Indefinite orthogonal group
Template:Short description In mathematics, the indefinite orthogonal group, Template:Nowrap is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature Template:Nowrap, where Template:Nowrap. It is also called the pseudo-orthogonal group<ref>Template:Harvnb</ref> or generalized orthogonal group.<ref>Template:Harnvb</ref> The dimension of the group is Template:Nowrap.
The indefinite special orthogonal group, Template:Nowrap is the subgroup of Template:Nowrap consisting of all elements with determinant 1. Unlike in the definite case, Template:Nowrap is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected Template:Nowrap and Template:Nowrap, which has 2 components – see Template:Slink for definition and discussion.
The signature of the form determines the group up to isomorphism; interchanging p with q amounts to replacing the metric by its negative, and so gives the same group. If either p or q equals zero, then the group is isomorphic to the ordinary orthogonal group O(n). We assume in what follows that both p and q are positive.
The group Template:Nowrap is defined for vector spaces over the reals. For complex spaces, all groups Template:Nowrap are isomorphic to the usual orthogonal group Template:Nowrap, since the transform <math>z_j \mapsto iz_j</math> changes the signature of a form. This should not be confused with the indefinite unitary group Template:Nowrap which preserves a sesquilinear form of signature Template:Nowrap.
In even dimension Template:Nowrap, Template:Nowrap is known as the split orthogonal group.
ExamplesEdit
The basic example is the squeeze mappings, which is the group Template:Nowrap of (the identity component of) linear transforms preserving the unit hyperbola. Concretely, these are the matrices <math>\left[\begin{smallmatrix} \cosh(\alpha) & \sinh(\alpha) \\ \sinh(\alpha) & \cosh(\alpha) \end{smallmatrix}\right],</math> and can be interpreted as hyperbolic rotations, just as the group SO(2) can be interpreted as circular rotations.
In physics, the Lorentz group Template:Nowrap is of central importance, being the setting for electromagnetism and special relativity. (Some texts use Template:Nowrap for the Lorentz group; however, Template:Nowrap is prevalent in quantum field theory because the geometric properties of the Dirac equation are more natural in Template:Nowrap.)
Matrix definitionEdit
One can define Template:Nowrap as a group of matrices, just as for the classical orthogonal group O(n). Consider the <math>(p+q)\times(p+q)</math> diagonal matrix <math>g</math> given by
- <math>g = \mathrm{diag}(\underbrace{1,\ldots,1}_{p},\underbrace{-1,\ldots,-1}_{q}) .</math>
Then we may define a symmetric bilinear form <math>[\cdot,\cdot]_{p,q}</math> on <math>\mathbb R^{p+q}</math> by the formula
- <math>[x,y]_{p,q}=\langle x,gy\rangle=x_1y_1+\cdots +x_py_p-x_{p+1}y_{p+1}-\cdots -x_{p+q}y_{p+q}</math>,
where <math>\langle\cdot,\cdot\rangle</math> is the standard inner product on <math>\mathbb R^{p+q}</math>.
We then define <math>\mathrm{O}(p,q)</math> to be the group of <math>(p+q)\times(p+q)</math> matrices that preserve this bilinear form:<ref>Template:Harvnb Section 1.2.3</ref>
- <math>\mathrm{O}(p,q)=\{A\in M_{p+q}(\mathbb R)|[Ax,Ay]_{p,q}=[x,y]_{p,q}\,\forall x,y\in\mathbb R^{p+q}\}</math>.
More explicitly, <math>\mathrm{O}(p,q)</math> consists of matrices <math>A</math> such that<ref>Template:Harvnb Chapter 1, Exercise 1</ref>
- <math>gA^{\mathrm{tr}}g = A^{-1}</math>,
where <math>A^{\mathrm{tr}}</math> is the transpose of <math>A</math>.
One obtains an isomorphic group (indeed, a conjugate subgroup of Template:Nowrap) by replacing g with any symmetric matrix with p positive eigenvalues and q negative ones. Diagonalizing this matrix gives a conjugation of this group with the standard group Template:Nowrap.
