經典羣

經典李羣(classical Lie groups) 係四無限咁長嘅系列嘅李羣，係一般線性羣嘅子羣（即係線性空間嘅對稱羣）。「經典」呢個名有少少靈活性；最初用呢個名嘅應該係 Hermann Weyl係佢喺1940年本叫《Classical Groups》嘅書；呢個名可能係反影佢同「經典幾何」嘅關係，類似克來恩(en:Felix Klein嘅Erlangen program。

相對於經典羣嘅係特殊李羣(en:exceptional Lie groups)；佢哋喺抽象嘅觀點上相似，但直覺上就唔係咁易明。

有時人會集中討論緊緻羣，因為佢哋嘅表示論同代數拓樸特別簡單。但咁就包唔到一般線性羣－一般認為最「經典」嘅羣。^[1]

同雙線性式(bilinear forms)嘅關係

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:

A_n = SU(n), the special unitary group of unitary n-by-n 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.

任何環或域上嘅經典羣

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.

一般同埋特殊線性羣

一般線性羣 GL_n(R) is the group of all automorphisms of some module. 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 2 cases when n=2 and the field has order 2 or 3.

幺正羣

[幺正羣]] U_n(R) is a group preserving a sesquilinear form on a module. There is a subgroup, the special 幺正羣 SU_n(R) and their quotients the projective 幺正羣 PU_n(R) = U_n(R)/Z(U_n(R)) and the projective special 幺正羣 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 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

正交羣 O_n(R) preserves a non-degenerate quadratic form on a module. There is a subgroup, the special 正交羣 SO_n(R) and quotients, the projective 正交羣 PO_n(R), and the projective special 正交羣 PSO_n(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 Ω_n(R) consisting of the elements of the 正交羣 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 正交羣, 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

參攷

E. Artin, Geometric algebra , Interscience (1957)
J.A. Dieudonné, La géométrie des groups classiques , Springer (1955)
V. L. Popov (2001), "Classical group", 出自 Hazewinkel, Michiel (編), Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Weyl, The classical groups, ISBN 0-691-05756-7

註

↑ 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.

[1] 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.

[1]