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H ® H viewed as a right coaction.

In the infinite-dimensional case) this should be imposed as an additional condition. 2 If H is finite dimensional then D(H) is a factorisable quasitriangular Hopf algebra, with R. = Y(fa ®1) 0(1 (9 ea), a where {ea} is a basis of H and {fa} is a dual basis. Proof We adopt the convention where repeated indices are summed. Then, to verify axiom 3 of a quasitriangular structure, we need (fa(1) ®1) ®(f a(2) ®1) ®(1® ea) _ (fa ®1) ®(fb ®1) ®(1® eaeb), (fa ®1) ®(1® ea(l)) 0(10 ea(2)) = (fbfa ®1) ®(1® eb) 0(10 ea), which are easily verified by evaluating against general elements: evaluating against 0 E H* in the third factor gives both sides of the first iden- tity as 0(1) 010 (2) ®1®1.

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A Quantum Groups Primer by Shahn Majid

by Kenneth

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