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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[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 edit

The classical groups are exactly the general linear groups over R, C and H 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, R) R SO(n)
Complex special linear SL(n, C) C SU(n) Complex Am, n = m + 1
Quaternionic special linear SL(n, H) =
SU(2n)
H Sp(n)
(Indefinite) special orthogonal SO(p, q) R Symmetric S(O(p) × O(q))
Complex special orthogonal SO(n, C) C Symmetric SO(n) Complex  
Symplectic Sp(n, R) R Skew-symmetric U(n)
Complex symplectic Sp(n, C) C Skew-symmetric Sp(n) Complex Cm, n = 2m
(Indefinite) special unitary SU(p, q) C Hermitian S(U(p) × U(q))
(Indefinite) quaternionic unitary Sp(p, q) H Hermitian Sp(p) × Sp(q)
Quaternionic orthogonal SO(2n) H Skew-Hermitian SO(2n)

The complex classical groups are SL(n, C), SO(n, C) and Sp(n, C). 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): Xu} 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:

The complex linear algebraic groups SL(n, C), SO(n, C), and Sp(n, C) together with their real forms.[7]

For instance, SO(2n) is a real form of SO(2n, C), SU(p, q) is a real form of SL(n, C), and SL(n, H) is a real form of SL(2n, C). 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 edit

The classical groups are defined in terms of forms defined on Rn, Cn, and Hn, 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 × VF on some finite-dimensional right vector space over F = R, C, or H 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 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:

A classical group is a group that preserves a bilinear or sesquilinear form on finite-dimensional vector spaces over R, C or H.

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]

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

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 pq, 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 pq 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.

Automorphism groups edit

 
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 R, C and H.

Aut(φ) – the automorphism group edit

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 AMn(V) has an adjoint Aφ 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 ξi, ηj 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 (φij). 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, Xaut(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 Xaut(φ). Then, using the above result, φ(Xx, y) = φ(x, Xφy) = −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.

Bilinear case edit

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]

Real case edit

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 edit

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 = Rn 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.

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

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 = Rn = R2m 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

 

Complex case edit

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 edit

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 = Cn 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 Bn and n even with root system Dn.

Sp(m, C) – the complex symplectic group edit

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

 

Sesquilinear case edit

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

 [14]
 
 

 

 

 

 

(6)

The real case, of course, provides nothing new. The complex and the quaternionic case will be considered below.

Complex case edit

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.

U(p, q) and U(n) – the unitary groups edit

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 = Cn, 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

 
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 edit

The space Hn 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 Hn was 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 αi and the lower n the βi, 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 JnT = TJn. With these identifications,

 

The space Mn(H) ⊂ M2n(C) is a real algebra, but it is not a complex subspace of M2n(C). Multiplication (from the left) by i in Mn(H) using entry-wise quaternionic multiplication and then mapping to the image in M2n(C) yields a different result than multiplying entry-wise by i directly in M2n(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 M2n(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 Mn(H) is embedded in M2n(C) is not unique, but all such embeddings are related through gAgA−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.

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

Under the identification above,

 

Its Lie algebra gl(n, H) is the set of all matrices in the image of the mapping Mn(H) ↔ M2n(C) of above,

 

The quaternionic special linear group is given by

 

where the determinant is taken on the matrices in C2n. Alternatively, one can define this as the kernel of the Dieudonné determinant  . The Lie algebra is

 
Sp(p, q) – the quaternionic unitary group edit

As above in the complex case, the normal form is

 

and the number of plus-signs is independent of basis. When V = Hn 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 wu + jv and zx + jy with u, v, x, y ∈ Cn. Then

 

viewed as a H-valued form on C2n.[19] Thus the elements of Sp(p, q), viewed as linear transformations of C2n, 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.

O(2n) = O(n, H)- quaternionic orthogonal group edit

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 Vo(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

 [21]

where the map θ: GL(2n, C) → GL(2n, C) is defined by g ↦ −J2ngJ2n.

Also, the form determining the group can be viewed as a H-valued form on C2n.[22] Make the substitutions xw1 + iw2 and yz1 + iz2 in the expression for the form. Then

 

The form φ1 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, (eif)/2) where e, f are the first and last n basis vectors respectively. The second form, φ2 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 over general fields or algebras edit

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 edit

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[clarification needed] 2 or 3.

Unitary groups edit

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 edit

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 edit

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 edit

Contrast with exceptional Lie groups edit

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.[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.

