Lie Groups

Lie Groups

Lie groups, named after the Norwegian mathematician Sophus Lie (1842-1899), stand at the intersection of several different branches of mathematics: algebra, analysis, geometry and topology. They are defined as groups that have the additional geometric structure of a smooth manifold. To tie the algebraic structure together with the geometric structure, the operations of multiplication and inversion are required to be differentiable.

At first glance, Lie groups might seem like the mathematical analogue of a centaur--an unnatural combination of two different creatures. But unlike centaurs, Lie groups really do exist. Some examples include:

• The Euclidean group, consisting of all rigid motions of the plane (including rotations, translations and reflections).
• The Lorentz group, consisting of all Lorentz transformations of special relativity that map one inertial reference frame to another.
• The general linear group GL(n), consisting of all n-by-n invertible matrices. The group operation is matrix multiplication, and the formulas for matrix multiplication and inversion from linear algebra prove that these operations are differentiable.
• The special orthogonal group SO(n), which consists of all rotations of n-dimensional space. These can be represented as n-by-n matrices.
• The unitary and special unitary groups U(n) and SU(n), which are analogous to SO(n) but contain n-by-n matrices of complex numbers instead.
• The syplectic groups Sp(n), whose elements can be thought of either as 4n-by-4n real matrices, or 2n-by-2n complex matrices, or n-by-n quaternionic matrices.

The crucial role of Lie groups stems from their interpretation as symmetries. Felix Klein, in 1872, argued that classical Euclidean geometry is simply the study of the Euclidean group. Similarly, the predictions of special relativity, such as time dilation and the impossibility of faster-than-light travel, all derive from the Lorentz group. In each case, the Lie group consists of all the transformations that preserve the essential features of the geometry being studied.

Lie groups are at the opposite extreme of abstract algebra from finite groups. Because Lie groups always contain infinitely many elements, the parts of group theory that deal with counting--for example, the order of an element, or the order and index of a subgroup--are nearly useless. But the geometric structure provides useful alternatives. The "size" of a group is now measured by its dimension as a manifold, in other words the number of degrees of freedom involved in specifying one element of the group. For example, the Lorentz group is 10-dimensional, because any inertial reference frame is defined by a location in spacetime (4 coordinates), a choice of coordinate axes (3 coordinates), and a velocity (3 coordinates). For compact Lie groups, a more significant measure of size is the rank of the group. For the classical matrix groups, which all represent rigid motions of Euclidean space, the rank can be interpreted as the largest number of independent motions that avoid interfering with one another. That is, they can be done in either order and produce the same result; in the language of abstract algebra, they commute. The rank of SO(3) is 1 because in 3-space (as anybody who has played with a Rubik's cube can tell) only rotations about a single axis will commute. In SO(4), on the other hand, the rank is 2: It is possible for 2 rotations in 4-space to commute, provided that they are rotations in completely orthogonal planes.

Finite-dimensional Lie groups can always be viewed as matrix groups. By the definition of a manifold, a Lie group G has a tangent plane V at the identity element 1. The elements of this tangent plane, called the Lie algebra of the group, can be thought of as "infinitesimal" rotations. The Lie group acts on itself by an operation called the adjoint map, ad_x(y) = xyx^-1, and hence it also acts on the infinitesimal rotations in the same way. But V is a vector space, and linear maps on finite-dimensional vector spaces can always be represented by matrices. Thus the group element x corresponds to the matrix that represents ad_x.

In fact, a Lie group typically has many different "representations" as matrix groups, which can be constructed from a small set of building blocks, called irreducible representations. The adjoint representation of a group, described above, is usually far from the simplest. Henri Cartan, in the 1910s, and Hermann Weyl, in the 1920s and 1930s, developed a beautiful theory to explain how to decompose any representation of a Lie group into irreducible ones.

As in finite group theory, a fundamental problem in Lie group theory is to classify the simple compact Lie groups--in other words, the ones that cannot be decomposed into smaller groups. The answer is again more elegant than one would have a right to hope for: There are only four families of simple compact Lie groups--namely the families SU(n), SO(n) for odd n, Sp(n), and SO(n) for even n. In addition, there is a rogue's gallery of exactly five exceptional Lie groups, called G₂, F₄, E₆, E₇, and E₈. (Here the subscripts refer to the rank of the groups.) The exceptional groups have dimensions 14, 52, 78, 133, and 248 respectively.

A surprising number of recent developments in mathematics and physics are tied to Lie groups. The classification of finite simple groups, completed in 1980, was certainly inspired by the much simpler classification of simple compact Lie groups. Moreover, many of the groups that appear in this classification are "finite groups of Lie type"--in other words, finite groups that are modeled after the classic matrix groups. The "fake R⁴"s discovered by Simon Donaldson in 1983 (in other words, four dimensional universes that are topologically the same as Euclidean four-space but cannot be mapped to it smoothly, like a shirt that is too wrinkled to be ironed flat again) were based on properties of the exceptional group E₈.

Finally, quantum physicists have used representations of Lie groups repeatedly to predict new subatomic particles. Perhaps the most spectacular example was Murray Gell-Mann's theory of the "eightfold way," which predicted the existence of quarks. He named the theory both for the eight paths to enlightenment in Buddhist philosophy, and for the eight dimensions of the group SU(3), which he used to describe the symmetries of the weak nuclear force. Each irreducible representation of this group corresponds to a different combination of quarks, or equivalently to a different subatomic particle. Gell-Mann received the Nobel Prize in Physics for his work in 1969.

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