Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 671whereC i (V ) = C 0 (V ) ⊕ C 1 (V ) ⊕ C 2 (v) ⊕ . . . ⊕ C i (V )and C 0 (V ) = R. It follows thatC i (V ) = span{ v 1 . . . v k | v j ∈ V ↩→ C(V ) for j = 1, . . . , k ≤ i }.The Clifford algebra C(V ) with this filtration is called the associatedgraded algebra of C(V ) and is denoted by gr C(V ). The ith grading ofgr C(V ) is denoted by gr i C(V ).The associated graded algebra gr C(V ) is naturally isomorphic tothe exterior algebra ΛV , where the isomorphism is given by sendinggr i (v 1 . . . v i ) ∈ gr i C(V ) to v 1 Λ . . . Λv i ∈ Λ i V . The symbol map σ extendsthe symbol mapσ i : C i (V ) −→ gr i C(V ) ∼ = Λ i V,in the sense that if a ∈ C i (V ), then σ(a) [i] = σ i (a). The filtration C i (V )may be written asC i (V ) =i∑q(Λ i V ).j=0Hence the Clifford algebra C(V ) may be identified with the exterior algebraΛV with a twisted, or quantized multiplication α ·Qβ.If v ∈ V ↩→ C(V ) and a ∈ C + (V ), then σ( [v, a] ) = −2 ι(v) σ(a).The space C 2 (V ) = q(Λ 2 V ) is a Lie subalgebra of C(V ), where theLie bracket is just the commutator in C(V ). It is isomorphic to the Liealgebra (V ), under the map τ : C 2 (V ) −→ so(V ), where any a ∈ C 2 (V ) ismapped into τ(a) which acts on any v ∈ V ∼ = C 1 (V ) by the adjoint action:τ(a) v = [ a , v ]. Here the bracket is the bracket of the Lie super–algebraC(V ), i.e.,[ a 1 , a 2 ] = a 1 a 2 −(−1) |a1| |a2| a 2 a 1 , (for a 1 ∈ C |a1| (V ), a 2 ∈ C |a2| (V )).It satisfies the following Axioms of Lie super–algebra:Supercommutativity [ a 1 , a 2 ] + (−1) |a1| |a2| [ a 2 , a 1 ] = 0, andJacobi’s identity[ a 1 , [a 2 , a 3 ] ] = [ [a 1 , a 2 ] , a 3 ] + (−1) |a1| |a2| [ a 2 , [a 1 , a 3 ] ].