SubgroupsEdit
The group Template:Nowrap and related subgroups of Template:Nowrap can be described algebraically. Partition a matrix L in Template:Nowrap as a block matrix:
- <math>L = \begin{pmatrix}
A & B \\ C & D \end{pmatrix} </math> where A, B, C, and D are p×p, p×q, q×p, and q×q blocks, respectively. It can be shown that the set of matrices in Template:Nowrap whose upper-left p×p block A has positive determinant is a subgroup. Or, to put it another way, if
- <math>L = \begin{pmatrix}
A & B \\ C & D \end{pmatrix} \;\mathrm{and}\; M = \begin{pmatrix} W & X \\ Y & Z \end{pmatrix}</math> are in Template:Nowrap, then
- <math>(\sgn \det A)(\sgn \det W) = \sgn \det (AW+BY).</math>
The analogous result for the bottom-right q×q block also holds. The subgroup Template:Nowrap consists of matrices L such that det A and det D are both positive.<ref name="lester">Template:Cite journal</ref><ref>Template:Harvnb</ref>
For all matrices L in Template:Nowrap, the determinants of A and D have the property that <math display="inline">\frac{\det A}{\det D} = \det L</math> and that <math>|{\det A}| = |{\det D}| \ge 1.</math><ref>Template:Harvnb</ref> In particular, the subgroup Template:Nowrap consists of matrices L such that det A and det D have the same sign.<ref name="lester" />
TopologyEdit
Assuming both p and q are positive, neither of the groups Template:Nowrap nor Template:Nowrap are connected, having 4 and 2 components respectively. Template:Nowrap is the Klein four-group, with each factor being whether an element preserves or reverses the respective orientations on the p and q dimensional subspaces on which the form is definite; note that reversing orientation on only one of these subspaces reverses orientation on the whole space. The special orthogonal group has components Template:Nowrap}, each of which either preserves both orientations or reverses both orientations, in either case preserving the overall orientation.Template:Clarify
The identity component of Template:Nowrap is often denoted Template:Nowrap and can be identified with the set of elements in Template:Nowrap that preserve both orientations. This notation is related to the notation Template:Nowrap for the orthochronous Lorentz group, where the + refers to preserving the orientation on the first (temporal) dimension.
The group Template:Nowrap is also not compact, but contains the compact subgroups O(p) and O(q) acting on the subspaces on which the form is definite. In fact, Template:Nowrap is a maximal compact subgroup of Template:Nowrap, while Template:Nowrap is a maximal compact subgroup of Template:Nowrap. Likewise, Template:Nowrap is a maximal compact subgroup of Template:Nowrap. Thus, the spaces are homotopy equivalent to products of (special) orthogonal groups, from which algebro-topological invariants can be computed. (See Maximal compact subgroup.)
In particular, the fundamental group of Template:Nowrap is the product of the fundamental groups of the components, Template:Nowrap, and is given by:
π1(SO+(p, q)) p = 1 p = 2 p ≥ 3 q = 1 C1 Z C2 q = 2 Z Z × Z Z × C2 q ≥ 3 C2 C2 × Z C2 × C2
Split orthogonal groupEdit
In even dimensions, the middle group Template:Nowrap is known as the split orthogonal group, and is of particular interest, as it occurs as the group of T-duality transformations in string theory, for example. It is the split Lie group corresponding to the complex Lie algebra so2n (the Lie group of the split real form of the Lie algebra); more precisely, the identity component is the split Lie group, as non-identity components cannot be reconstructed from the Lie algebra. In this sense it is opposite to the definite orthogonal group Template:Nowrap, which is the compact real form of the complex Lie algebra.
The group Template:Nowrap may be identified with the group of unit split-complex numbers.
In terms of being a group of Lie type – i.e., construction of an algebraic group from a Lie algebra – split orthogonal groups are Chevalley groups, while the non-split orthogonal groups require a slightly more complicated construction, and are Steinberg groups.
Split orthogonal groups are used to construct the generalized flag variety over non-algebraically closed fields.
See alsoEdit
ReferencesEdit
SourcesEdit
Template:Sfn whitelist Template:Refbegin
- Template:Citation
- Anthony Knapp, Lie Groups Beyond an Introduction, Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. Template:ISBN – see page 372 for a description of the indefinite orthogonal group
- Template:Springer
- Template:Cite journal
- Joseph A. Wolf, Spaces of constant curvature, (1967) page. 335.