Classical group

In mathematics, the classical groups are defined as the special linear groups over the reals $ℝ$, the complex numbers $ℂ$ and the quaternions $ℍ$ 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 Template:Math is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group Template:Math is a symmetry group of spacetime of special relativity. The special unitary group Template:Math is the symmetry group of quantum chromodynamics and the symplectic group Template:Math finds application in Hamiltonian mechanics and quantum mechanical versions of it.

The classical groups

The classical groups are exactly the general linear groups over Template:Mathbb, Template:Mathbb and Template:Mathbb 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.

Name	Group	Field	Form	Maximal compact subgroup	Lie algebra	Root system
Special linear	[[Special linear group|Template:Math]]	Template:Mathbb	—	Template:Math
Complex special linear	[[Special linear group|Template:Math]]	Template:Mathbb	—	[[SU(n)|Template:Math]]	Complex	[[Root system#Explicit construction of the irreducible root systems|Template:Math, Template:Math]]
Quaternionic special linear	Template:Math Template:Math	Template:Mathbb	—	Template:Math
(Indefinite) special orthogonal	[[Indefinite orthogonal group|Template:Math]]	Template:Mathbb	Symmetric	Template:Math
Complex special orthogonal	[[Special orthogonal group|Template:Math]]	Template:Mathbb	Symmetric	[[SO(n)|Template:Math]]	Complex	$\{\begin{array}{cc}{B}_m,& n=2m+1\\ D_m,& n=2m\end{array}$
Symplectic	[[Symplectic group|Template:Math]]	Template:Mathbb	Skew-symmetric	Template:Math
Complex symplectic	[[Symplectic group|Template:Math]]	Template:Mathbb	Skew-symmetric	[[Sp(n)|Template:Math]]	Complex	[[Root system#Explicit construction of the irreducible root systems|Template:Math, Template:Math]]
(Indefinite) special unitary	[[Special unitary group|Template:Math]]	Template:Mathbb	Hermitian	Template:Math
(Indefinite) quaternionic unitary	Template:Math	Template:Mathbb	Hermitian	Template:Math
Quaternionic orthogonal	Template:Math	Template:Mathbb	Skew-Hermitian	Template:Math

The complex classical groups are Template:Math, Template:Math and Template:Math. 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, Template:Math, Template:Math and Template:Math. One characterization of the compact real form is in terms of the Lie algebra Template:Math. If Template:Math, the complexification of Template:Math, and if the connected group Template:Math generated by Template:Math is compact, then Template:Math 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 Template:Math, and Template:Math together with their real forms.

For instance, Template:Math is a real form of Template:Math, Template:Math is a real form of Template:Math, and Template:Math is a real form of Template:Math. 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

Template:Main The classical groups are defined in terms of forms defined on Template:Math, Template:Math, and Template:Math, where Template:Math and Template:Math are the fields of the real and complex numbers. The quaternions, Template:Math, 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 Template:Math is allowed to be defined over Template:Math, Template:Math, as well as Template:Math below. In the case of Template:Math, Template:Math is a right vector space to make possible the representation of the group action as matrix multiplication from the left, just as for Template:Math and Template:Math.

A form Template:Math on some finite-dimensional right vector space over Template:Math, or Template:Math is bilinear if

$\phi (x\alpha ,y\beta )=\alpha \phi (x,y)\beta ,\mathrm{\forall }x,y\in V,\mathrm{\forall }\alpha ,\beta \in F.$ and if

$\phi (x_1+x_2,y_1+y_2)=\phi (x_1,y_1)+\phi (x_1,y_2)+\phi (x_2,y_1)+\phi (x_2,y_2),\mathrm{\forall }{x}_1,x_2,y_1,y_2\in V.$

It is called sesquilinear if

$\phi (x\alpha ,y\beta )=\overline{\alpha }\phi (x,y)\beta ,\mathrm{\forall }x,y\in V,\mathrm{\forall }\alpha ,\beta \in F.$ and if