Notes edit

  1. ^ Here, special means the subgroup of the full automorphism group whose elements have determinant 1.
  2. ^ Rossmann 2002 p. 94.
  3. ^ Weyl 1939
  4. ^ Rossmann 2002 p. 91.
  5. ^ Rossmann 2002 p. 94
  6. ^ Rossmann 2002 p. 103
  7. ^ Goodman & Wallach 2009 See end of chapter 1
  8. ^ Rossmann 2002p. 93.
  9. ^ Rossmann 2002 p. 105
  10. ^ Rossmann 2002 p. 91
  11. ^ Rossmann 2002 p. 92
  12. ^ Rossmann 2002 p. 105
  13. ^ Rossmann 2002 p. 107.
  14. ^ Rossmann 2002 p. 93
  15. ^ Rossmann 2002 p. 95.
  16. ^ Rossmann 2002 p. 94.
  17. ^ Goodman & Wallach 2009 Exercise 14, Section 1.1.
  18. ^ Rossmann 2002 p. 94.
  19. ^ Goodman & Wallach 2009Exercise 11, Chapter 1.
  20. ^ Rossmann 2002 p. 94.
  21. ^ Goodman & Wallach 2009 p.11.
  22. ^ Goodman & Wallach 2009 Exercise 12 Chapter 1.
  23. ^ Wybourne, B. G. (1974). Classical Groups for Physicists, Wiley-Interscience. ISBN 0471965057.

