In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with special^{[1]} 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.^{[2]} 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.^{[3]}
The classical groups form the deepest and most useful part of the subject of linear Lie groups.^{[4]} Most types of classical groups find application in classical and modern physics. A few examples are the following. The rotation group SO(3) is a symmetry of Euclidean space and all fundamental laws of physics, the Lorentz group O(3,1) is a symmetry group of spacetime of special relativity. The special unitary group SU(3) is the symmetry group of quantum chromodynamics and the symplectic group Sp(m) finds application in Hamiltonian mechanics and quantum mechanical versions of it.
The classical groups are exactly the general linear groups over , and together with the automorphism groups of non-degenerate forms discussed below.^{[5]} 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 | SL(n, ) | — | SO(n) | |||
Complex special linear | SL(n, ) | — | SU(n) | Complex | A_{m}, n = m + 1 | |
Quaternionic special linear | SL(n, ) = SU^{∗}(2n) |
— | Sp(n) | |||
(Indefinite) special orthogonal | SO(p, q) | Symmetric | S(O(p) × O(q)) | |||
Complex special orthogonal | SO(n, ) | Symmetric | SO(n) | Complex | ||
Symplectic | Sp(n, ) | Skew-symmetric | U(n) | |||
Complex symplectic | Sp(n, ) | Skew-symmetric | Sp(n) | Complex | C_{m}, n = 2m | |
(Indefinite) special unitary | SU(p, q) | Hermitian | S(U(p) × U(q)) | |||
(Indefinite) quaternionic unitary | Sp(p, q) | Hermitian | Sp(p) × Sp(q) | |||
Quaternionic orthogonal | SO^{∗}(2n) | Skew-Hermitian | SO(2n) |
The complex classical groups are SL(n, ), SO(n, ) and Sp(n, ). 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, SU(n), SO(n) and Sp(n). One characterization of the compact real form is in terms of the Lie algebra g. If g = u + iu, the complexification of u, and if the connected group K generated by {exp(X): X ∈ u} is compact, then K is a compact real form.^{[6]}
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:
For instance, SO^{∗}(2n) is a real form of SO(2n, ), SU(p, q) is a real form of SL(n, ), and SL(n, ) is a real form of SL(2n, ). 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".
The classical groups are defined in terms of forms defined on R^{n}, C^{n}, and H^{n}, where R and C are the fields of the real and complex numbers. The quaternions, H, 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 V is allowed to be defined over R, C, as well as H below. In the case of H, V is a right vector space to make possible the representation of the group action as matrix multiplication from the left, just as for R and C.^{[8]}
A form φ: V × V → F on some finite-dimensional right vector space over F = R, C, or H is bilinear if
It is called sesquilinear if
These conventions are chosen because they work in all cases considered. An automorphism of φ is a map Α in the set of linear operators on V such that
(1) |
The set of all automorphisms of φ form a group, it is called the automorphism group of φ, denoted Aut(φ). This leads to a preliminary definition of a classical group:
This definition has some redundancy. In the case of F = R, bilinear is equivalent to sesquilinear. In the case of F = H, there are no non-zero bilinear forms.^{[9]}
A form is symmetric if
It is skew-symmetric if
It is Hermitian if
Finally, it is skew-Hermitian if
A bilinear form φ is uniquely a sum of a symmetric form and a skew-symmetric form. A transformation preserving φ 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 j in the skew-Hermitian form is the third basis element in the basis (1, i, j, k) for H. Proof of existence of these bases and Sylvester's law of inertia, the independence of the number of plus- and minus-signs, p and q, in the symmetric and Hermitian forms, as well as the presence or absence of the fields in each expression, can be found in Rossmann (2002) or Goodman & Wallach (2009). The pair (p, q), and sometimes p − q, is called the signature of the form.
Explanation of occurrence of the fields R, C, H: There are no nontrivial bilinear forms over H. In the symmetric bilinear case, only forms over R have a signature. In other words, a complex bilinear form with "signature" (p, q) 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 p − q 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 i, so in this case, only H is interesting.
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 R, C and H.
