Indefinite orthogonal group

Squeeze mappings, here r = 3/2, are the basic hyperbolic symmetries.

The basic example is the squeeze mappings, which is the group SO⁺(1, 1) of (the identity component of) linear transforms preserving the unit hyperbola. Concretely, these are the matrices ${\displaystyle \left[{\begin{smallmatrix}\cosh(\alpha )&\sinh(\alpha )\\\sinh(\alpha )&\cosh(\alpha )\end{smallmatrix}}\right],}$ and can be interpreted as hyperbolic rotations, just as the group SO(2) can be interpreted as circular rotations.

In physics, the Lorentz group O(1,3) is of central importance, being the setting for electromagnetism and special relativity. (Some texts use O(3,1) for the Lorentz group; however, O(1,3) is prevalent in quantum field theory because the geometric properties of the Dirac equation are more natural in O(1,3).)

One can define O(p, q) as a group of matrices, just as for the classical orthogonal group O(n). Consider the ${\displaystyle (p+q)\times (p+q)}$ diagonal matrix ${\displaystyle g}$ given by

${\displaystyle g=\mathrm {diag} (\underbrace {1,\ldots ,1} _{p},\underbrace {-1,\ldots ,-1} _{q}).}$

Then we may define a symmetric bilinear form ${\displaystyle [\cdot ,\cdot ]_{p,q}}$ on ${\displaystyle \mathbb {R} ^{p+q}}$ by the formula

${\displaystyle [x,y]_{p,q}=\langle x,gy\rangle =x_{1}y_{1}+\cdots +x_{p}y_{p}-x_{p+1}y_{p+1}-\cdots -x_{p+q}y_{p+q}}$,

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ is the standard inner product on ${\displaystyle \mathbb {R} ^{p+q}}$.

We then define ${\displaystyle \mathrm {O} (p,q)}$ to be the group of ${\displaystyle (p+q)\times (p+q)}$ matrices that preserve this bilinear form:[1]

${\displaystyle \mathrm {O} (p,q)=\{A\in M_{p+q}(\mathbb {R} )|[Ax,Ay]_{p,q}=[x,y]_{p,q}\,\forall x,y\in \mathbb {R} ^{p+q}\}}$.

More explicitly, ${\displaystyle \mathrm {O} (p,q)}$ consists of matrices ${\displaystyle A}$ such that[2]

${\displaystyle g^{-1}A^{\mathrm {tr} }g=A^{-1}}$,

where ${\displaystyle A^{\mathrm {tr} }}$ is the transpose of ${\displaystyle A}$.

One obtains an isomorphic group (indeed, a conjugate subgroup of GL(p + q)) 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 O(p, q).

Assuming both p and q are positive, neither of the groups O(p, q) nor SO(p, q) are connected, having four and two components respectively. π₀(O(p, q)) ≅ C₂ × C₂ 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 π₀(SO(p, q)) = {(1, 1), (−1, −1)}, each of which either preserves both orientations or reverses both orientations, in either case preserving the overall orientation.

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) that preserve both orientations. This notation is related to the notation O⁺(1, 3) for the orthochronous Lorentz group, where the + refers to preserving the orientation on the first (temporal) dimension.

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, O(p) × O(q) is a maximal compact subgroup of O(p, q), while S(O(p) × O(q)) is a maximal compact subgroup of SO(p, q). Likewise, SO(p) × SO(q) is a maximal compact subgroup of SO⁺(p, q). Thus, the spaces are homotopy equivalent to products of (special) orthogonal groups, from which algebro-topological invariants can be computed. (See https://en.wikipedia.org/wiki/Maximal_compact_subgroup#Topology.)

In particular, the fundamental group of SO⁺(p, q) is the product of the fundamental groups of the components, π₁(SO⁺(p, q)) = π₁(SO(p)) × π₁(SO(q)), 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

In even dimensions, the middle group O(n, n) 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 so_2n (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 O(n) := O(n, 0) = O(0, n), which is the compact real form of the complex Lie algebra.

The case (1, 1) corresponds to the multiplicative group of the 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.

• Orthogonal group
• Lorentz group
• Poincaré group
• Symmetric bilinear form