References edit

classical, group, book, weyl, classical, groups, mathematics, classical, groups, defined, special, linear, groups, over, reals, complex, numbers, quaternions, together, with, special, automorphism, groups, symmetric, skew, symmetric, bilinear, forms, hermitian. For the book by Weyl see The Classical Groups 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 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 Contents 1 The classical groups 2 Bilinear and sesquilinear forms 2 1 Symmetric skew symmetric Hermitian and skew Hermitian forms 3 Automorphism groups 3 1 Aut f the automorphism group 3 2 Bilinear case 3 2 1 Real case 3 2 1 1 O p q and O n the orthogonal groups 3 2 1 2 Sp m R the real symplectic group 3 2 2 Complex case 3 2 2 1 O n C the complex orthogonal group 3 2 2 2 Sp m C the complex symplectic group 3 3 Sesquilinear case 3 3 1 Complex case 3 3 1 1 U p q and U n the unitary groups 3 3 2 Quaternionic case 3 3 2 1 GL n H and SL n H 3 3 2 2 Sp p q the quaternionic unitary group 3 3 2 3 O 2n O n H quaternionic orthogonal group 4 Classical groups over general fields or algebras 4 1 General and special linear groups 4 2 Unitary groups 4 3 Symplectic groups 4 4 Orthogonal groups 4 5 Notational conventions 5 Contrast with exceptional Lie groups 6 Notes 7 ReferencesThe classical groups editThe classical groups are exactly the general linear groups over R C and H 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 systemSpecial linear SL n R R SO n Complex special linear SL n C C SU n Complex Am n m 1Quaternionic special linear SL n H SU 2n H Sp n Indefinite special orthogonal SO p q R Symmetric S O p O q Complex special orthogonal SO n C C Symmetric SO n Complex B m n 2 m 1 D m n 2 m displaystyle color Blue begin cases B m amp n 2m 1 D m amp n 2m end cases nbsp Symplectic Sp n R R Skew symmetric U n Complex symplectic Sp n C C Skew symmetric Sp n Complex Cm n 2m Indefinite special unitary SU p q C Hermitian S U p U q Indefinite quaternionic unitary Sp p q H Hermitian Sp p Sp q Quaternionic orthogonal SO 2n H Skew Hermitian SO 2n The complex classical groups are SL n C SO n C and Sp n C 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 The complex linear algebraic groups SL n C SO n C and Sp n C together with their real forms 7 For instance SO 2n is a real form of SO 2n C SU p q is a real form of SL n C and SL n H is a real form of SL 2n C 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 editMain articles Bilinear form and Sesquilinear form The classical groups are defined in terms of forms defined on Rn Cn and Hn 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 f V V F on some finite dimensional right vector space over F R C or H is bilinear if f x a y b a f x y b x y V a b F displaystyle varphi x alpha y beta alpha varphi x y beta quad forall x y in V forall alpha beta in F nbsp and if f x 1 x 2 y 1 y 2 f x 1 y 1 f x 1 y 2 f x 2 y 1 f x 2 y 2 x 1 x 2 y 1 y 2 V displaystyle varphi x 1 x 2 y 1 y 2 varphi x 1 y 1 varphi x 1 y 2 varphi x 2 y 1 varphi x 2 y 2 quad forall x 1 x 2 y 1 y 2 in V nbsp It is called sesquilinear if f x a y b a f x y b x y V a b F displaystyle varphi x alpha y beta bar alpha varphi x y beta quad forall x y in V forall alpha beta in F nbsp and if f x 1 x 2 y 1 y 2 f x 1 y 1 f x 1 y 2 f x 2 y 1 f x 2 y 2 x 1 x 2 y 1 y 2 V displaystyle varphi x 1 x 2 y 1 y 2 varphi x 1 y 1 varphi x 1 y 2 varphi x 2 y 1 varphi x 2 y 2 quad forall x 1 x 2 y 1 y 2 in V nbsp These conventions are chosen because they work in all cases considered An automorphism of f is a map A in the set of linear operators on V such that f A x A y f x y x y V displaystyle varphi Ax Ay varphi x y quad forall x y in V nbsp 1 The set of all automorphisms of f form a group it is called the automorphism group of f denoted Aut f 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 R C or H 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 Symmetric skew symmetric Hermitian and skew Hermitian forms edit A form is symmetric if f x y f y x displaystyle varphi x y varphi y x nbsp It is skew symmetric if f x y f y x displaystyle varphi x y varphi y x nbsp It is Hermitian if f x y f y x displaystyle varphi x y overline varphi y x nbsp Finally it is skew Hermitian if f x y f y x displaystyle varphi x y overline varphi y x nbsp A bilinear form f is uniquely a sum of a symmetric form and a skew symmetric form A transformation preserving f 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 Bilinear symmetric form in pseudo orthonormal basis f x y 3 1 h 1 3 2 h 2 3 n h n R Bilinear symmetric form in orthonormal basis f x y 