Algebraic topology: Proc. conf. Goettingen 1984 by Larry Smith

By Larry Smith

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Example text

One can just take it as a definition even when the Gaussian and themselves are not known. It is worth noting that the quantum metric needed in these constructions can be recovered if one knows the norm element x · x in the braided covector algebra, by partial differentiation. For example, in the setting (104)–(105) we have the identity ηij = ∂ i ∂ j (1 + q−2)−1 x · x. (110) The same idea applies more generally to generate R -symmetric tensors with more indices[52]. 38 It is clear that we have the ingredients now to do most of the constructions of classical scalar field theory in our general braided setting.

Suppose it is true for m − 1, then (a1 + x1 ) · · · (am + xm ) = (a1 + x1) · · · (am−1 + xm−1 )am + (a1 + x1 ) · · · (am−1 + xm−1 )xm m−1 = r=0 m = r=1 a1 · · · ar xr+1 · · · xm−1 am m−1 ;R r a1 · · · ar−1 xr · · · xm−1 am m−1 ;R r−1 m−1 a1 · · · ar xr+1 · · · xm + r=0 1···m−1 + (a1 + x1 ) · · · (am−1 + xm−1 )xm 1···m−1 m−1 ;R r 1···m−1 m a1 · · · ar−1 ar xr+1 · · · xm (P R)r,r+1 · · · (P R)m−1,m = r=1 m−1 + r=0 a1 · · · ar xr+1 · · · xm m a1 · · · ar xr+1 · · · xm = r=1 m ;R r m−1 ;R r 1···m m−1 ;R r−1 1···m−1 1···m−1 + x1 · · · x m m−1 ;R 0 1···m−1 using the induction hypothesis and (84).

Moreover, the braided approach allows us to bosonise all the other braided covector algebras in this paper just as well and obtain their natural induced ‘Poincar´e’ quantum groups and their coactions. For example, when R is Hecke, we can apply the bosonisation to the additive braided groups ¯ A(R), A(R) and B(R) just as well. We just use R, R from (27), (22), (36) respectively. The Poincare groups consist of adjoining the Lorentz generators t I J obtained from A(R). We can also give a spinorial or matrix version using the quantum symmetry in the matrix form, as given ¯ for A(R) and B(R) in (139) and (141) respectively.

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Algebraic topology: Proc. conf. Goettingen 1984 by Larry Smith
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