Generalized orthogonal group

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In mathematics, the generalized orthogonal group, O(p,q) is the Lie group of all linear transformations of a n = p + q dimensional real vector space which leave invariant a nondegenerate, symmetric bilinear form of signature (p, q). The dimension of the group is

n(n − 1)/2.

The generalized special orthogonal group, SO(p,q) is the subgroup of O(p,q) consisting of all elements with determinant 1.

The signature of the metric (p positive and q negative eigenvalues) 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 O(p,q) is defined for vector spaces over the reals. For complex spaces, all groups O(p,q; C) are isomorphic to the usual orthogonal group O(p + q; C).

Contents

1 Matrix definition
2 Topology
3 References
4 See also

Matrix definition

One can define O(p,q) as a group of matrices, just as for the classical orthogonal group O(n). The standard inner product on R^p,q is given in coordinates by the diagonal matrix:

[\eta = \mathrm(\underbrace_,\underbrace_).]

The group O(p,q) is then the group of a n×n matrices M (where n = p+q) such that:

[M^T\eta M = \eta.\,]

Here M^T denotes the transpose of the matrix M. One can easily verify that the set of all such matrices forms a group. The inverse of M is given by

[M^ = \eta^M^T\eta.\,]

One obtains an isomorphic group by replacing η with any symmetric matrix with p positive eigenvalues and q negative ones. Such a matrix is necessarily nonsingular.

Topology

Neither of the groups O(p,q) or SO(p,q) are connected, having 4 and 2 components respectively. The identity component of O(p,q) is often denoted SO⁺(p,q) and can be identified with the set of elements in SO(p,q) which preserve the respective orientations of the p and q dimensional subspaces on which the form is definite.

The group O(p,q) 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, the maximal compact subgroup of O(p,q) is given by O(p)×O(q). Likewise, the maximal compact subgroup of SO⁺(p, q) is SO(p)×SO(q). It follows that the fundamental group of SO⁺(p, q) is given by

[\pi_1(\mbox^(p,q)) = \pi_1(\mbox(p))\times\pi_1(\mbox(q))\;]

or (for p ≥ q):

[\pi_1(\mbox^(p,q)) = \begin\ & p=q=1 \\\mathbb & p=2, q=1 \\\mathbb_2 & p \ge 3, q=1 \\\mathbb\times\mathbb & p=q=2 \\\mathbb\times\mathbb_2 & p \ge 3, q=2 \\\mathbb_2\times\mathbb_2 & p,q \ge 3\end]

References

Anthony Knapp, Lie Groups Beyond an Introduction, Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. ISBN 0-8176-4259-5. (see page 372 for a description of the generalized orthogonal group)

Wolf, Spaces of constant curvature, (1967) page. 335.

Generalized orthogonal group

Matrix definition

Topology

References

See also