$\phi (x_1+x_2,y_1+y_2)=\phi (x_1,y_1)+\phi (x_1,y_2)+\phi (x_2,y_1)+\phi (x_2,y_2),\mathrm{\forall }{x}_1,x_2,y_1,y_2\in V.$

These conventions are chosen because they work in all cases considered. An automorphism of Template:Math is a map Template:Math in the set of linear operators on Template:Math such that Template:NumBlk The set of all automorphisms of Template:Math form a group, it is called the automorphism group of Template:Math, denoted Template:Math. 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 Template:Math, Template:Math or Template:Math.

This definition has some redundancy. In the case of Template:Math, bilinear is equivalent to sesquilinear. In the case of Template:Math, there are no non-zero bilinear forms.

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

A form is symmetric if

$\phi (x,y)=\phi (y,x).$

It is skew-symmetric if

$\phi (x,y)=-\phi (y,x).$

It is Hermitian if

$\phi (x,y)=\overline{\phi (y,x)}$

Finally, it is skew-Hermitian if

$\phi (x,y)=-\overline{\phi (y,x)}.$

A bilinear form Template:Math is uniquely a sum of a symmetric form and a skew-symmetric form. A transformation preserving Template:Math 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:

$\begin{array}{cccc}\text{Bilinear symmetric form in (pseudo-)orthonormal basis:}\phi (x,y)=& \pm {\xi }_1{\eta }_1\pm {\xi }_2{\eta }_2\pm \cdots \pm {\xi }_n{\eta }_n,& & (𝐑)\\ \text{Bilinear symmetric form in orthonormal basis:}\phi (x,y)=& {\xi }_1{\eta }_1+{\xi }_2{\eta }_2+\cdots +{\xi }_n{\eta }_n,& & (𝐂)\\ \text{Bilinear skew-symmetric in symplectic basis:}\phi (x,y)=& {\xi }_1{\eta }_{m+1}+{\xi }_2{\eta }_{m+2}+\cdots +{\xi }_m{\eta }_{2m=n}\\ & -{\xi }_{m+1}{\eta }_1-{\xi }_{m+2}{\eta }_2-\cdots -{\xi }_{2m=n}{\eta }_m,& & (𝐑,𝐂)\\ \text{Sesquilinear Hermitian:}\phi (x,y)=& \pm \overline{{\xi }_1}{\eta }_1\pm \overline{{\xi }_2}{\eta }_2\pm \cdots \pm \overline{{\xi }_n}{\eta }_n,& & (𝐂,𝐇)\\ \text{Sesquilinear skew-Hermitian:}\phi (x,y)=& \overline{{\xi }_1}𝐣{\eta }_1+\overline{{\xi }_2}𝐣{\eta }_2+\cdots +\overline{{\xi }_n}𝐣{\eta }_n,& & (𝐇)\end{array}$

The Template:Math in the skew-Hermitian form is the third basis element in the basis Template:Math for Template:Math. Proof of existence of these bases and Sylvester's law of inertia, the independence of the number of plus- and minus-signs, Template:Math and Template:Math, in the symmetric and Hermitian forms, as well as the presence or absence of the fields in each expression, can be found in Template:Harvtxt or Template:Harvtxt. The pair Template:Math, and sometimes Template:Math, is called the signature of the form.

Explanation of occurrence of the fields Template:Math: There are no nontrivial bilinear forms over Template:Math. In the symmetric bilinear case, only forms over Template:Math have a signature. In other words, a complex bilinear form with "signature" Template:Math can, by a change of basis, be reduced to a form where all signs are "Template:Math" in the above expression, whereas this is impossible in the real case, in which Template:Math 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 Template:Mvar, so in this case, only Template:Math is interesting.

Automorphism groups

Hermann Weyl, the author of The Classical Groups. Weyl made substantial contributions to the representation theory of the classical 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 Template:Math, Template:Math and Template:Math.

