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Simple Lie groups Unfortunately, there is no generally accepted definition of a simple Lie group. In particular, it is not necessarily defined as a Lie group that is simple as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether R is a simple Lie group. The most common definition implies that simple Lie groups must be connected, and non-abelian, but are allowed to have a non-trivial center. In this article the connected simple Lie groups with trivial center are listed. Once these are known, the ones with non-trivial center are easy to list as follows. Any simple Lie group with trivial center has a universal cover, whose center is the fundamental group of the simple Lie group. The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the center. Simple Lie algebras The Lie algebra of a simple Lie group is a simple Lie algebra. This is a one-to-one correspondence between connected simple Lie groups with trivial center and simple Lie algebras of dimension greater than 1. (Authors differ on whether the one-dimensional Lie algebra should be counted as simple.) Over the complex numbers the simple Lie algebras are given by the usual "ABCDEFG" classification. If L is a real simple Lie algebra, its complexification is a simple complex Lie algebra, unless L is already the complexification of a Lie algebra, in which case the complexification of L is a product of two copies of L. This reduces the problem of classifying the real simple Lie algebras to that of finding all the real forms of each complex simple Lie algebra (i.e., real Lie algebras whose complexification is the given complex Lie algebra). There are always at least 2 such forms: a split form and a compact form, and there are usually a few others. The different real forms correspond to the classes of automorphisms of order at most 2 of the complex Lie algebra. Symmetric spaces Symmetric spaces are classified as follows. First, the universal cover of a symmetric space is still symmetric, so we can reduce to the case of simply connected symmetric spaces. (For example, the universal cover of a real projective plane is a sphere.) Second, the product of symmetric spaces is symmetric, so we may as well just classify the irreducible simply connected ones (where irreducible means they cannot be written as a product of smaller symmetric spaces). The irreducible simply connected symmetric spaces are the real line, and exactly two symmetric spaces corresponding to each non-compact simple Lie group G, one compact and one non-compact. The non-compact one is a cover of the quotient of G by a maximal compact subgroup H, and the compact one is a cover of the quotient of the compact form of G by the same subgroup H. This duality between compact and non-compact symmetric spaces is a generalization of the well known duality between spherical and hyperbolic geometry. Hermitian symmetric spaces A symmetric space with a compatible complex structure is called Hermitian. The compact simply connected irreducible Hermitian symmetric spaces fall into 4 infinite families with 2 exceptional ones left over, and each has a non-compact dual. In addition the complex plane is also a Hermitian symmetric space; this gives the complete list of irreducible Hermitian symmetric spaces. The four families are the types A III, B I and D I for p=2, D III, and C I, and the two exceptional ones are types E III and E VII of complex dimensions 16 and 27. Notation R, C, H, and O stand for the real numbers, complex numbers, quaternions, and octonions. In the symbols such as E6−26 for the exceptional groups, the exponent −26 is the signature of an invariant symmetric bilinear form that is negative definite on the maximal compact subgroup. It is equal to the dimension of the group minus twice the dimension of a maximal compact subgroup. The fundamental group listed in the table below is the fundamental group of the simple group with trivial center. Other simple groups with the same Lie algebra correspond to subgroups of this fundamental group (modulo the action of the outer automorphism group). |
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