Help in writing the basis for Clifford algebra $Cl^2 (W)$ and the quotient $Cl^3 (W) / Cl^2 (W)$

Question

Consider a basis $(c_1 ^ {\dagger}, c_2 ^ {\dagger}, c_1 ^ {\dagger}, c_1, c_2, c_3 )$ of creation and annihilation operators for $W=V \oplus V^*$.

I need help to write the basis for Clifford algebra $Cl^2 (V \oplus V^*)$=$Cl^2 (W)$. By definition, $$ Cl^2 (V \oplus V^*) := \bigoplus_{l =0 }^2 T^l(W)/I, $$ where $I$ is two sided ideal generated by $(xy+yx)-b(x,y).1$ where $b$ is the canonical bilinear form attached to $W.$

I wrote $$ Cl^2 (V \oplus V^*) = \mathbb{C}/I \oplus W / I \oplus W \otimes W/I. $$

Is this correct?

What is the basis for $Cl^2 (V \oplus V^*)$? Also, how can I write basis for quotient space $Cl^3(W)/Cl^2(W)$? For the latter, I am assuming that it is enough to understand the basis for $Cl^3(W)$. Then I can simply consider the coset of each basis element. Right?

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Added: Here is the context of the question:

Let $V$ be a 3-dimensional vector space, $W=V\oplus V^*$ and $C(W)$ be the Clifford algebra defined by $W$ and its symmetric form. Take the basis $(c_1 ^ {\dagger}, c_2 ^ {\dagger}, c_1 ^ {\dagger}, c_1, c_2, c_3 )$ of creation and annihilation operators for $W$. The Lie algebra $\mathfrak{so}(6)$ is embedded as a Lie subalgebra $\mathfrak{s}$ in $C(W)$ by sending a matrix $$\begin{pmatrix} A & B \\ C & -A^t \end{pmatrix}$$ with $B,C$ anti-symmetric to the element $$ \sum_{i,j} A_{i,j} (c_i^{\dagger}c_j-c_jc_i ^{\dagger})+B_{i,j}c_i ^{\dagger}c_j^{\dagger} +C_{i,j}c_ic_j. $$

1) First thing is to explain why $\mathfrak{s}$ is represented on the quotient vector space $C^3 / C^2$. I have explained why $\mathfrak{so}(6)$ acts on $C^3/C^2$. But I am unable to explain why $\mathfrak{s}$ acts on $C^3/C^2.$

2) Secondly, I need to choose a basis of this quotient such that matrices with $A=Diag(\lambda_1,\lambda_2,\lambda_3)$ and $B=C=0$ are diagonalized in this action. I don't get what I am supposed to do in this part.

The reason I asked the original question was to understand how does the basis of $C^2$ and $C^3/C^2$ look like because so far I don't understand this. If you are willing to help with actual questions 1) and 2), that is fine too.

This post imported from StackExchange Physics at 2014-12-28 09:00 (UTC), posted by SE-user monomorphic

Comments

Interesting question!

Could you give more info on why/where such a construction makes sense. What is the role of the creation/annihilation operators?