Assume that φ is a non-degenerate form on a finite-dimensional vector space V over R, C or H. The automorphism group is defined, based on condition (1), as
Every A ∈ M_{n}(V) has an adjoint A^{φ} with respect to φ defined by
(2) |
Using this definition in condition (1), the automorphism group is seen to be given by
^{[10]} | (3) |
Fix a basis for V. In terms of this basis, put
where ξ_{i}, η_{j} are the components of x, y. This is appropriate for the bilinear forms. Sesquilinear forms have similar expressions and are treated separately later. In matrix notation one finds
and
^{[11]} | (4) |
from (2) where Φ is the matrix (φ_{ij}). The non-degeneracy condition means precisely that Φ is invertible, so the adjoint always exists. Aut(φ) expressed with this becomes
The Lie algebra aut(φ) of the automorphism groups can be written down immediately. Abstractly, X ∈ aut(φ) if and only if
for all t, corresponding to the condition in (3) under the exponential mapping of Lie algebras, so that
or in a basis
(5) |
as is seen using the power series expansion of the exponential mapping and the linearity of the involved operations. Conversely, suppose that X ∈ aut(φ). Then, using the above result, φ(Xx, y) = φ(x, X^{φ}y) = −φ(x, Xy). Thus the Lie algebra can be characterized without reference to a basis, or the adjoint, as
The normal form for φ will be given for each classical group below. From that normal form, the matrix Φ can be read off directly. Consequently, expressions for the adjoint and the Lie algebras can be obtained using formulas (4) and (5). This is demonstrated below in most of the non-trivial cases.
When the form is symmetric, Aut(φ) is called O(φ). When it is skew-symmetric then Aut(φ) is called Sp(φ). This applies to the real and the complex cases. The quaternionic case is empty since no nonzero bilinear forms exists on quaternionic vector spaces.^{[12]}
The real case breaks up into two cases, the symmetric and the antisymmetric forms that should be treated separately.
If φ 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.^{[13]} In the case V = R^{n} one writes O(φ) = O(p, q) where p is the number of plus signs and q is the number of minus-signs, p + q = n. If q = 0 the notation is O(n). The matrix Φ is in this case
after reordering the basis if necessary. The adjoint operation (4) then becomes
which reduces to the usual transpose when p or q is 0. The Lie algebra is found using equation (5) and a suitable ansatz (this is detailed for the case of Sp(m, R) below),
and the group according to (3) is given by
The groups O(p, q) and O(q, p) 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 q-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.
If φ is skew-symmetric and the vector space is real, there is a basis giving
where n = 2m. For Aut(φ) one writes Sp(φ) = Sp(V) In case V = R^{n} = R^{2m} one writes Sp(m, R) or Sp(2m, R). From the normal form one reads off
By making the ansatz
where X, Y, Z, W are m-dimensional matrices and considering (5),
one finds the Lie algebra of Sp(m, R),
and the group is given by
Like in the real case, there are two cases, the symmetric and the antisymmetric case that each yield a family of classical groups.
If case φ 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 V = C^{n} called O(n, C). The lie algebra is simply a special case of that for o(p, q),
and the group is given by
In terms of classification of simple Lie algebras, the so(n) are split into two classes, those with n odd with root system B_{n} and n even with root system D_{n}.
For φ skew-symmetric and the vector space complex, the same formula,
applies as in the real case. For Aut(φ) one writes Sp(φ) = Sp(V). In the case one writes Sp(m, ) or Sp(2m, ). The Lie algebra parallels that of sp(m, ),
and the group is given by
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
(6) |
The real case, of course, provides nothing new. The complex and the quaternionic case will be considered below.
From a qualitative point of view, consideration of skew-Hermitian forms (up to isomorphism) provide no new groups; multiplication by i renders a skew-Hermitian form Hermitian, and vice versa. Thus only the Hermitian case needs to be considered.
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 U(V), or, in the case of V = C^{n}, U(p, q). If q = 0 the notation is U(n). In this case, Φ takes the form
and the Lie algebra is given by
The group is given by
As a comparison, a Unitary matrix U(n) is defined as
We note that is the same as
The space H^{n} is considered as a right vector space over H. This way, A(vh) = (Av)h for a quaternion h, a quaternion column vector v and quaternion matrix A. If H^{n} were a left vector space over H, 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 V is henceforth a right vector space over H. Even so, care must be taken due to the non-commutative nature of H. 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 2×2-matrices,
^{[15]} | (7) |
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 q = x + jy is given as a column vector (x, y)^{T}, 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 (αα + ββ = 1 = det Q) is isomorphic to SU(2).
