The uniting feature of classical Lie groups is that they are close to the isometry groups of a certain bilinear or sesquilinear forms. The four series are labelled by the Dynkin diagram attached to it, with subscript n ≥ 1. The families may be represented as follows:
- An = SU(n), the special unitary group of unitary n-by-n 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.
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.
一般線性羣 GLn(R) is the group of all automorphisms of some module. 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(R) over a field R is simple for n≥2, except for the 2 cases when n=2 and the field has order 2 or 3.
[幺正羣]] Un(R) is a group preserving a sesquilinear form on a module. There is a subgroup, the special 幺正羣 SUn(R) and their quotients the projective 幺正羣 PUn(R) = Un(R)/Z(Un(R)) and the projective special 幺正羣 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(R) over a field R is simple for n≥1, except for the 2 cases when n=1 and the field has order 2 or 3.
正交羣 On(R) preserves a non-degenerate quadratic form on a module. There is a subgroup, the special 正交羣 SOn(R) and quotients, the projective 正交羣 POn(R), and the projective special 正交羣 PSOn(R). (In characteristic 2 the determinant is always 1, so the special 正交羣 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 正交羣 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 正交羣, 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.
- E. Artin, Geometric algebra , Interscience (1957)
- J.A. Dieudonné, La géométrie des groups classiques , Springer (1955)
- V. L. Popov (2001), "Classical group", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- Weyl, The classical groups, 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. Subsequently these groups were considered algebraic groups. In the development of the Langlands program, the general linear groups became central as the simplest and most universal cases.