In mathematics, the indefinite orthogonal group is the Lie group of all linear transformations-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature. It is also called the pseudo-orthogonal group or generalized orthogonal group..

The indefinite special orthogonal group is the subgroup consisting of all elements with determinant is not connected – it has 2 components – and there are two additional finite index, which has 2 components – see for definition and discussion.

The signature of the form determines the group up to isomorphism equals zero, then the group is isomorphic to the ordinary orthogonal group are positive.

is defined for vector spaces over the reals. On complex spaces, all nondegenerate symmetric bilinear forms are the same up to change of coordinates; however, one can define the indefinite unitary group which preserves a sesquilinear form.

is known as the split orthogonal group.

Examples

The basic example is the squeeze mappings of (the identity component of) linear transforms preserving the unit hyperbola. Concretely, these are the matrices and can be interpreted as hyperbolic rotations, can be interpreted as circular rotations.

In physics, the Lorentz group is of central importance, being the setting for electromagnetism and special relativity is prevalent in quantum field theory because the geometric properties of the Dirac equation.)

Matrix definition

as a group of matrices, just as for the classical orthogonal group diagonal matrix Then we may define a symmetric bilinear form is the standard inner product.

matrices that preserve this bilinear form:

such that is the transpose.

with any symmetric matrix positive eigenvalues negative ones. Diagonalizing.

Subgroups

as a block matrix

are both positive.

. have the same sign.

Topology

are connected is the Klein four-group, each of which either preserves both orientations or reverses both orientations, in either case preserving the overall orientation.

The identity component for the orthochronous Lorentz group refers to preserving the orientation on the first (temporal) dimension.

is also not compact is a maximal compact subgroup. 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, and is given by:

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\mathbb{Z}\times C_2

C_2\times C_2

|}

Split orthogonal group

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 (the Lie group of the split real form of the Lie algebra); more precisely, the identity component, which is the compact real form of the complex Lie algebra.

may be identified with the unit hyperbola group, a subgroup of the group of units in 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 also

• Orthogonal group

• Lorentz group

• Poincaré group

• Symmetric bilinear form

References

Sources

• Anthony Knapp, Lie Groups Beyond an Introduction, Second Edition, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 2002. – see page 372 for a description of the indefinite orthogonal group

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