Consider a basis (c†1,c†2,c†1,c1,c2,c3) of creation and annihilation operators for W=V⊕V∗.
I need help to write the basis for Clifford algebra Cl2(V⊕V∗)=Cl2(W). By definition, Cl2(V⊕V∗):=2⨁l=0Tl(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 Cl2(V⊕V∗)=C/I⊕W/I⊕W⊗W/I.
Is this correct?
What is the basis for Cl2(V⊕V∗)? Also, how can I write basis for quotient space Cl3(W)/Cl2(W)? For the latter, I am assuming that it is enough to understand the basis for Cl3(W). Then I can simply consider the coset of each basis element. Right?
Added: Here is the context of the question:
Let V be a 3-dimensional vector space, W=V⊕V∗ and C(W) be the Clifford algebra defined by W and its symmetric form. Take the basis (c†1,c†2,c†1,c1,c2,c3) of creation and annihilation operators for W. The Lie algebra so(6) is embedded as a Lie subalgebra s in C(W) by sending a matrix (ABC−At)
with
B,C anti-symmetric to the element
∑i,jAi,j(c†icj−cjc†i)+Bi,jc†ic†j+Ci,jcicj.
1) First thing is to explain why s is represented on the quotient vector space C3/C2. I have explained why so(6) acts on C3/C2. But I am unable to explain why s acts on C3/C2.
2) Secondly, I need to choose a basis of this quotient such that matrices with A=Diag(λ1,λ2,λ3) 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 C2 and C3/C2 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