3 1 h 1 3 2 h 2 3 n h n C Bilinear skew symmetric in symplectic basis f x y 3 1 h m 1 3 2 h m 2 3 m h 2 m n 3 m 1 h 1 3 m 2 h 2 3 2 m n h m R C Sesquilinear Hermitian f x y 3 1 h 1 3 2 h 2 3 n h n C H Sesquilinear skew Hermitian f x y 3 1 j h 1 3 2 j h 2 3 n j h n H displaystyle begin aligned text Bilinear symmetric form in pseudo orthonormal basis quad varphi x y amp pm xi 1 eta 1 pm xi 2 eta 2 pm cdots pm xi n eta n amp amp mathbf R text Bilinear symmetric form in orthonormal basis quad varphi x y amp xi 1 eta 1 xi 2 eta 2 cdots xi n eta n amp amp mathbf C text Bilinear skew symmetric in symplectic basis quad varphi x y amp xi 1 eta m 1 xi 2 eta m 2 cdots xi m eta 2m n amp xi m 1 eta 1 xi m 2 eta 2 cdots xi 2m n eta m amp amp mathbf R mathbf C text Sesquilinear Hermitian quad varphi x y amp pm bar xi 1 eta 1 pm bar xi 2 eta 2 pm cdots pm bar xi n eta n amp amp mathbf C mathbf H text Sesquilinear skew Hermitian quad varphi x y amp bar xi 1 mathbf j eta 1 bar xi 2 mathbf j eta 2 cdots bar xi n mathbf j eta n amp amp mathbf H end aligned nbsp 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 amp 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 Automorphism groups edit nbsp 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 R C and H Aut f the automorphism group edit Assume that f 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 A u t f A G L V f A x A y f x y x y V displaystyle mathrm Aut varphi A in mathrm GL V varphi Ax Ay varphi x y quad forall x y in V nbsp Every A Mn V has an adjoint Af with respect to f defined by f A x y f x A f y x y V displaystyle varphi Ax y varphi x A varphi y qquad x y in V nbsp 2 Using this definition in condition 1 the automorphism group is seen to be given by Aut f A GL V A f A 1 displaystyle operatorname Aut varphi A in operatorname GL V A varphi A 1 nbsp 10 3 Fix a basis for V In terms of this basis put f x y 3 i f i j h j displaystyle varphi x y sum xi i varphi ij eta j nbsp where 3i hj 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 f x y x T F y displaystyle varphi x y x mathrm T Phi y nbsp and A f F 1 A T F displaystyle A varphi Phi 1 A mathrm T Phi nbsp 11 4 from 2 where F is the matrix fij The non degeneracy condition means precisely that F is invertible so the adjoint always exists Aut f expressed with this becomes Aut f A GL V F 1 A T F A 1 displaystyle operatorname Aut varphi left A in operatorname GL V Phi 1 A mathrm T Phi A 1 right nbsp The Lie algebra aut f of the automorphism groups can be written down immediately Abstractly X aut f if and only if e t X f e t X 1 displaystyle e tX varphi e tX 1 nbsp for all t corresponding to the condition in 3 under the exponential mapping of Lie algebras so that a u t f X M n V X f X displaystyle mathfrak aut varphi left X in M n V X varphi X right nbsp or in a basis a u t f X M n V F 1 X T F X displaystyle mathfrak aut varphi left X in M n V Phi 1 X mathrm T Phi X right nbsp 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 f Then using the above result f Xx y f x Xfy f x Xy Thus the Lie algebra can be characterized without reference to a basis or the adjoint as a u t f X M n V f X x y f x X y x y V displaystyle mathfrak aut varphi X in M n V varphi Xx y varphi x Xy quad forall x y in V nbsp The normal form for f will be given for each classical group below From that normal form the matrix F 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 Bilinear case edit When the form is symmetric Aut f is called O f When it is skew symmetric then Aut f is called Sp f 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 Real case edit 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 edit Main articles Orthogonal group and Indefinite orthogonal group If f is symmetric and the vector space is real a basis may be chosen so that f x y 3 1 h 1 3 2 h 2 3 n h n displaystyle varphi x y pm xi 1 eta 1 pm xi 2 eta 2 cdots pm xi n eta n nbsp The number of plus and minus signs is independent of the particular basis 13 In the case V Rn one writes O f 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 F is in this case F I p 0 0 I q I p q displaystyle Phi left begin matrix I p amp 0 0 amp I q end matrix right equiv I p q nbsp after reordering the basis if necessary The adjoint operation 4 then becomes A f I p 0 0 I q A 11 A n n T I p 0 0 I q displaystyle A varphi left begin matrix I p amp 0 0 amp I q end matrix right left begin matrix A 11 amp cdots cdots amp A nn end matrix right mathrm T left begin matrix I p amp 0 0 amp I q end matrix right nbsp 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 o p q X p p Y p q Y T W q q X T X W T