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Graded C_n modules and ungraded C_(n−1) modules

Statement

For $n \geq 1$ , taking the even part is an equivalence from finite-dimensional graded $C_{n}$ -modules to finite-dimensional ungraded $C_{n - 1}$ -modules, where $C_{n} = C l (R^{n}, - ∥ \cdot ∥^{2})$ . An inverse is induction from the even subalgebra. For $n = 0$ , retain both parity types and treat the restriction cokernel separately.

Facts & Assumptions

Given: $n \geq 1$ and a finite-dimensional graded left $C_{n}$ -module $M = M^{0} \oplus M^{1}$ or a finite-dimensional ungraded left module $N$ over $C_{n - 1}$ .

[F1]

A graded Clifford module has $C_{n}^{i} M^{j} \subseteq M^{i + j m o d 2}$ (Real graded Clifford-module bundles and their Grothendieck groups).

[F2]

The even subalgebra of the negative-square algebra $C_{n} = {C l}_{0, n}$ is isomorphic to $C_{n - 1} = {C l}_{0, n - 1}$ (Even subalgebras of real and complex Clifford algebras).

[F3]

$C_{n}$ is finite-dimensional and has its even/odd word grading (PBW basis and exterior associated graded for a Clifford algebra).

[F4]

$M_{0}$ retains the even and odd dimension coordinates separately (Clifford-module restriction cokernels A_n).

Proof

technique · explicit induction and evaluation

1.1givenF1F2construct

Let $e_{n}$ be the last standard generator. It is odd and invertible because $e_{n}^{2} = - 1$ . Multiplication by $e_{n}$ maps $C_{n}^{0}$ bijectively onto $C_{n}^{1}$ , with inverse multiplication by $- e_{n}$ ; thus $C_{n}^{1} = e_{n} C_{n}^{0}$ . By [F2], the action of $C_{n}^{0}$ on $M^{0}$ makes it an ungraded $C_{n - 1}$ -module.

2.1F2F3step 1.1construct

Define $G (N) = C_{n} \otimes_{C_{n}^{0}} N$ , with parity inherited from $C_{n}$ and the $C_{n - 1}$ -action transported along the algebra isomorphism in [F2]. Because $C_{n} = C_{n}^{0} \oplus e_{n} C_{n}^{0}$ as a right $C_{n}^{0}$ -module, its even part is canonically $N$ and its odd part is $e_{n} \otimes N$ . Hence $G (N)$ is finite-dimensional and taking its even part returns $N$ .

2.2givenF1step 1.1algebra

For $M$ , define evaluation $ε : C_{n} \otimes_{C_{n}^{0}} M^{0} \to M$ by $a \otimes m \mapsto a m$ . It is well-defined because $a b \otimes m = a \otimes b m$ for $b \in C_{n}^{0}$ . On the even part it is the canonical isomorphism $C_{n}^{0} \otimes_{C_{n}^{0}} M^{0} ≅ M^{0}$ . On the odd part it sends $e_{n} \otimes m$ to $e_{n} m$ ; this is bijective since $e_{n} : M^{0} \to M^{1}$ has inverse $- e_{n}$ . Therefore evaluation is an even graded $C_{n}$ -module isomorphism.

3.1F1F4step 2.1step 2.2algebra∎

The restriction functor $M \mapsto M^{0}$ and induction $N \mapsto C_{n} \otimes_{C_{n}^{0}} N$ are inverse up to the natural isomorphisms in steps 2.1 and 2.2, so they give an equivalence. At $n = 0$ , $C_{0} = R$ is purely even; a one-dimensional module concentrated in odd parity has even part zero, while a one-dimensional even module has even part $R$ . Thus the same equivalence fails at $n = 0$ , and the two parity generators must remain distinct in $M_{0} ≅ Z \oplus Z$ .

Depends on

Used by

Restriction matrices for negative-square graded real Clifford modules Lemma

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. F. Atiyah, R. Bott, and A. Shapiro, Clifford Modules, Topology 3, Supplement 1 (1964) (standard reference, not scraped)
Jost-Hinrich Eschenburg and Bernhard Hanke, Bott–Thom isomorphism, Hopf bundles and Morse theory (2021) (standard reference, not scraped)
M. F. Atiyah, K-Theory (lectures, notes by D. W. Anderson, 1967), with K-Theory and Reality reprint (standard reference, not scraped)