rotation of a heavy rigid body around its fixed center of mass without any forces acting on the body. Equation (41) possess a bi-Hamiltonian structure The Poisson brackets {·, ·} 1 and {·, ·} 2 are defined by Equation (21), where we restrict our considerations to diagonal matrices S 1 = 1 and S 1 = diag(s 11 , s 22 , s 33 ). The Casimirs for these structures assume the following form (see Equation (27)) respectively. Choosing as the Hamiltonian H 1 the Casimir C 2 and as Hamiltonian H 2 the Casimir C 1 , we obtain Equation (42). Of course, the first Poisson bracket is associated with the Lie algebra so(3). We can look at the second Poisson bracket as a collection of three brackets corresponding to the Lie algebra g 3,1 for matrices: respectively (see Table 1). Now, we discuss another example of a Lie bundle. We now turn to the Clebsch system of the motion of a rigid body in an ideal fluid dy 3 dt = (s 33 − s 44 )x 2 y 1 .
Choosing as the first Hamiltonian the function H 1 and as Hamiltonian H 2 the Casimir C 1 , we obtain Equation (48). The first Poisson bracket is associated with the Lie algebra e(3).
We can look at the second Poisson bracket as a collection of four brackets corresponding to the nilpotent Lie algebra g 6,3 (we use the classification from [30]