W displaystyle mathfrak o p q left left left begin matrix X p times p amp Y p times q Y mathrm T amp W q times q end matrix right right X mathrm T X quad W mathrm T W right nbsp and the group according to 3 is given by O p q g G L n R I p q 1 g T I p q g I displaystyle mathrm O p q g in mathrm GL n mathbb R I p q 1 g mathrm T I p q g I nbsp The groups O p q and O q p are isomorphic through the map O p q O q p g s g s 1 s 0 0 1 0 1 0 1 0 0 displaystyle mathrm O p q rightarrow mathrm O q p quad g rightarrow sigma g sigma 1 quad sigma left begin smallmatrix 0 amp 0 amp cdots amp 1 vdots amp vdots amp ddots amp vdots 0 amp 1 amp cdots amp 0 1 amp 0 amp cdots amp 0 end smallmatrix right nbsp For example the Lie algebra of the Lorentz group could be written as o 3 1 s p a n 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 displaystyle mathfrak o 3 1 mathrm span left left begin smallmatrix 0 amp 1 amp 0 amp 0 1 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 end smallmatrix right left begin smallmatrix 0 amp 0 amp 1 amp 0 0 amp 0 amp 0 amp 0 1 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 end smallmatrix right left begin smallmatrix 0 amp 0 amp 0 amp 0 0 amp 0 amp 1 amp 0 0 amp 1 amp 0 amp 0 0 amp 0 amp 0 amp 0 end smallmatrix right left begin smallmatrix 0 amp 0 amp 0 amp 1 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 1 amp 0 amp 0 amp 0 end smallmatrix right left begin smallmatrix 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 1 0 amp 0 amp 0 amp 0 0 amp 1 amp 0 amp 0 end smallmatrix right left begin smallmatrix 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 1 0 amp 0 amp 1 amp 0 end smallmatrix right right nbsp 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 Sp m R the real symplectic group edit Main article Symplectic group If f is skew symmetric and the vector space is real there is a basis giving f x y 3 1 h m 1 3 2 h m 2 3 m h 2 m n 3 m 1 h 1 3 m 2 h 2 3 2 m n h m displaystyle varphi 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 nbsp where n 2m For Aut f one writes Sp f Sp V In case V Rn R2m one writes Sp m R or Sp 2m R From the normal form one reads off F 0 m I m I m 0 m J m displaystyle Phi left begin matrix 0 m amp I m I m amp 0 m end matrix right J m nbsp By making the ansatz V X Y Z W displaystyle V left begin matrix X amp Y Z amp W end matrix right nbsp where X Y Z W are m dimensional matrices and considering 5 0 m I m I m 0 m X Y Z W T 0 m I m I m 0 m X Y Z W displaystyle left begin matrix 0 m amp I m I m amp 0 m end matrix right left begin matrix X amp Y Z amp W end matrix right mathrm T left begin matrix 0 m amp I m I m amp 0 m end matrix right left begin matrix X amp Y Z amp W end matrix right nbsp one finds the Lie algebra of Sp m R s p m R X M n R J m X X T J m 0 X Y Z X T Y T Y Z T Z displaystyle mathfrak sp m mathbb R X in M n mathbb R J m X X mathrm T J m 0 left left left begin matrix X amp Y Z amp X mathrm T end matrix right right Y mathrm T Y Z mathrm T Z right nbsp and the group is given by S p m R g M n R g T J m g J m displaystyle mathrm Sp m mathbb R g in M n mathbb R g mathrm T J m g J m nbsp Complex case edit 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 edit Main article Complex orthogonal group If case f is symmetric and the vector space is complex a basis f x y 3 1 h 1 3 1 h 1 3 n h n displaystyle varphi x y xi 1 eta 1 xi 1 eta 1 cdots xi n eta n nbsp with only plus signs can be used The automorphism group is in the case of V Cn called O n C The lie algebra is simply a special case of that for o p q o n C s o n C X X T X displaystyle mathfrak o n mathbb C mathfrak so n mathbb C X X mathrm T X nbsp and the group is given by O n C g g T g I n displaystyle mathrm O n mathbb C g g mathrm T g I n nbsp In terms of classification of simple Lie algebras the so n are split into two classes those with n odd with root system Bn and n even with root system Dn Sp m C the complex symplectic group edit Main article Symplectic group For f skew symmetric and the vector space complex the same formula f x y 3 1 h m 1 3 2 h m 2 3 m h 2 m n 3 m 1 h 1 3 m 2 h 2 3 2 m n h m displaystyle varphi 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 nbsp applies as in the real case For Aut f one writes Sp f Sp V In the case V C n C 2 m displaystyle V mathbb C n mathbb C 2m nbsp one writes Sp m C displaystyle mathbb C nbsp or Sp 2m C displaystyle mathbb C nbsp The Lie algebra parallels that of sp m R displaystyle mathbb R nbsp s p m C X M n C J m X X T J m 0 X Y Z X T Y T Y Z T Z displaystyle mathfrak sp m mathbb C X in M n mathbb C J m X X mathrm T J m 0 left left left