Figure 1: Sophus Lie, Norway, 1842–1899

Figure 1: Sophus Lie, Norway, 1842–1899.

Some general definitions In mathematics, a Lie algebra is an algebraic structure used to study geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations associated to Lie groups. The term “Lie algebra” (after the Norwegian mathematician Sophus Lie, 1842–1899) was introduced by Hermann Weyl in the 1930s. In older texts, the name infinitesimal group is used.

Topological spaces Topological spaces are mathematical structures that allow the formal deﬁnition of concepts such as convergence, connectedness, and continuity; they appear in virtually every branch of modern mathematics and are a central unifying notion. The branch of mathematics that studies topological spaces is called topology. A topological space is a set X together with T , a collection of subsets of X, satisfying the following axioms:

• The empty set {0} and the whole set X are in T . • The union of any collection of sets in T is also in T . • The intersection of any ﬁnite collection of sets in T is also in T .

The collection T is called a topology on X. The elements of X are usually called points, though they can be any mathematical objects. A topological space in which the points are functions is called a function space.

Lie groups A Lie group is a group G that is also a topological space or more exactly, a smooth manifold, on which the group operations are analytic (holomorphic). The group operation, denoted by (g,h) 7→ gh where g,h ∈ G, is a smooth map, and also g 7→ g−1 is a smooth map (i.e., C∞-mappings). A Lie group is abelian iff gh = hg In other words, a Lie group is a group that is also a differentiable (i.e., smooth) manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are smooth manifolds and, therefore, can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups, is to replace the global object, the group, with its local or linearized version, which Lie himself called its infinitesimal group.

Lie algebras The tangent space to the manifold, and in particular the tangent space at the identity of the group, is called the Lie algebra associated to the Lie group. In the case of linear groups, it can be computed explicitly by diﬀerentiating curves through the identity. A real Lie algebra g is a vector space L over R, with an operation [·, ·] : L×L→L, called Lie bracket, satisfying:

1. Bilinearity over R: [ αiXi, βiYi]= αiβj [Xi, Yj ], for αi,βj ∈ R and Xi, Yj ∈ L, P P P 2. Skew-symmetry: [X, Y ]= −[Y,X], for X, Y ∈ L, 3. The Jacobi identity: [X, [Y,Z]] + [Y, [Z,X]] + [Z, [X, Y ]] = 0, for X,Y,Z ∈ L.

The simplest examples of Lie algebras are matrix Lie algebras, i.e., those deriving from matrix Lie groups, where the group is the set of n × n matrices, closed under products, inverses and nonsingular limits. The plane rotation matrices form a subgroup, denoted by SO2(R) or SO(2), that is a Lie group in its own right: indeed it is a one-dimensional compact connected Lie group which is diﬀeomorphic to the circle. Using the rotation angle θ as a parameter, this group can be parameterized as follows

cos θ − sin θ SO(2) = ; θ = R/2πZ sin θ cos θ

Another example of Lie group is the special orthogonal group SO(3), whose elements R(u,θ) ∈ SO(3) are rigid rotations and can be thought as curves through the identity I ∈ SO(3). Any Lie group G deﬁnes an associated real Lie algebra g

G → g = Lie(G)

The deﬁnition in general is somewhat technical, but in the case of real matrix groups, it can be formulated via the exponential map, or the matrix exponent. The Lie algebra consists of those matrices M for which exp(M t) ∈ G; t ∈ R The Lie bracket of g is given by the commutator of matrices

[M a, M b]= M˙ bM a − M˙ aM b

As a concrete example, consider the special linear group SL(3, R), consisting of all 3 × 3 matrices with real entries and determinant 1 (rotation matrices). This is a matrix Lie group, and its Lie algebra consists of all 3 × 3 matrices with real entries and trace 0. The distributivity, linearity and Jacobi identity show that R3 together with vector addition and cross product forms a Lie algebra. In fact, the Lie algebra is that of the real orthogonal group in 3 dimensions, SO(3).

2 References [01] D.H. Sattinger, O.L. Weaver, Lie Groups and Algebras with Applications to Physics, Ge- ometry and Mechanics, Springer, 1986. [02] J. Gallier, Geometric Methods and Applications, Springer, 2001. [03] J. Stilwell, Naive Lie Theory, Springer, 2008. [04] J.C. Monforte, Geometric, Control and Numerical Aspects of Nonholonomic Systems, Springer, 2002.