 Classical group

For the book by Weyl, see The Classical Groups.
Lie groups Classical groupsGeneral linear group GL(n)
Special linear group SL(n)
Orthogonal group O(n)
Special orthogonal group SO(n)
Unitary group U(n)
Special unitary group SU(n)
Symplectic group Sp(n)Exponential map
Adjoint representation of a Lie group
Adjoint representation of a Lie algebra
Killing form
Lie point symmetryStructure of semisimple Lie groupsDynkin diagrams
Cartan subalgebra
Root system
Real form
Complexification
Split Lie algebra
Compact Lie algebraRepresentation of a Lie group
Representation of a Lie algebrav · 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). Contrasting with the classical Lie groups are the exceptional Lie groups, which share their abstract properties, but not their familiarity.
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.^{[1]}
Contents
Relationship with bilinear forms
The unifying feature of classical Lie groups is that they are close to the isometry groups of certain bilinear or sesquilinear forms. The four series are labelled by the Dynkin diagram attached to them, with subscript n ≥ 1. The families may be represented as follows:
 A_{n} = SU(n + 1), the special unitary group of unitary n+1byn+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 nbyn quaternionic matrices that preserve the usual inner product on H^{n}.
 D_{n} = SO(2n), the special orthogonal group of orthogonal 2nby2n 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 socalled connected compact real forms of the groups; they have closely related complex analogues and various noncompact 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.
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 Rlinear automorphisms of R^{n}. 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 nondegenerate 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
For more details on this topic, see Group of Lie type#Notation issues.Notes
 ^ 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.
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: SpringerVerlag, ISBN 9780387053912, MR0072144, http://books.google.com/books?id=AfYZAQAAIAAJ
 V. L. Popov (2001), "Classical group", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, http://eom.springer.de/C/c022410.htm
 Weyl, The classical groups, ISBN 0691057567
 R.Slansky, Group theory for unified model building, Physics Reports, Volume 79, Issue 1, p. 1–128
Categories: Lie groups
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Classical group
 Classical group

For the book by Weyl, see The Classical Groups.
Lie groups Classical groupsGeneral linear group GL(n)
Special linear group SL(n)
Orthogonal group O(n)
Special orthogonal group SO(n)
Unitary group U(n)
Special unitary group SU(n)
Symplectic group Sp(n)Exponential map
Adjoint representation of a Lie group
Adjoint representation of a Lie algebra
Killing form
Lie point symmetryStructure of semisimple Lie groupsDynkin diagrams
Cartan subalgebra
Root system
Real form
Complexification
Split Lie algebra
Compact Lie algebraRepresentation of a Lie group
Representation of a Lie algebra