Aut(φ) – the automorphism group

Assume that Template:Math is a non-degenerate form on a finite-dimensional vector space Template:Math over Template:Math or Template:Math. The automorphism group is defined, based on condition (Template:EquationNote), as

$\mathrm{A}\mathrm{u}\mathrm{t}(\phi )=\{A\in \mathrm{G}\mathrm{L}(V):\phi (Ax,Ay)=\phi (x,y),\mathrm{\forall }x,y\in V\}.$

Every Template:Math has an adjoint Template:Math with respect to Template:Math defined by Template:NumBlk

Using this definition in condition (Template:EquationNote), the automorphism group is seen to be given by Template:NumBlk

Fix a basis for Template:Math. In terms of this basis, put

$\phi (x,y)=\sum {\xi }_i{\phi }_{ij}{\eta }_j$

where Template:Math are the components of Template:Math. This is appropriate for the bilinear forms. Sesquilinear forms have similar expressions and are treated separately later. In matrix notation one finds

$\phi (x,y)=x^{\mathrm{T}}\mathrm{\Phi }y$

and Template:NumBlk from (Template:EquationNote) where Template:Math is the matrix Template:Math. The non-degeneracy condition means precisely that Template:Math is invertible, so the adjoint always exists. Template:Math expressed with this becomes

$\mathrm{Aut}(\phi )=\{A\in \mathrm{GL}(V):{\mathrm{\Phi }}^{-1}{A}^{\mathrm{T}}\mathrm{\Phi }A=1\}.$

The Lie algebra Template:Math of the automorphism groups can be written down immediately. Abstractly, Template:Math if and only if

$(e^{tX}{)}^{\phi }{e}^{tX}=1$

for all Template:Math, corresponding to the condition in (Template:EquationNote) under the exponential mapping of Lie algebras, so that

$𝔞𝔲𝔱(\phi )=\{X\in M_n(V):X^{\phi }=-X\},$

or in a basis Template:NumBlk as is seen using the power series expansion of the exponential mapping and the linearity of the involved operations. Conversely, suppose that Template:Math. Then, using the above result, Template:Math. Thus the Lie algebra can be characterized without reference to a basis, or the adjoint, as

$𝔞𝔲𝔱(\phi )=\{X\in M_n(V):\phi (Xx,y)=-\phi (x,Xy),\mathrm{\forall }x,y\in V\}.$

The normal form for Template:Math will be given for each classical group below. From that normal form, the matrix Template:Math can be read off directly. Consequently, expressions for the adjoint and the Lie algebras can be obtained using formulas (Template:EquationNote) and (Template:EquationNote). This is demonstrated below in most of the non-trivial cases.

Bilinear case

When the form is symmetric, Template:Math is called Template:Math. When it is skew-symmetric then Template:Math is called Template:Math. 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

Template:Main If Template:Math is symmetric and the vector space is real, a basis may be chosen so that

$\phi (x,y)=\pm {\xi }_1{\eta }_1\pm {\xi }_2{\eta }_2\cdots \pm {\xi }_n{\eta }_n.$

The number of plus and minus-signs is independent of the particular basis. In the case Template:Math one writes Template:Math where Template:Math is the number of plus signs and Template:Math is the number of minus-signs, Template:Math. If Template:Math the notation is Template:Math. The matrix Template:Math is in this case

$\mathrm{\Phi }=\left(\begin{array}{cc}{I}_p& 0\\ 0& -I_q\end{array}\right)\equiv I_{p,q}$

after reordering the basis if necessary. The adjoint operation (Template:EquationNote) then becomes

