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Local Spin lift from the matrix logarithm

Statement

Let $(V, B)$ be a finite-dimensional nondegenerate real quadratic space, let $J = diag (ϵ_{1}, \dots, ϵ_{n})$ be its matrix in an orthogonal anisotropic basis with $ϵ_{i} = Q (e_{i}) \in {1, - 1}$ , and let $S O (V, B)$ have the subspace topology from matrices. There is a neighborhood $U$ of the identity in $S O (V, B)$ and a continuous map $s : U \to {Spin}_{N = 1} (V, Q)$ with $s (I) = 1$ and $ρ (s (A)) = A$ .

Facts & Assumptions

Given: The Clifford relation, group conventions, and PBW theorem cited below.

[F1]

$e_{i} e_{j} + e_{j} e_{i} = 2 B (e_{i}, e_{j}) 1$ and $e_{i}^{2} = Q (e_{i}) 1$ (Universal Clifford algebra of a quadratic space, Polarized relations in a Clifford algebra).

[F2]

The orthogonal square-free monomials form a basis; in particular the bivectors $e_{i} e_{j}$ are linearly independent (PBW basis and exterior associated graded for a Clifford algebra).

[F3]

$α$ , reversion, Clifford conjugation and $N (x) = x \overset{ˉ}{x}$ have the conventions used here (Clifford involutions and the norm of a vector product).

[F4]

$Γ$ , $ρ$ , and ${Spin}_{N = 1}$ are defined as in Real Clifford group, Pin, and the two Spin subgroups, and the norm-one product characterization is in Normalized Pin products and Clifford norm.

Proof

technique · direct

1.1givenconstructalgebra

If $n = 0$ , set $U = {I_{0}}$ , $L = 0$ , and $s (I_{0}) = 1$ ; this is a continuous section with norm one. For $n > 0$ , choose a submultiplicative matrix norm and shrink a neighborhood $U$ of $I$ in $S O (V, B)$ until $∥ A - I ∥$ , $∥ A^{T} - I ∥$ , and $∥ A^{- 1} - I ∥$ are all less than $1$ . Then the series $L = \log A = \sum_{m \geq 1} (- 1)^{m + 1} (A - I)^{m} / m$ and the corresponding series for $A^{T}$ and $A^{- 1}$ converge absolutely and continuously. The scalar power-series identities, by absolute Cauchy products, give $\exp L = A$ , $\log (A^{T}) = L^{T}$ , and $\log (A^{- 1}) = - L$ . From $A^{T} J A = J$ one gets $J A J = A^{- T}$ , so conjugating the logarithm series gives $J L J = - L^{T}$ , equivalently $L^{T} J + J L = 0$ .

2.1F1F2step 1.1algebra

For $i < j$ , direct use of [F1] gives $[e_{i} e_{j}, e_{k}] = 2 (B (e_{j}, e_{k}) e_{i} - B (e_{i}, e_{k}) e_{j})$ . The $n (n - 1) / 2$ bivectors therefore map under $b \mapsto {ad}_{b} ∣_{V}$ to a basis of the matrices $L$ satisfying $L^{T} J + J L = 0$ : each coefficient $c_{i j}$ independently gives entries $L_{i j} = 2 c_{i j} ϵ_{j}$ and $L_{j i} = - 2 c_{i j} ϵ_{i}$ . Thus there is a unique bivector $b (L) = \sum_{i < j} c_{i j} e_{i} e_{j}$ with $[b (L), v] = L v$ for every $v \in V$ , and it depends linearly and continuously on $L$ .

3.1F3F4step 2.1construct

Set $s (A) = \exp (b (L))$ in the finite-dimensional Clifford algebra. Since $b (L)$ is even, $α (s (A)) = s (A)$ . The convergent commutator series gives $s (A) v s (A)^{- 1} = \exp ({ad}_{b (L)}) (v) = \exp (L) v = A v$ , so $s (A) \in Γ$ and $ρ (s (A)) = A$ .

4.1F3F4step 3.1algebra∎

Reversion negates each bivector, hence Clifford conjugation sends $b (L)$ to $- b (L)$ and $\overline{\exp (b (L))} = \exp (- b (L))$ . Therefore $N (s (A)) = \exp (b (L)) \exp (- b (L)) = 1$ . By [F4], $s (A) \in {Spin}_{N = 1}$ . Continuity of matrix logarithm on $U$ , of $b (L)$ , and of the finite-dimensional algebra exponential proves continuity of $s$ , with $s (I) = 1$ .

Remarks

This explicit local section proves the infinitesimal-to-local group lift needed for a topological covering; the abstract exact sequence alone would not do so.

Depends on

Used by

Dependency tree · two levels

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Sources

Eckhard Meinrenken, Lie groups and Clifford algebras, Chapter 2 §4.1 (standard reference, not scraped)
Jean Gallier, Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions: The Pin and Spin Groups, §§1.5 and 1.8 (standard reference, not scraped)