Classical group

Group theory → Lie groups Lie groups
Classical groups General linear GL(n) Special linear SL(n) Orthogonal O(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n)
Simple Lie groups List of simple Lie groups Classical An Bn Cn Dn Exceptional G2 F4 E6 E7 E8
Other Lie groups Circle Lorentz Poincaré Conformal group Diffeomorphism Loop
Lie algebras Exponential map Adjoint representation (group algebra) Killing form Index Lie point symmetry Simple Lie algebra
Semisimple Lie algebra Dynkin diagrams Cartan subalgebra Root system Weyl group Real form Complexification Split Lie algebra Compact Lie algebra
Homogeneous spaces Closed subgroup Parabolic subgroup Symmetric space Hermitian symmetric space Restricted root system
Representation theory Lie group representation Lie algebra representation
Lie groups in physics Particle physics and representation Lorentz group representation Poincaré group representation Galilean group representation
Scientists Sophus Lie Henri Poincaré Wilhelm Killing Élie Cartan Hermann Weyl Claude Chevalley Harish-Chandra Armand Borel
Table of Lie groups
v t e

In mathematics, the classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces. Their finite analogues are the classical groups of Lie type. The term was coined by Hermann Weyl (as seen in the title of his 1939 monograph The Classical Groups).^[1]

Sometimes classical groups are discussed in the restricted setting of compact groups, a formulation which makes their representation theory and algebraic topology easiest to handle. It does, however, exclude the general linear group.^[2]

Relationship with bilinear forms

A common feature of classical Lie groups is that each are close to isometry groups of certain bilinear or sesquilinear forms. The four series of each classical Lie group type are labelled by the Dynkin diagram attached to them, with subscript n ≥ 1. The families may be represented as follows:^[3]

• A_n = SU(n + 1), the special unitary group of unitary (n + 1)-by-(n + 1) complex matrices with determinant 1.
• B_n = SO(2n + 1), the special orthogonal group of orthogonal (2n + 1)-by-(2n + 1) real matrices with determinant 1.
• C_n = Sp(n), the symplectic group of n-by-n quaternionic matrices that preserve the usual inner product on Hⁿ.
• D_n = SO(2n), the special orthogonal group of orthogonal 2n-by-2n real matrices with determinant 1.

For certain purposes it is also natural to drop the condition that the determinant be 1 and consider unitary groups and (disconnected) orthogonal groups. The table lists the so-called connected compact real forms of the groups; they have closely related complex analogues and various non-compact forms, for example, together with compact orthogonal groups one considers indefinite orthogonal groups. The Lie algebras corresponding to these groups are known as the classical Lie algebras. All classical Lie algebras may be fit into infinite dimensional Lie algebras, such as the Moyal algebra.^[4]

Viewing a classical group G as a subgroup of GL(n) via its definition as automorphisms of a vector space preserving some involution provides a representation of G called the standard representation.

Classical groups over general fields or rings

Classical groups, more broadly considered in algebra, provide particularly interesting matrix groups. When the ring of coefficients of the matrix group is the real number or complex number field, these groups are just certain of the classical Lie groups.

When the underlying ring is a finite field the classical groups are groups of Lie type. These groups play an important role in the classification of finite simple groups. Considering their abstract group theory, many linear groups have a "special" subgroup, usually consisting of the elements of determinant 1 (for orthogonal groups in characteristic 2 it consists of the elements of Dickson invariant 0), and most of them have associated "projective" quotients, which are the quotients by the center of the group.

The word "general" in front of a group name usually means that the group is allowed to multiply some sort of form by a constant, rather than leaving it fixed. The subscript n usually indicates the dimension of the module on which the group is acting. Caveat: this notation clashes somewhat with the n of Dynkin diagrams, which is the rank.