$A^{\phi }=\left(\begin{array}{cc}{I}_p& 0\\ 0& -I_q\end{array}\right){\left(\begin{array}{cc}{A}_{11}& \cdots \\ \cdots & A_{nn}\end{array}\right)}^{\mathrm{T}}\left(\begin{array}{cc}{I}_p& 0\\ 0& -I_q\end{array}\right),$

which reduces to the usual transpose when Template:Math or Template:Math is 0. The Lie algebra is found using equation (Template:EquationNote) and a suitable ansatz (this is detailed for the case of Template:Math below),

$𝔬(p,q)=\{\left(\begin{array}{cc}{X}_{p\times p}& Y_{p\times q}\\ Y^{\mathrm{T}}& W_{q\times q}\end{array}\right)|X^{\mathrm{T}}=-X,W^{\mathrm{T}}=-W\},$

and the group according to (Template:EquationNote) is given by

$\mathrm{O}(p,q)=\{g\in \mathrm{G}\mathrm{L}(n,ℝ)|I_{p,q}^{-1}{g}^{\mathrm{T}}{I}_{p,q}g=I\}.$

The groups Template:Math and Template:Math are isomorphic through the map

$\mathrm{O}(p,q)\to \mathrm{O}(q,p),g\to \sigma g{\sigma }^{-1},\sigma =\left[\begin{array}{cccc}0& 0& \cdots & 1\\ ⋮& ⋮& \ddots & ⋮\\ 0& 1& \cdots & 0\\ 1& 0& \cdots & 0\end{array}\right].$

For example, the Lie algebra of the Lorentz group could be written as

$𝔬(3,1)=\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}\{\left(\begin{array}{cccc}0& 1& 0& 0\\ -1& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\left(\begin{array}{cccc}0& 0& -1& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right),\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 1& 0\\ 0& -1& 0& 0\\ 0& 0& 0& 0\end{array}\right),\left(\begin{array}{cccc}0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 0\end{array}\right),\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 1& 0& 0\end{array}\right),\left(\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 1\\ 0& 0& 1& 0\end{array}\right)\}.$

Naturally, it is possible to rearrange so that the Template:Math-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

Template:Main If Template:Math is skew-symmetric and the vector space is real, there is a basis giving

$\phi (x,y)={\xi }_1{\eta }_{m+1}+{\xi }_2{\eta }_{m+2}\cdots +{\xi }_m{\eta }_{2m=n}-{\xi }_{m+1}{\eta }_1-{\xi }_{m+2}{\eta }_2\cdots -{\xi }_{2m=n}{\eta }_m,$

where Template:Math. For Template:Math one writes Template:Math In case Template:Math one writes Template:Math or Template:Math. From the normal form one reads off

$\mathrm{\Phi }=\left(\begin{array}{cc}{0}_m& I_m\\ -I_m& 0_m\end{array}\right)=J_m.$

By making the ansatz

$V=\left(\begin{array}{cc}X& Y\\ Z& W\end{array}\right),$

where Template:Math are Template:Math-dimensional matrices and considering (Template:EquationNote),

$\left(\begin{array}{cc}{0}_m& -I_m\\ I_m& 0_m\end{array}\right){\left(\begin{array}{cc}X& Y\\ Z& W\end{array}\right)}^{\mathrm{T}}\left(\begin{array}{cc}{0}_m& I_m\\ -I_m& 0_m\end{array}\right)=-\left(\begin{array}{cc}X& Y\\ Z& W\end{array}\right)$

one finds the Lie algebra of Template:Math,

$𝔰𝔭(m,ℝ)=\{X\in M_n(ℝ):J_{m}X+X^{\mathrm{T}}{J}_m=0\}=\{\left(\begin{array}{cc}X& Y\\ Z& -X^{\mathrm{T}}\end{array}\right)|Y^{\mathrm{T}}=Y,Z^{\mathrm{T}}=Z\},$

and the group is given by

$\mathrm{S}\mathrm{p}(m,ℝ)=\{g\in M_n(ℝ)|g^{\mathrm{T}}{J}_{m}g=J_m\}.$

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

Template:Main If case Template:Math is symmetric and the vector space is complex, a basis