begin matrix X amp Y Z amp X mathrm T end matrix right right Y mathrm T Y Z mathrm T Z right nbsp and the group is given by S p m C g M n C g T J m g J m displaystyle mathrm Sp m mathbb C g in M n mathbb C g mathrm T J m g J m nbsp Sesquilinear case edit In the sesquilinear case one makes a slightly different approach for the form in terms of a basis f x y 3 i f i j h j displaystyle varphi x y sum bar xi i varphi ij eta j nbsp The other expressions that get modified are f x y x F y A f F 1 A F displaystyle varphi x y x Phi y qquad A varphi Phi 1 A Phi nbsp 14 Aut f A GL V F 1 A F A 1 displaystyle operatorname Aut varphi A in operatorname GL V Phi 1 A Phi A 1 nbsp a u t f X M n V F 1 X F X displaystyle mathfrak aut varphi X in M n V Phi 1 X Phi X nbsp 6 The real case of course provides nothing new The complex and the quaternionic case will be considered below Complex case edit 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 U p q and U n the unitary groups edit Main article Unitary group A non degenerate hermitian form has the normal form f x y 3 1 h 1 3 2 h 2 3 n h n displaystyle varphi x y pm bar xi 1 eta 1 pm bar xi 2 eta 2 cdots pm bar xi n eta n nbsp 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 Cn U p q If q 0 the notation is U n In this case F takes the form F 1 p 0 0 1 q I p q displaystyle Phi left begin matrix 1 p amp 0 0 amp 1 q end matrix right I p q nbsp and the Lie algebra is given by u p q X p p Z p q Z p q T Y q q X T X Y T Y displaystyle mathfrak u p q left left left begin matrix X p times p amp Z p times q overline Z p times q mathrm T amp Y q times q end matrix right right overline X mathrm T X quad overline Y mathrm T Y right nbsp The group is given by U p q g I p q 1 g I p q g I displaystyle mathrm U p q g I p q 1 g I p q g I nbsp where g is a general n x n complex matrix and g displaystyle g nbsp is defined as the conjugate transpose of g what physicists call g displaystyle g dagger nbsp As a comparison a Unitary matrix U n is defined asU n g g g I displaystyle mathrm U n g g g I nbsp We note that U n displaystyle mathrm U n nbsp is the same as U n 0 displaystyle mathrm U n 0 nbsp Quaternionic case edit The space Hn 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 Hn was 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 q a 1 b i c j d k a j b a b b a Q q H a b c d R a b C displaystyle q a mathrm 1 b mathrm i c mathrm j d mathrm k alpha j beta leftrightarrow begin bmatrix alpha amp overline beta beta amp overline alpha end bmatrix Q quad q in mathbb H quad a b c d in mathbb R quad alpha beta in mathbb C nbsp 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 aa bb 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 ai and the lower n the bi then a quaternionic n n matrix becomes a complex 2n 2n matrix exactly of the form given above but now with a and b n n matrices More formally Q n n X n n j Y n n X Y Y X 2 n 2 n displaystyle left Q right n times n left X right n times n mathrm j left Y right n times n leftrightarrow left begin matrix X amp bar Y Y amp bar X end matrix right 2n times 2n nbsp 8 A matrix T GL 2n C has the form displayed in 8 if and only if JnT TJn With these identifications H n C 2 n M n H T M 2 n C J n T T J n J n 0 I n I n 0 displaystyle mathbb H n approx mathbb C 2n M n mathbb H approx left left T in M 2n mathbb C right J n T overline T J n quad J n left begin matrix 0 amp I n I n amp 0 end matrix right right nbsp The space Mn H M2n C is a real algebra but it is not a complex subspace of M2n C Multiplication from the left by i in Mn H using entry wise quaternionic multiplication and then mapping to the image in M2n C yields a different result than multiplying entry wise by i directly in M2n 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 M2n 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 Mn H is embedded in M2n 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 GL n H and SL n H edit Under the identification above G L n H g G L 2 n C J g g J d e t g 0 U 2 n displaystyle mathrm GL n mathbb H g in mathrm GL 2n mathbb C Jg overline g J mathrm det quad g neq 0 equiv mathrm U 2n nbsp Its Lie algebra gl n H is the set of all matrices in the image of the mapping Mn H M2n C of above g l n H X Y Y X X Y g l n C u 2 n displaystyle mathfrak gl n mathbb H left left left begin matrix X amp overline Y Y amp overline X end matrix right right X Y in mathfrak gl n mathbb C right equiv mathfrak u 2n nbsp The quaternionic special linear group is given by S L n H g G L n H d e t g 1 S U 2 n displaystyle mathrm SL n mathbb H g in mathrm GL n mathbb H mathrm det g 1 equiv