General and special linear groups

The general linear group GL_n(R) is the group of all R-linear automorphisms of Rⁿ. There is a subgroup: the special linear group SL_n(R), and their quotients: the projective general linear group PGL_n(R) = GL_n(R)/Z(GL_n(R)) and the projective special linear group PSL_n(R) = SL_n(R)/Z(SL_n(R)). The projective special linear group PSL_n(R) over a field R is simple for n ≥ 2, except for the two cases when n = 2 and the field has order 2 or 3.

Unitary groups

The unitary group U_n(R) is a group preserving a sesquilinear form on a module. There is a subgroup, the special unitary group SU_n(R) and their quotients the projective unitary group PU_n(R) = U_n(R)/Z(U_n(R)) and the projective special unitary group PSU_n(R) = SU_n(R)/Z(SU_n(R))

Symplectic groups

The symplectic group Sp_2n(R) preserves a skew symmetric form on a module. It has a quotient, the projective symplectic group PSp_2n(R). The general symplectic group GSp_2n(R) consists of the automorphisms of a module multiplying a skew symmetric form by some invertible scalar. The projective symplectic group PSp_2n(R) over a finite field R is simple for n ≥ 1, except for the two cases when n = 1 and the field has order 2 or 3.

Orthogonal groups

The orthogonal group O_n(R) preserves a non-degenerate quadratic form on a module. There is a subgroup, the special orthogonal group SO_n(R) and quotients, the projective orthogonal group PO_n(R), and the projective special orthogonal group PSO_n(R). (In characteristic 2 the determinant is always 1, so the special orthogonal group is often defined as the subgroup of elements of Dickson invariant 1.)

There is a nameless group often denoted by Ω_n(R) consisting of the elements of the orthogonal group of elements of spinor norm 1, with corresponding subgroup and quotient groups SΩ_n(R), PΩ_n(R), PSΩ_n(R). (For positive definite quadratic forms over the reals, the group Ω happens to be the same as the orthogonal group, but in general it is smaller.) There is also a double cover of Ω_n(R), called the pin group Pin_n(R), and it has a subgroup called the spin group Spin_n(R). The general orthogonal group GO_n(R) consists of the automorphisms of a module multiplying a quadratic form by some invertible scalar.

Notational conventions

Contrast with exceptional Lie groups

Contrasting with the classical Lie groups are the exceptional Lie groups, G₂, F₄, E₆, E₇, E₈, which share their abstract properties, but not their familiarity.^[5] These were only discovered around 1890 in the classification of the simple Lie algebras over the complex numbers by Wilhelm Killing and Élie Cartan.

Notes

1. ↑ Weyl, H. (1939). The classical groups: Their Invariants and Representations, ISBN 0-691-05756-7
2. ↑ Historically, in Klein's time, the most obvious example would have been the complex projective linear group, because it was the symmetry group of complex projective space, the dominant geometric concept of the nineteenth century. Vector spaces came later (indeed at the hands of Weyl, as an abstract algebraic notion), referring attention to their symmetry groups, the general linear groups. These groups are algebraic groups. In the development of the Langlands program, the general linear groups became central as the simplest and most universal cases.
3. ↑ R.Slansky, Group theory for unified model building, Physics Reports 79: Issue 1, pp. 1–128
4. ↑ Fairlie, D. B.; Fletcher, P.; Zachos, C. K. (1990). "Infinite-dimensional algebras and a trigonometric basis for the classical Lie algebras". Journal of Mathematical Physics 31 (5): 1088. Bibcode:1990JMP....31.1088F. doi:10.1063/1.528788. 
5. ↑ Wybourne, B. G. (1974). Classical Groups for Physicists, Wiley-Interscience. ISBN 0471965057 .

References

• E. Artin, Geometric algebra, Interscience (1957)
• Dieudonné, Jean (1955), La géométrie des groupes classiques, Ergebnisse der Mathematik und ihrer Grenzgebiete (N.F.), Heft 5, Berlin, New York: Springer-Verlag, ISBN 978-0-387-05391-2, MR 0072144 
• V. L. Popov (2001), "Classical group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4