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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).
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.
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:
- An = SU(n + 1), the special unitary group of unitary (n + 1)-by-(n + 1) complex matrices with determinant 1.
- Bn = SO(2n + 1), the special orthogonal group of orthogonal (2n + 1)-by-(2n + 1) real matrices with determinant 1.
- Cn = Sp(n), the symplectic group of n-by-n quaternionic matrices that preserve the usual inner product on Hn.
- Dn = 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.
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 algebras
Classical groups, more broadly considered in algebra, provide particularly interesting matrix groups. When the field F of coefficients of the matrix group is either real number or complex numbers, these groups are just certain of the classical Lie groups. When the ground field is a finite field, then the classical groups are groups of Lie type. These groups play an important role in the classification of finite simple groups. Also, one may consider classical groups over a unital associative algebra R over F; where R = H (an algebra over reals) represents an important case. For the sake of generality the article will refer to groups over R, where R may be the ground field F itself.
Considering their abstract group theory, many linear groups have a "special" subgroup, usually consisting of the elements of determinant 1 over the ground field, and most of them have associated "projective" quotients, which are the quotients by the center of the group. For orthogonal groups in characteristic 2 "S" has a different meaning.
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; it is a vector space if R = F. Caveat: this notation clashes somewhat with the n of Dynkin diagrams, which is the rank.
General and special linear groups
The general linear group GLn(R) is the group of all R-linear automorphisms of Rn. There is a subgroup: the special linear group SLn(R), and their quotients: the projective general linear group PGLn(R) = GLn(R)/Z(GLn(R)) and the projective special linear group PSLn(R) = SLn(R)/Z(SLn(R)). The projective special linear group PSLn(F) over a field F is simple for n ≥ 2, except for the two cases when n = 2 and the field has order[clarification needed] 2 or 3.
The unitary group Un(R) is a group preserving a sesquilinear form on a module. There is a subgroup, the special unitary group SUn(R) and their quotients the projective unitary group PUn(R) = Un(R)/Z(Un(R)) and the projective special unitary group PSUn(R) = SUn(R)/Z(SUn(R))
The symplectic group Sp2n(R) preserves a skew symmetric form on a module. It has a quotient, the projective symplectic group PSp2n(R). The general symplectic group GSp2n(R) consists of the automorphisms of a module multiplying a skew symmetric form by some invertible scalar. The projective symplectic group PSp2n(Fq) over a finite field is simple for n ≥ 1, except for the two cases when n = 1 and the field has order[clarification needed] 2 or 3.
The orthogonal group On(R) preserves a non-degenerate quadratic form on a module. There is a subgroup, the special orthogonal group SOn(R) and quotients, the projective orthogonal group POn(R), and the projective special orthogonal group PSOn(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 Ωn(R) consisting of the elements of the orthogonal group of elements of spinor norm 1, with corresponding subgroup and quotient groups SΩn(R), PΩn(R), PSΩn(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 Ωn(R), called the pin group Pinn(R), and it has a subgroup called the spin group Spinn(R). The general orthogonal group GOn(R) consists of the automorphisms of a module multiplying a quadratic form by some invertible scalar.
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. These were only discovered around 1890 in the classification of the simple Lie algebras over the complex numbers by Wilhelm Killing and Élie Cartan.
- Weyl, H. (1939). The classical groups: Their Invariants and Representations, ISBN 0-691-05756-7
- 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.
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