Classical group

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.

[5] These groups are usually additionally restricted to the subgroups whose elements have determinant 1, so that their centers are discrete.

[6] The classical groups can uniformly be characterized in a different way using real forms.

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.

For this reason, for the purposes of classification, only purely symmetric, skew-symmetric, Hermitian, or skew-Hermitian forms are considered.

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).

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 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 ∈ Mn(V) has an adjoint Aφ with respect to φ defined by Using this definition in condition (1), the automorphism group is seen to be given by Fix a basis for V. In terms of this basis, put where ξi, ηj are the components of x, y.

The non-degeneracy condition means precisely that Φ is invertible, so the adjoint always exists.

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 as is seen using the power series expansion of the exponential mapping and the linearity of the involved operations.

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.

Here the "time component" end up as the fourth coordinate in a physical interpretation, and not the first as may be more common.

If case φ is symmetric and the vector space is complex, a basis with only plus-signs can be used.

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.

For φ skew-symmetric and the vector space complex, the same formula, applies as in the real case.

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

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.

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).

[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.

The quaternionic multiplication rules give i(X + jY) = (iX) + j(−iY) where the new X and Y are inside the parentheses.

The Lie algebra is As above in the complex case, the normal form is and the number of plus-signs is independent of basis.

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 ∈ Cn.

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 ↦ −J2ngJ2n.

For the sake of generality the article will refer to groups over R, where R may be the ground field F itself.

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 symplectic group GSp2n(R) consists of the automorphisms of a module multiplying a skew symmetric form by some invertible scalar.

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.

The general orthogonal group GOn(R) consists of the automorphisms of a module multiplying a quadratic form by some invertible scalar.

[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.