mathrm SU 2n nbsp where the determinant is taken on the matrices in C2n Alternatively one can define this as the kernel of the Dieudonne determinant G L n H H H H R gt 0 displaystyle mathrm GL n mathbb H rightarrow mathbb H mathbb H mathbb H simeq mathbb R gt 0 nbsp The Lie algebra is s l n H X Y Y X R e Tr X 0 s u 2 n displaystyle mathfrak sl n mathbb H left left left begin matrix X amp overline Y Y amp overline X end matrix right right Re operatorname Tr X 0 right equiv mathfrak su 2n nbsp Sp p q the quaternionic unitary group edit As above in the complex case the normal form is f x y 3 1 h 1 3 2 h 2 3 n h n displaystyle varphi x y pm bar xi 1 eta 1 pm bar xi 2 eta 2 cdots pm bar xi n eta n nbsp and the number of plus signs is independent of basis When V Hn with this form Sp f 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 F I p 0 0 I q I p q displaystyle Phi begin pmatrix I p amp 0 0 amp I q end pmatrix I p q nbsp meaning that quaternionic matrices of the form Q X p p Z p q Z Y q q X X Y Y displaystyle mathcal Q begin pmatrix mathcal X p times p amp mathcal Z p times q mathcal Z amp mathcal Y q times q end pmatrix quad mathcal X mathcal X quad mathcal Y mathcal Y nbsp 9 will satisfy F 1 Q F Q displaystyle Phi 1 mathcal Q Phi mathcal Q nbsp 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 X X 1 p p X 2 X 2 X 1 Y Y 1 q q Y 2 Y 2 Y 1 Z Z 1 p q Z 2 Z 2 Z 1 displaystyle mathcal X begin pmatrix X 1 p times p amp overline X 2 X 2 amp overline X 1 end pmatrix quad mathcal Y begin pmatrix Y 1 q times q amp overline Y 2 Y 2 amp overline Y 1 end pmatrix quad mathcal Z begin pmatrix Z 1 p times q amp overline Z 2 Z 2 amp overline Z 1 end pmatrix nbsp and the relations in 9 will be satisfied if X 1 X 1 Y 1 Y 1 displaystyle X 1 X 1 quad Y 1 Y 1 nbsp The Lie algebra becomes s p p q X 1 p p X 2 X 2 X 1 Z 1 p q Z 2 Z 2 Z 1 Z 1 p q Z 2 Z 2 Z 1 Y 1 q q Y 2 Y 2 Y 1 X 1 X 1 Y 1 Y 1 displaystyle mathfrak sp p q left left begin pmatrix begin bmatrix X 1 p times p amp overline X 2 X 2 amp overline X 1 end bmatrix amp begin bmatrix Z 1 p times q amp overline Z 2 Z 2 amp overline Z 1 end bmatrix begin bmatrix Z 1 p times q amp overline Z 2 Z 2 amp overline Z 1 end bmatrix amp begin bmatrix Y 1 q times q amp overline Y 2 Y 2 amp overline Y 1 end bmatrix end pmatrix right X 1 X 1 quad Y 1 Y 1 right nbsp The group is given by S p p q g G L n H I p q 1 g I p q g I p q g G L 2 n C K p q 1 g K p q g I 2 p q K diag I p q I p q displaystyle mathrm Sp p q left g in mathrm GL n mathbb H mid I p q 1 g I p q g I p q right left g in mathrm GL 2n mathbb C mid K p q 1 g K p q g I 2 p q quad K operatorname diag left I p q I p q right right nbsp Returning to the normal form of f w z for Sp p q make the substitutions w u jv and z x jy with u v x y Cn Then f w z u v K p q x y j u v K p q y x f 1 w z j f 2 w z K p q d i a g I p q I p q displaystyle varphi w z begin bmatrix u amp v end bmatrix K p q begin bmatrix x y end bmatrix j begin bmatrix u amp v end bmatrix K p q begin bmatrix y x end bmatrix varphi 1 w z mathbf j varphi 2 w z quad K p q mathrm diag left I p q I p q right nbsp viewed as a H valued form on C2n 19 Thus the elements of Sp p q viewed as linear transformations of C2n 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 S p p q U C 2 n f 1 S p C 2 n f 2 displaystyle mathrm Sp p q mathrm U left mathbb C 2n varphi 1 right cap mathrm Sp left mathbb C 2n varphi 2 right nbsp and this explains both the name of the group and the notation O 2n O n H quaternionic orthogonal group edit The normal form for a skew hermitian form is given by f x y 3 1 j h 1 3 2 j h 2 3 n j h n displaystyle varphi x y bar xi 1 mathbf j eta 1 bar xi 2 mathbf j eta 2 cdots bar xi n mathbf j eta n nbsp where j is the third basis quaternion in the ordered listing 1 i j k In this case Aut f 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 F j 0 0 0 j 0 0 j j n displaystyle Phi left begin smallmatrix mathbf j amp 0 amp cdots amp 0 0 amp mathbf j amp cdots amp vdots vdots amp amp ddots amp amp 0 amp cdots amp 0 amp mathbf j end smallmatrix right equiv mathrm j n nbsp and from 6 follows that F V F V V j n V j n displaystyle Phi V Phi V Leftrightarrow V mathrm j n V mathrm j n nbsp 9 for V o 2n Now put V X j Y X Y Y X displaystyle V X mathbf j Y leftrightarrow left begin matrix X amp overline Y Y amp overline X end matrix right nbsp according to prescription 8 The same prescription yields for F F 0 I n I n 0 J n displaystyle Phi leftrightarrow left begin matrix 0 amp I n I n amp 0 end matrix right equiv J n nbsp Now the last condition in 9 in complex notation reads X Y Y X 0 I n I