Quaternionic n×n-matrices can, by obvious extension, be represented by 2n×2n block-matrices of complex numbers.^{[16]} If one agrees to represent a quaternionic n×1 column vector by a 2n×1 column vector with complex numbers according to the encoding of above, with the upper n numbers being the α_{i} and the lower n the β_{i}, then a quaternionic n×n-matrix becomes a complex 2n×2n-matrix exactly of the form given above, but now with α and β n×n-matrices. More formally
(8) |
A matrix T ∈ GL(2n, C) has the form displayed in (8) if and only if J_{n}T = TJ_{n}. With these identifications,
The space M_{n}(H) ⊂ M_{2n}(C) is a real algebra, but it is not a complex subspace of M_{2n}(C). Multiplication (from the left) by i in M_{n}(H) using entry-wise quaternionic multiplication and then mapping to the image in M_{2n}(C) yields a different result than multiplying entry-wise by i directly in M_{2n}(C). The quaternionic multiplication rules give i(X + jY) = (iX) + j(−iY) where the new X and Y 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 M_{2n}(C) where n 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 M_{n}(H) is embedded in M_{2n}(C) is not unique, but all such embeddings are related through g ↦ AgA^{−1}, g ∈ GL(2n, C) for A ∈ O(2n, C), leaving the determinant unaffected.^{[17]} The name of SL(n, H) in this complex guise is SU^{∗}(2n).
As opposed to in the case of C, both the Hermitian and the skew-Hermitian case bring in something new when H is considered, so these cases are considered separately.
Under the identification above,
Its Lie algebra gl(n, H) is the set of all matrices in the image of the mapping M_{n}(H) ↔ M_{2n}(C) of above,
The quaternionic special linear group is given by
where the determinant is taken on the matrices in C^{2n}. Alternatively, one can define this as the kernel of the Dieudonné determinant . The Lie algebra is
As above in the complex case, the normal form is
and the number of plus-signs is independent of basis. When V = H^{n} with this form, Sp(φ) = Sp(p, q). The reason for the notation is that the group can be represented, using the above prescription, as a subgroup of Sp(n, C) preserving a complex-hermitian form of signature (2p, 2q)^{[18]} If p or q = 0 the group is denoted U(n, H). It is sometimes called the hyperunitary group.
In quaternionic notation,
meaning that quaternionic matrices of the form
(9) |
will satisfy
see the section about u(p, q). Caution needs to be exercised when dealing with quaternionic matrix multiplication, but here only I and -I are involved and these commute with every quaternion matrix. Now apply prescription (8) to each block,
and the relations in (9) will be satisfied if
The Lie algebra becomes
The group is given by
Returning to the normal form of φ(w, z) for Sp(p, q), make the substitutions w → u + jv and z → x + jy with u, v, x, y ∈ C^{n}. Then
viewed as a H-valued form on C^{2n}.^{[19]} Thus the elements of Sp(p, q), viewed as linear transformations of C^{2n}, preserve both a Hermitian form of signature (2p, 2q) and a non-degenerate skew-symmetric form. Both forms take purely complex values and due to the prefactor of j of the second form, they are separately conserved. This means that
and this explains both the name of the group and the notation.
The normal form for a skew-hermitian form is given by
where j is the third basis quaternion in the ordered listing (1, i, j, k). In this case, Aut(φ) = O^{∗}(2n) may be realized, using the complex matrix encoding of above, as a subgroup of O(2n, C) which preserves a non-degenerate complex skew-hermitian form of signature (n, n).^{[20]} From the normal form one sees that in quaternionic notation
and from (6) follows that
(9) |
for V ∈ o(2n). Now put
according to prescription (8). The same prescription yields for Φ,
Now the last condition in (9) in complex notation reads
The Lie algebra becomes
and the group is given by
The group SO^{∗}(2n) can be characterized as
where the map θ: GL(2n, C) → GL(2n, C) is defined by g ↦ −J_{2n}gJ_{2n}.
Also, the form determining the group can be viewed as a H-valued form on C^{2n}.^{[22]} Make the substitutions x → w_{1} + iw_{2} and y → z_{1} + iz_{2} in the expression for the form. Then
The form φ_{1} is Hermitian (while the first form on the left hand side is skew-Hermitian) of signature (n, n). The signature is made evident by a change of basis from (e, f) to ((e + if)/√2, (e − if)/√2) where e, f are the first and last n basis vectors respectively. The second form, φ_{2} is symmetric positive definite. Thus, due to the factor j, O^{∗}(2n) preserves both separately and it may be concluded that
and the notation "O" is explained.
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.
The general linear group GL_{n}(R) is the group of all R-linear automorphisms of R^{n}. 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 order^{[clarification needed]} 2 or 3.
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))
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 PSp_{2} over the fields of two and three elements.
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.
Contrasting with the classical Lie groups are the exceptional Lie groups, G_{2}, F_{4}, E_{6}, E_{7}, E_{8}, which share their abstract properties, but not their familiarity.^{[23]} These were only discovered around 1890 in the classification of the simple Lie algebras over the complex numbers by Wilhelm Killing and Élie Cartan.