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 (here with the determinant 1 condition, but this is not necessary) are the following:
The complex linear algebraic groups , and  together with their real forms. 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 
 and if 

It is called sesquilinear if 
 and 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:
A classical group is a group that preserves a bilinear or sesquilinear form on finite-dimensional vector spaces over ,  or . 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. (The real case reduces to the symmetric 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 (this is detailed for the case of  below),

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 (or any other block). 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 . Sp(m, C) – the complex symplectic group

For  skew-symmetric and the vector space complex, the same formula,

applies as in the real case. For  one writes . In the case  one writes  or . The Lie algebra parallels that of ,

and the group is given by

Sesquilinear case
In the sesquilinear case, one makes a slightly different approach for the form in terms of a basis,

The other expressions that get modified are

The real case, of course, provides nothing new. The complex and the quaternionic case will be considered below. Complex case 
From a qualitative point of view, consideration of skew-Hermitian forms (up to isomorphism) provide no new groups; multiplication by  renders a skew-Hermitian form Hermitian, and vice versa. Thus only the Hermitian case needs to be considered. U(p, q) and U(n) – the unitary groups

A non-degenerate hermitian form has the normal form

As in the bilinear case, the signature (p, q) is independent of the basis. The automorphism group is denoted , or, in the case of , . If  the notation is . In this case,  takes the form

and the Lie algebra is given by

The group is given by

where g is a general n x n complex matrix and  is defined as the conjugate transpose of g, what physicists call . As a comparison, a Unitary matrix U(n) is defined as

We note that  is the same as

Quaternionic case 
The space  is considered as a right vector space over . This way,  for a quaternion , a quaternion column vector  and quaternion matrix . If  was a left vector space over , then matrix multiplication from the right on row vectors would be required to maintain linearity. This does not correspond to the usual linear operation of a group on a vector space when a basis is given, which is matrix multiplication from the left on column vectors. Thus  is henceforth a right vector space over . Even so, care must be taken due to the non-commutative nature of . The (mostly obvious) details are skipped because complex representations will be used. When dealing with quaternionic groups it is convenient to represent quaternions using complex ,

With this representation, quaternionic multiplication becomes matrix multiplication and quaternionic conjugation becomes taking the Hermitian adjoint. Moreover, if a quaternion according to the complex encoding  is given as a column vector , then multiplication from the left by a matrix representation of a quaternion produces a new column vector representing the correct quaternion. This representation differs slightly from a more common representation found in the quaternion article. The more common convention would force multiplication from the right on a row matrix to achieve the same thing. Incidentally, the representation above makes it clear that the group of unit quaternions () is isomorphic to . Quaternionic -matrices can, by obvious extension, be represented by  block-matrices of complex numbers. If one agrees to represent a quaternionic  column vector by a  column vector with complex numbers according to the encoding of above, with the upper  numbers being the  and the lower  the , then a quaternionic -matrix becomes a complex -matrix exactly of the form given above, but now with α and β -matrices. More formally

A matrix  has the form displayed in () if and only if . With these identifications,

The space  is a real algebra, but it is not a complex subspace of . Multiplication (from the left) by  in  using entry-wise quaternionic multiplication and then mapping to the image in  yields a different result than multiplying entry-wise by  directly in . The quaternionic multiplication rules give  where the new  and  are inside the parentheses. The action of the quaternionic matrices on quaternionic vectors is now represented by complex quantities, but otherwise it is the same as for "ordinary" matrices and vectors. The quaternionic groups are thus embedded in  where  is the dimension of the quaternionic matrices. The determinant of a quaternionic matrix is defined in this representation as being the ordinary complex determinant of its representative matrix. The non-commutative nature of quaternionic multiplication would, in the quaternionic representation of matrices, be ambiguous. The way  is embedded in  is not unique, but all such embeddings are related through  for , leaving the determinant unaffected. The name of  in this complex guise is . As opposed to in the case of , both the Hermitian and the skew-Hermitian case bring in something new when  is considered, so these cases are considered separately. GL(n, H) and SL(n, H)
Under the identification above,

Its Lie algebra  is the set of all matrices in the image of the mapping  of above,

The quaternionic special linear group is given by

where the determinant is taken on the matrices in . Alternatively, one can define this as the kernel of the Dieudonné determinant . The Lie algebra is

Sp(p, q) – the quaternionic unitary group
As above in the complex case, the normal form is

and the number of plus-signs is independent of basis. When  with this form, . The reason for the notation is that the group can be represented, using the above prescription, as a subgroup of  preserving a complex-hermitian form of signature  If  or  the group is denoted . It is sometimes called the hyperunitary group. In quaternionic notation,

meaning that quaternionic matrices of the form

will satisfy

see the section about . Caution needs to be exercised when dealing with quaternionic matrix multiplication, but here only  and  are involved and these commute with every quaternion matrix. Now apply prescription () to each block,

and the relations in () will be satisfied if

The Lie algebra becomes

The group is given by

Returning to the normal form of  for , make the substitutions  and  with . Then

viewed as a -valued form on . Thus the elements of , viewed as linear transformations of , preserve both a Hermitian form of signature  and a non-degenerate skew-symmetric form. Both forms take purely complex values and due to the prefactor of  of the second form, they are separately conserved. This means that

and this explains both the name of the group and the notation. O∗(2n) = O(n, H)- quaternionic orthogonal group 
The normal form for a skew-hermitian form is given by

where  is the third basis quaternion in the ordered listing . In this case,  may be realized, using the complex matrix encoding of above, as a subgroup of  which preserves a non-degenerate complex skew-hermitian form of signature . From the normal form one sees that in quaternionic notation

and from () follows that

for . Now put 

according to prescription (). The same prescription yields for ,

Now the last condition in () in complex notation reads

The Lie algebra becomes

and the group is given by

The group  can be characterized as

where the map  is defined by . Also, the form determining the group can be viewed as a -valued form on . Make the substitutions  and  in the expression for the form. Then
 
The form  is Hermitian (while the first form on the left hand side is skew-Hermitian) of signature . The signature is made evident by a change of basis from  to  where  are the first and last  basis vectors respectively. The second form,  is symmetric positive definite. Thus, due to the factor ,  preserves both separately and it may be concluded that

and the notation "O" is explained. 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 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 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(Fq) over a finite field is simple for n ≥ 1, except for the cases of PSp2 over the fields of two and three elements. 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

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. Notes

References
E. Artin (1957) Geometric Algebra, chapters III, IV, & V via Internet Archive

Lie groups