$\phi (x,y)={\xi }_1{\eta }_1+{\xi }_1{\eta }_1\cdots +{\xi }_n{\eta }_n$

with only plus-signs can be used. The automorphism group is in the case of Template:Math called Template:Math. The lie algebra is simply a special case of that for Template:Math,

$𝔬(n,ℂ)=𝔰𝔬(n,ℂ)=\{X|X^{\mathrm{T}}=-X\},$

and the group is given by

$\mathrm{O}(n,ℂ)=\{g|g^{\mathrm{T}}g=I_n\}.$

In terms of classification of simple Lie algebras, the Template:Math are split into two classes, those with Template:Math odd with root system Template:Math and Template:Math even with root system Template:Math.

Sp(m, C) – the complex symplectic group

Template:Main For Template:Math skew-symmetric and the vector space complex, the same formula,

$\phi (x,y)={\xi }_1{\eta }_{m+1}+{\xi }_2{\eta }_{m+2}\cdots +{\xi }_m{\eta }_{2m=n}-{\xi }_{m+1}{\eta }_1-{\xi }_{m+2}{\eta }_2\cdots -{\xi }_{2m=n}{\eta }_m,$

applies as in the real case. For Template:Math one writes Template:Math. In the case $V={ℂ}^n={ℂ}^{2m}$ one writes Template:Math or Template:Math. The Lie algebra parallels that of Template:Math,

$𝔰𝔭(m,ℂ)=\{X\in M_n(ℂ):J_{m}X+X^{\mathrm{T}}{J}_m=0\}=\{\left(\begin{array}{cc}X& Y\\ Z& -X^{\mathrm{T}}\end{array}\right)|Y^{\mathrm{T}}=Y,Z^{\mathrm{T}}=Z\},$

and the group is given by

$\mathrm{S}\mathrm{p}(m,ℂ)=\{g\in M_n(ℂ)|g^{\mathrm{T}}{J}_{m}g=J_m\}.$

Sesquilinear case

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

$\phi (x,y)=\sum {\overline{\xi }}_i{\phi }_{ij}{\eta }_j.$

The other expressions that get modified are

$\phi (x,y)=x^{*}\mathrm{\Phi }y,A^{\phi }={\mathrm{\Phi }}^{-1}{A}^{*}\mathrm{\Phi },$

$\mathrm{Aut}(\phi )=\{A\in \mathrm{GL}(V):{\mathrm{\Phi }}^{-1}{A}^{*}\mathrm{\Phi }A=1\},$

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 Template:Math 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

Template:Main A non-degenerate hermitian form has the normal form

$\phi (x,y)=\pm \overline{{\xi }_1}{\eta }_1\pm \overline{{\xi }_2}{\eta }_2\cdots \pm \overline{{\xi }_n}{\eta }_n.$

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

$\mathrm{\Phi }=\left(\begin{array}{cc}{1}_p& 0\\ 0& -1_q\end{array}\right)=I_{p,q},$

and the Lie algebra is given by

$𝔲(p,q)=\{\left(\begin{array}{cc}{X}_{p\times p}& Z_{p\times q}\\ {\overline{Z_{p\times q}}}^{\mathrm{T}}& Y_{q\times q}\end{array}\right)|{\bar{X}}^{\mathrm{T}}=-X,{\bar{Y}}^{\mathrm{T}}=-Y\}.$

The group is given by

$\mathrm{U}(p,q)=\{g|I_{p,q}^{-1}{g}^{*}{I}_{p,q}g=I\}.$

where g is a general n x n complex matrix and $g^{*}$ is defined as the conjugate transpose of g, what physicists call $g^{†}$.