n 0 X Y Y X 0 I n I n 0 X T X Y T Y displaystyle left begin matrix X amp overline Y Y amp overline X end matrix right left begin matrix 0 amp I n I n amp 0 end matrix right left begin matrix X amp overline Y Y amp overline X end matrix right left begin matrix 0 amp I n I n amp 0 end matrix right Leftrightarrow X mathrm T X quad overline Y mathrm T Y nbsp The Lie algebra becomes o 2 n X Y Y X X T X Y T Y displaystyle mathfrak o 2n left left left begin matrix X amp overline Y Y amp overline X end matrix right right X mathrm T X quad overline Y mathrm T Y right nbsp and the group is given by O 2 n g G L n H j n 1 g j n g I n g G L 2 n C J n 1 g J n g I 2 n displaystyle mathrm O 2n left g in mathrm GL n mathbb H mid mathrm j n 1 g mathrm j n g I n right left g in mathrm GL 2n mathbb C mid J n 1 g J n g I 2n right nbsp The group SO 2n can be characterized as O 2 n g O 2 n C 8 g g displaystyle mathrm O 2n left g in mathrm O 2n mathbb C mid theta left overline g right g right nbsp 21 where the map 8 GL 2n C GL 2n C is defined by g J2ngJ2n Also the form determining the group can be viewed as a H valued form on C2n 22 Make the substitutions x w1 iw2 and y z1 iz2 in the expression for the form Then f x y w 2 I n z 1 w 1 I n z 2 j w 1 I n z 1 w 2 I n z 2 f 1 w z j f 2 w z displaystyle varphi x y overline w 2 I n z 1 overline w 1 I n z 2 mathbf j w 1 I n z 1 w 2 I n z 2 overline varphi 1 w z mathbf j varphi 2 w z nbsp The form f1 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 f2 is symmetric positive definite Thus due to the factor j O 2n preserves both separately and it may be concluded that O 2 n O 2 n C U C 2 n f 1 displaystyle mathrm O 2n mathrm O 2n mathbb C cap mathrm U left mathbb C 2n varphi 1 right nbsp and the notation O is explained Classical groups over general fields or algebras editClassical 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 edit 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 clarification needed 2 or 3 Unitary groups edit 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 edit 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 edit 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 Wn R consisting of the elements of the orthogonal group of elements of spinor norm 1 with corresponding subgroup and quotient groups SWn R PWn R PSWn R For positive definite quadratic forms over the reals the group W happens to be the same as the orthogonal group but in general it is smaller There is also a double cover of Wn 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 edit Further information Group of Lie type Notation issuesContrast with exceptional Lie groups editContrasting with the classical Lie groups are the exceptional Lie groups G2 F4 E6 E7 E8 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 Elie Cartan Notes edit Here special means the subgroup of the full automorphism group whose elements have determinant 1 Rossmann 2002 p 94 Weyl 1939 Rossmann 2002 p 91 Rossmann 2002 p 94 Rossmann 2002 p 103 Goodman amp Wallach 2009 See end of chapter 1 Rossmann 2002p 93 Rossmann 2002 p 105 Rossmann 2002 p 91 Rossmann 2002 p 92 Rossmann 2002 p 105 Rossmann 2002 p 107 Rossmann 2002 p 93 Rossmann 2002 p 95 Rossmann 2002 p 94 Goodman amp Wallach 2009 Exercise 14 Section 1 1 Rossmann 2002 p 94 Goodman amp Wallach 2009Exercise 11 Chapter 1 Rossmann 2002 p 94 Goodman amp Wallach 2009 p 11 Goodman amp Wallach 2009 Exercise 12 Chapter 1 Wybourne B G 1974 Classical Groups for Physicists Wiley Interscience ISBN 0471965057 References editE Artin 1957 Geometric Algebra chapters III IV amp V via Internet Archive Dieudonne Jean 1955 La geometrie des groupes classiques Ergebnisse der Mathematik und ihrer Grenzgebiete N F Heft 5 Berlin New York Springer Verlag ISBN 978 0 387 05391 2 MR 0072144 Goodman Roe Wallach Nolan R 2009 Symmetry Representations and Invariants Graduate texts in mathematics vol 255 Springer Verlag ISBN 978 0 387 79851 6 Knapp A W 2002 Lie groups beyond an introduction Progress in Mathematics Vol 120 2nd ed Boston Basel Berlin Birkhauser ISBN 0 8176 4259 5 V L Popov 2001 1994 Classical group Encyclopedia of Mathematics EMS Press Rossmann Wulf 2002 Lie Groups An Introduction Through Linear Groups Oxford Graduate Texts in Mathematics Oxford Science Publications ISBN 0 19 859683 9 Weyl Hermann 1939 The Classical Groups Their Invariants and Representations Princeton University Press ISBN 978 0 691 05756 9 MR 0000255 Retrieved from https en wikipedia org w index php title Classical group amp oldid 1150578908, wikipedia, wiki, book, books, library,

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