By Shahn Majid
Here's a self-contained creation to quantum teams as algebraic gadgets. according to the author's lecture notes for the half III natural arithmetic path at Cambridge collage, the e-book is acceptable as a chief textual content for graduate classes in quantum teams or supplementary examining for contemporary classes in complicated algebra. the fabric assumes wisdom of easy and linear algebra. a few familiarity with semisimple Lie algebras might even be useful. the amount is a primer for mathematicians however it may also be important for mathematical physicists.
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Similarly for IF-1 with inverse braid crossings.
Therefore, we can set En = 0. Similarly for Fn = 0. The relation gn = 1 is clearly compatible with A. This completes the proof that uq(sl2) is a Hopf algebra. Finally, we turn to the quasitriangular structure. For the quasicocommutativity axiom 2 on DE, we need R(E ® g + 1® E) = (E ®1 + g,& E)R. But R9(1®E) = (g ®E)R9 by the gE = g2Eg relations and a change of variable in the sum for R,. Similarly for E ®1 since Rg is 38 6 Roots of unity. uq(sl2) symmetric. Hence (E®1+g®E)Rg =Rg(E®g-1+1®E). In view of this, we require eq 2 (B + C) = (C + D)eq 2, where A=(q-q-1)E®F,B=E®g,C=1®E,D=E®g-1.
Iii) Next, we define S-1(h) = u-1(Sh)u and verify that for all h E H we have (S-1h(2))h(1) = u-1(Sh(2))uh(1) = u-'(Sh(2))(S2h(1))u = u-1(S((Sh(1))h(2)))u = e(h)u-lu = e(h), using parts (i) and (ii) and the properties of S. Likewise, we verify that h(2)S-1h(1) = h(2)u 1(Sh(1))u = U_ 1(S2h(2))(Sh(1))u = u- 1(S(h(1)Sh(2)))u = e(h)u-1u = e(h). From this it follows that SS-1h = (SS-1h(1))(Sh(2))h(3) = S-'Sh = (S-1Sh(3))(S-'h(2))h(1) = (S(h(2)S-1h(1)))h(3) (S-1(h(2)Sh(3)))h(1) = S(1)h = h, = S-'(1)h = h for all h E H, using antimultiplicativity of S (and hence of S-1).
A Quantum Groups Primer by Shahn Majid