As a comparison, a Unitary matrix U(n) is defined as

$\mathrm{U}(n)=\{g|g^{*}g=I\}.$

We note that $\mathrm{U}(n)$ is the same as $\mathrm{U}(n,0)$

Quaternionic case

The space Template:Math is considered as a right vector space over Template:Math. This way, Template:Math for a quaternion Template:Math, a quaternion column vector Template:Math and quaternion matrix Template:Math. If Template:Math were a left vector space over Template:Math, 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 Template:Math is henceforth a right vector space over Template:Math. Even so, care must be taken due to the non-commutative nature of Template:Math. 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 Template:Nowrap, Template:NumBlk 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 Template:Math is given as a column vector Template:Math, 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 (Template:Math) is isomorphic to Template:Math.

Quaternionic Template:Math-matrices can, by obvious extension, be represented by Template:Math block-matrices of complex numbers. If one agrees to represent a quaternionic Template:Nowrap column vector by a Template:Nowrap column vector with complex numbers according to the encoding of above, with the upper Template:Math numbers being the Template:Math and the lower Template:Math the Template:Math, then a quaternionic Template:Math-matrix becomes a complex Template:Math-matrix exactly of the form given above, but now with α and β Template:Math-matrices. More formally Template:NumBlk

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

${ℍ}^n\approx {ℂ}^{2n},M_n(ℍ)\approx \{T\in M_{2n}(ℂ)|J_{n}T=\bar{T}{J}_n,J_n=\left(\begin{array}{cc}0& I_n\\ -I_n& 0\end{array}\right)\}.$

The space Template:Math is a real algebra, but it is not a complex subspace of Template:Math. Multiplication (from the left) by Template:Math in Template:Math using entry-wise quaternionic multiplication and then mapping to the image in Template:Math yields a different result than multiplying entry-wise by Template:Math directly in Template:Math. The quaternionic multiplication rules give Template:Math where the new Template:Math and Template:Math 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 Template:Math where Template:Math 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 Template:Math is embedded in Template:Math is not unique, but all such embeddings are related through Template:Math for Template:Math, leaving the determinant unaffected. The name of Template:Math in this complex guise is Template:Math.

As opposed to in the case of Template:Math, both the Hermitian and the skew-Hermitian case bring in something new when Template:Math is considered, so these cases are considered separately.

GL(n, H) and SL(n, H)

Under the identification above,

$\mathrm{G}\mathrm{L}(n,ℍ)=\{g\in \mathrm{G}\mathrm{L}(2n,ℂ)|Jg=\bar{g}J,\mathrm{d}\mathrm{e}\mathrm{t}g\ne 0\}\equiv {\mathrm{U}}^{*}(2n).$

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

$𝔤𝔩(n,ℍ)=\{\left(\begin{array}{cc}X& -\bar{Y}\\ Y& \bar{X}\end{array}\right)|X,Y\in 𝔤𝔩(n,ℂ)\}\equiv {𝔲}^{*}(2n).$

The quaternionic special linear group is given by

$\mathrm{S}\mathrm{L}(n,ℍ)=\{g\in \mathrm{G}\mathrm{L}(n,ℍ)|\mathrm{d}\mathrm{e}\mathrm{t}g=1\}\equiv {\mathrm{S}\mathrm{U}}^{*}(2n),$

where the determinant is taken on the matrices in Template:Math. Alternatively, one can define this as the kernel of the Dieudonné determinant $\mathrm{G}\mathrm{L}(n,ℍ)\to {ℍ}^{*}/[{ℍ}^{*},{ℍ}^{*}]\simeq {ℝ}_{>0}^{*}$. The Lie algebra is

$𝔰𝔩(n,ℍ)=\{\left(\begin{array}{cc}X& -\bar{Y}\\ Y& \bar{X}\end{array}\right)|Re(\mathrm{Tr}X)=0\}\equiv {𝔰𝔲}^{*}(2n).$

Sp(p, q) – the quaternionic unitary group

As above in the complex case, the normal form is

$\phi (x,y)=\pm \overline{{\xi }_1}{\eta }_1\pm \overline{{\xi }_2}{\eta }_2\cdots \pm \overline{{\xi }_n}{\eta }_n$

