CLIFFORD AND GRASSMANN HOPF ALGEBRAS VIA THE ...

S_Gr:=matrix(2^dim_V,2^dim_V,(i,j)->EV(bas[i],gantipode(bas[j]))):
> BS:=evalm(BW &* S_Gr):
> BW=evalm(BW),S_gr=evalm(S_Gr),BS=evalm(BS);
⎡
⎤
⎡
⎤
⎢ 1 0 0
⎢ 0 b1, 1 b1, 2
BW = ⎢ 0 b2, 1 b2, 2
⎣
0
0
0
⎥ ,
⎥
⎦
⎢ 1 0 0
⎢ 0 −1 0
S gr = ⎢ 0 0 −1
⎣
0 ⎥
0 ⎥ ,
0 ⎥
⎦
0 0 0 b2, 1 b1, 2 − b2, 2 b1, 1
0 0 0 1
⎡
⎤
⎢ 1 0 0
⎢ 0 −b1, 1 −b1, 2
BS = ⎢ 0 −b2, 1 −b2, 2
⎣
0
0
0
⎥
⎦
0 0 0 b2, 1 b1, 2 − b2, 2 b1, 1
Note that the convolutive inverse BS is just the scalar product w.r.t the negative
bilinear form equal to the product of matrices BW and S_gr.
BS
BW
Tangle 2. Convolutive inverse of BW and BS
Proposition 2 Every exponentially generated endomorphism B has a convolutive
inverse B S = B ◦ S = S ◦ B.
A quasi triangular structure is an element R ∈ V ⊗ V which satisfies, among
others, the following condition:
sw ˆ ◦ ∆Cℓ(x) =(R(1) ⊗ R(2)) ◦ ∆Gr(x) (2)
∆Cℓ
sw
=
∆Gr
R
∧ ∧
Tangle 3. Definition of the quasi triangular structure R. Note that R = BS.
see, e.g., [15]. Note that our definition is somewhat different due to our reversed
ordering of tensor products of dual elements. The task is now to define a general
14