Classical group

In mathematics, the classical groups are defined as the special linear groups over the reals, the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.
The classical groups form the deepest and most useful part of the subject of linear Lie groups. Most types of classical groups find application in classical and modern physics. A few examples are the following. The rotation group is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group is a symmetry group of spacetime of special relativity. The special unitary group is the symmetry group of quantum chromodynamics and the symplectic group finds application in Hamiltonian mechanics and quantum mechanical versions of it.

The classical groups

The classical groups are exactly the general linear groups over, and together with the automorphism groups of non-degenerate forms discussed below. These groups are usually additionally restricted to the subgroups whose elements have determinant 1, so that their centers are discrete. The classical groups, with the determinant 1 condition, are listed in the table below. In the sequel, the determinant 1 condition is not used consistently in the interest of greater generality.
The complex classical groups are, and. A group is complex according to whether its Lie algebra is complex. The real classical groups refers to all of the classical groups since any Lie algebra is a real algebra. The compact classical groups are the compact real forms of the complex classical groups. These are, in turn,, and. One characterization of the compact real form is in terms of the Lie algebra. If, the complexification of, and if the connected group generated by is compact, then is a compact real form.
The classical groups can uniformly be characterized in a different way using real forms. The classical groups are the following:
For instance, is a real form of, is a real form of, and is a real form of. Without the determinant 1 condition, replace the special linear groups with the corresponding general linear groups in the characterization. The algebraic groups in question are Lie groups, but the "algebraic" qualifier is needed to get the right notion of "real form".

Bilinear and sesquilinear forms

The classical groups are defined in terms of forms defined on,, and, where and are the fields of the real and complex numbers. The quaternions,, do not constitute a field because multiplication does not commute; they form a division ring or a skew field or non-commutative field. However, it is still possible to define matrix quaternionic groups. For this reason, a vector space is allowed to be defined over,, as well as below. In the case of, is a right vector space to make possible the representation of the group action as matrix multiplication from the left, just as for and.
A form on some finite-dimensional right vector space over, or is bilinear if
It is called sesquilinear if
These conventions are chosen because they work in all cases considered. An automorphism of is a map in the set of linear operators on such that
The set of all automorphisms of form a group, it is called the automorphism group of, denoted. This leads to a preliminary definition of a classical group:
This definition has some redundancy. In the case of, bilinear is equivalent to sesquilinear. In the case of, there are no non-zero bilinear forms.

Symmetric, skew-symmetric, Hermitian, and skew-Hermitian forms

A form is symmetric if
It is skew-symmetric if
It is Hermitian if
Finally, it is skew-Hermitian if
A bilinear form is uniquely a sum of a symmetric form and a skew-symmetric form. A transformation preserving preserves both parts separately. The groups preserving symmetric and skew-symmetric forms can thus be studied separately. The same applies, mutatis mutandis, to Hermitian and skew-Hermitian forms. For this reason, for the purposes of classification, only purely symmetric, skew-symmetric, Hermitian, or skew-Hermitian forms are considered. The normal forms of the forms correspond to specific suitable choices of bases. These are bases giving the following normal forms in coordinates:
The in the skew-Hermitian form is the third basis element in the basis for. Proof of existence of these bases and Sylvester's law of inertia, the independence of the number of plus- and minus-signs, and, in the symmetric and Hermitian forms, as well as the presence or absence of the fields in each expression, can be found in or. The pair, and sometimes, is called the signature of the form.
Explanation of occurrence of the fields : There are no nontrivial bilinear forms over. In the symmetric bilinear case, only forms over have a signature. In other words, a complex bilinear form with "signature" can, by a change of basis, be reduced to a form where all signs are "" in the above expression, whereas this is impossible in the real case, in which is independent of the basis when put into this form. However, Hermitian forms have basis-independent signature in both the complex and the quaternionic case. A skew-Hermitian form on a complex vector space is rendered Hermitian by multiplication by, so in this case, only is interesting.

Automorphism groups

The first section presents the general framework. The other sections exhaust the qualitatively different cases that arise as automorphism groups of bilinear and sesquilinear forms on finite-dimensional vector spaces over, and.

Aut(''φ'') – the automorphism group

Assume that is a non-degenerate form on a finite-dimensional vector space over or. The automorphism group is defined, based on condition, as
Every has an adjoint with respect to defined by
Using this definition in condition, the automorphism group is seen to be given by
Fix a basis for. In terms of this basis, put
where are the components of. This is appropriate for the bilinear forms. Sesquilinear forms have similar expressions and are treated separately later. In matrix notation one finds
and
from where is the matrix. The non-degeneracy condition means precisely that is invertible, so the adjoint always exists. expressed with this becomes
The Lie algebra of the automorphism groups can be written down immediately. Abstractly, if and only if
for all, corresponding to the condition in under the exponential mapping of Lie algebras, so that
or in a basis
as is seen using the power series expansion of the exponential mapping and the linearity of the involved operations. Conversely, suppose that. Then, using the above result,. Thus the Lie algebra can be characterized without reference to a basis, or the adjoint, as
The normal form for will be given for each classical group below. From that normal form, the matrix can be read off directly. Consequently, expressions for the adjoint and the Lie algebras can be obtained using formulas and. This is demonstrated below in most of the non-trivial cases.

Bilinear case

When the form is symmetric, is called. When it is skew-symmetric then is called. This applies to the real and the complex cases. The quaternionic case is empty since no nonzero bilinear forms exists on quaternionic vector spaces.

Real case

The real case breaks up into two cases, the symmetric and the antisymmetric forms that should be treated separately.

O(''p'', ''q'') and O(''n'') – the orthogonal groups

If is symmetric and the vector space is real, a basis may be chosen so that
The number of plus and minus-signs is independent of the particular basis. In the case one writes where is the number of plus signs and is the number of minus-signs,. If the notation is. The matrix is in this case
after reordering the basis if necessary. The adjoint operation then becomes
which reduces to the usual transpose when or is 0. The Lie algebra is found using equation and a suitable ansatz,
and the group according to is given by
The groups and are isomorphic through the map
For example, the Lie algebra of the Lorentz group could be written as
Naturally, it is possible to rearrange so that the -block is the upper left. Here the "time component" end up as the fourth coordinate in a physical interpretation, and not the first as may be more common.

Sp(''m'', R) – the real symplectic group

If is skew-symmetric and the vector space is real, there is a basis giving
where. For one writes In case one writes or. From the normal form one reads off
By making the ansatz
where are -dimensional matrices and considering,
one finds the Lie algebra of,
and the group is given by

Complex case

Like in the real case, there are two cases, the symmetric and the antisymmetric case that each yield a family of classical groups.

O(''n'', C) – the complex orthogonal group

If case is symmetric and the vector space is complex, a basis
with only plus-signs can be used. The automorphism group is in the case of called. The Lie algebra is simply a special case of that for,
and the group is given by
In terms of classification of simple Lie algebras, the are split into two classes, those with odd with root system and even with root system.