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

In quaternionic notation,

$\mathrm{\Phi }=\left(\begin{array}{cc}{I}_p& 0\\ 0& -I_q\end{array}\right)=I_{p,q}$

meaning that quaternionic matrices of the form Template:NumBlk will satisfy

${\mathrm{\Phi }}^{-1}{𝒬}^{*}\mathrm{\Phi }=-𝒬,$

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

$𝒳=\left(\begin{array}{cc}{X}_{1(p\times p)}& -{\bar{X}}_2\\ X_2& {\bar{X}}_1\end{array}\right),𝒴=\left(\begin{array}{cc}{Y}_{1(q\times q)}& -{\bar{Y}}_2\\ Y_2& {\bar{Y}}_1\end{array}\right),𝒵=\left(\begin{array}{cc}{Z}_{1(p\times q)}& -{\bar{Z}}_2\\ Z_2& {\bar{Z}}_1\end{array}\right),$

and the relations in (Template:EquationNote) will be satisfied if

$X_1^{*}=-X_1,Y_1^{*}=-Y_1.$

The Lie algebra becomes

$𝔰𝔭(p,q)=\{\left(\begin{array}{cc}\left[\begin{array}{cc}{X}_{1(p\times p)}& -{\bar{X}}_2\\ X_2& {\bar{X}}_1\end{array}\right]& \left[\begin{array}{cc}{Z}_{1(p\times q)}& -{\bar{Z}}_2\\ Z_2& {\bar{Z}}_1\end{array}\right]\\ {\left[\begin{array}{cc}{Z}_{1(p\times q)}& -{\bar{Z}}_2\\ Z_2& {\bar{Z}}_1\end{array}\right]}^{*}& \left[\begin{array}{cc}{Y}_{1(q\times q)}& -{\bar{Y}}_2\\ Y_2& {\bar{Y}}_1\end{array}\right]\end{array}\right)|X_1^{*}=-X_1,Y_1^{*}=-Y_1\}.$

The group is given by

$\mathrm{S}\mathrm{p}(p,q)=\{g\in \mathrm{G}\mathrm{L}(n,ℍ)\mid I_{p,q}^{-1}{g}^{*}{I}_{p,q}g=I_{p+q}\}=\{g\in \mathrm{G}\mathrm{L}(2n,ℂ)\mid K_{p,q}^{-1}{g}^{*}{K}_{p,q}g=I_{2(p+q)},K=\mathrm{diag}(I_{p,q},I_{p,q})\}.$

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

$\phi (w,z)=\left[\begin{array}{cc}{u}^{*}& v^{*}\end{array}\right]K_{p,q}\left[\begin{array}{c}x\\ y\end{array}\right]+j\left[\begin{array}{cc}u& -v\end{array}\right]K_{p,q}\left[\begin{array}{c}y\\ x\end{array}\right]={\phi }_1(w,z)+𝐣{\phi }_2(w,z),K_{p,q}=\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(I_{p,q},I_{p,q})$

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

$\mathrm{S}\mathrm{p}(p,q)=\mathrm{U}({ℂ}^{2n},{\phi }_1)\cap \mathrm{S}\mathrm{p}({ℂ}^{2n},{\phi }_2)$

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

$\phi (x,y)=\overline{{\xi }_1}𝐣{\eta }_1+\overline{{\xi }_2}𝐣{\eta }_2\cdots +\overline{{\xi }_n}𝐣{\eta }_n,$

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

$\mathrm{\Phi }=\left(\begin{array}{cccc}𝐣& 0& \cdots & 0\\ 0& 𝐣& \cdots & ⋮\\ ⋮& & \ddots & & \\ 0& \cdots & 0& 𝐣\end{array}\right)\equiv {\mathrm{j}}_n$

and from (Template:EquationNote) follows that Template:NumBlk for Template:Math. Now put

