Spin and SO groups associated to a degenerate symmetric bilinear form

2012-07-31T20:56:11Z

In "Spin geometry" by Lawson and Michelsohn it is defined the Clifford algebra $Cl(g)$ associated to a symmetric bilinear form $g$ in general, including the degenerate case. But the rest of the book is devoted exclusively to the non-degenerate case.

Are there any references concerned with the spin group $Spin(g)$, the group $SO(g)$, their representations, corresponding to a degenerate symmetric bilinear form $g$?

Answer by Qiaochu Yuan for Spin and SO groups associated to a degenerate symmetric bilinear form

2012-07-31T22:01:16Z

Some naive comments. Any real vector space $V$ with a symmetric bilinear form $g$ admits an orthogonal direct sum decomposition $V_0 \oplus V_1$ where $V_0$ consists of the vectors $v$ such that $g(v, -) = 0$ and $g$ is nondegenerate on $V_1$ (e.g. by the spectral theorem). An inspection of the defining relation $$\frac{uv + vu}{2} = g(u, v)$$

of the Clifford algebra shows that $\text{Cl}(V, g)$ is the (graded) tensor product $\Lambda(V_0) \otimes \text{Cl}(V_1, g)$. So this is not too bad.

The corresponding special orthogonal group is more complicated; $\text{SO}(g)$ consists of block matrices $$\left[ \begin{array}{cc} A & B \\ 0 & C \end{array} \right]$$

where $A \in \text{GL}(V_0), C \in \text{O}(V_1, g)$, $B$ is an arbitrary linear map $V_1 \to V_0$, and $\det(A) \det(C) = 1$. This does not seem like a very nice group to work with and I have no comment on what the corresponding spin groups might look like.

Spin and SO groups associated to a degenerate symmetric bilinear form - MathOverflow

Spin and SO groups associated to a degenerate symmetric bilinear form

Answer by Qiaochu Yuan for Spin and SO groups associated to a degenerate symmetric bilinear form