Classical group

Classical group
For the book by Weyl, see The Classical Groups.
Lie groups
E8PetrieFull.svg
v · 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:

  • 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.

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 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(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 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))

Symplectic groups

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 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 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.

Notational conventions

For more details on this topic, see Group of Lie type#Notation issues.

Notes

  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. 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

  • R.Slansky, Group theory for unified model building, Physics Reports, Volume 79, Issue 1, p. 1–128

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Group of Lie type — In mathematics, a group of Lie type G(k) is a (not necessarily finite) group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups.… …   Wikipedia

  • Classical elements in popular culture — Classical Elements v · d · e …   Wikipedia

  • Classical Latin — Latinitas Latin inscription in the Colosseum …   Wikipedia

  • Classical Arabic — Spoken in Historically in the Middle East, now used as a liturgical language of Islam Language family Afro Asiatic Semitic Central Semitic …   Wikipedia

  • Classical Hindu law in practice — originates from community, not a state polity. In this way, particular groups of society began to gain influence in the creation and administration of law. Primary corporate groups, Kingships, and Brahmins were the factions, which conveyed Hindu… …   Wikipedia

  • Classical theory of growth and stagnation — Classical economics refers to work done by a group of economists in the eighteenth and nineteenth centuries. The theories developed mainly focused on the way market economies functioned. Classical Economics study mainly concentrates on the… …   Wikipedia

  • Classical swine fever — Virus classification Group: Group IV ((+)ssRNA) Family: Flaviviridae Genus …   Wikipedia

  • Classical Archives — Homepage, June 21, 2010 Opened August, 2000 (as Classical Archives, LLC) Pricing model Downloads for purchase, unlimited streaming by subscription …   Wikipedia

  • Classical Latin Literature in the Church — • This article deals only with the relations of the classical literature, chiefly Latin, to the Catholic Church Catholic Encyclopedia. Kevin Knight. 2006. Classical Latin Literature in the Church     Classic …   Catholic encyclopedia

  • Classical 24 — is a syndicated, satellite delivered public radio service providing classical music to its carrying stations. It generally airs overnights on many non commercial and a handful of commercial classical music stations. However, the service is… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”