$V=X+𝐣Y\leftrightarrow \left(\begin{array}{cc}X& -\bar{Y}\\ Y& \bar{X}\end{array}\right)$

according to prescription (Template:EquationNote). The same prescription yields for Template:Math,

$\mathrm{\Phi }\leftrightarrow \left(\begin{array}{cc}0& -I_n\\ I_n& 0\end{array}\right)\equiv J_n.$

Now the last condition in (Template:EquationNote) in complex notation reads

${\left(\begin{array}{cc}X& -\bar{Y}\\ Y& \bar{X}\end{array}\right)}^{*}=\left(\begin{array}{cc}0& -I_n\\ I_n& 0\end{array}\right)\left(\begin{array}{cc}X& -\bar{Y}\\ Y& \bar{X}\end{array}\right)\left(\begin{array}{cc}0& -I_n\\ I_n& 0\end{array}\right)\iff X^{\mathrm{T}}=-X,\stackrel{\mathrm{T}}{\bar{Y}}=Y.$

The Lie algebra becomes

${𝔬}^{*}(2n)=\{\left(\begin{array}{cc}X& -\bar{Y}\\ Y& \bar{X}\end{array}\right)|X^{\mathrm{T}}=-X,\stackrel{\mathrm{T}}{\bar{Y}}=Y\},$

and the group is given by

${\mathrm{O}}^{*}(2n)=\{g\in \mathrm{G}\mathrm{L}(n,ℍ)\mid {\mathrm{j}}_n^{-1}{g}^{*}{\mathrm{j}}_{n}g=I_n\}=\{g\in \mathrm{G}\mathrm{L}(2n,ℂ)\mid J_n^{-1}{g}^{*}{J}_{n}g=I_{2n}\}.$

The group Template:Math can be characterized as

${\mathrm{O}}^{*}(2n)=\{g\in \mathrm{O}(2n,ℂ)\mid \theta \left(\bar{g}\right)=g\},$

where the map Template:Math is defined by Template:Math.

Also, the form determining the group can be viewed as a Template:Math-valued form on Template:Math. Make the substitutions Template:Math and Template:Math in the expression for the form. Then

$\phi (x,y)={\bar{w}}_2{I}_n{z}_1-{\bar{w}}_1{I}_n{z}_2+𝐣(w_1{I}_n{z}_1+w_2{I}_n{z}_2)=\overline{{\phi }_1(w,z)}+𝐣{\phi }_2(w,z).$

The form Template:Math is Hermitian (while the first form on the left hand side is skew-Hermitian) of signature Template:Math. The signature is made evident by a change of basis from Template:Math to Template:Math where Template:Math are the first and last Template:Math basis vectors respectively. The second form, Template:Math is symmetric positive definite. Thus, due to the factor Template:Math, Template:Math preserves both separately and it may be concluded that

${\mathrm{O}}^{*}(2n)=\mathrm{O}(2n,ℂ)\cap \mathrm{U}({ℂ}^{2n},{\phi }_1),$

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 GL_n(R) is the group of all R-linear automorphisms of Rⁿ. 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(F) over a field F is simple for n ≥ 2, except for the two cases when n = 2 and the field has orderTemplate:Clarify 2 or 3.

Unitary groups

The unitary group U_n(R) is a group preserving a sesquilinear form on a module. There is a subgroup, the special unitary group SU_n(R) and their quotients the projective unitary group PU_n(R) = U_n(R)/Z(U_n(R)) and the projective special unitary group 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(F_q) 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 O_n(R) preserves a non-degenerate quadratic form on a module. There is a subgroup, the special orthogonal group SO_n(R) and quotients, the projective orthogonal group PO_n(R), and the projective special orthogonal group PSO_n(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 Ω_n(R) consisting of the elements of the orthogonal group 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 orthogonal group, 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.

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.

References

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