What does the notation $[,]$, $\{,\}$ in the Clifford algebra mean?

Question

From Charbonneau, Harland: Deformations of nearly Kähler instantons:

It is explained in the previous paragraph that the authors use the canonical identification $\operatorname{Cl}(V,g)=\Lambda^*V$.

Question: What do the symbols $[,]$, $\{,\}$, and $\lrcorner$ mean?

I guess that $\alpha \lrcorner \beta$ means interior product of the vector $\alpha* \in V$ and $\beta$, where $\alpha*$ is defined via the bilinear form on $V$. I further thought that $[\alpha,\beta]$ might mean $\alpha \beta- \beta \alpha$ (multiplication in the Clifford algebra), but then the first claim isn't true.

For $\{,\}$ I have no clue what it could mean.

Answer 1

The symbols mean:

• $[\alpha,\beta]=\alpha \beta - \beta \alpha$
• $\{ \alpha,\beta \} = \alpha \beta + \beta \alpha$
• $\alpha \lrcorner \beta = \alpha^\# \lrcorner \beta$, where $\alpha ^\#$ denotes the dual vector of $\alpha$ with respect to the metric $g$ on $V$

The notation $\lrcorner$ appears in this context in Baum, Friedrich: Twistor and Killing spinors on Riemannian manifolds, p.15. (It is not used in exactly the same way there, but it strongly suggests that the interpretation given above is correct) $[,]$ and $\{,\}$ follows from this. As an example, check $[e_1,e_1e_2]=e_1e_1e_2-e_1e_2e_1=2e_2=2 e_1 \lrcorner e_1e_2$.