By Shahn Majid

Here's a self-contained advent to quantum teams as algebraic gadgets. in accordance with the author's lecture notes for the half III natural arithmetic direction at Cambridge college, the ebook is appropriate as a first-rate textual content for graduate classes in quantum teams or supplementary analyzing for contemporary classes in complex algebra. the cloth assumes wisdom of easy and linear algebra. a few familiarity with semisimple Lie algebras could even be necessary. the amount is a primer for mathematicians however it may also be priceless for mathematical physicists.

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**Example text**

The natural transformation 0 is called a `natural isomorphism' if each Ov is an isomorphism. M of modules over an algebra A. The morphisms are maps commuting with the action of A. We have a lot of structure in this category: direct sums, k-linearity etc. 1 A monoidal category is (C, ®, 1, 1P,1, r), where 1. C is a category. 2. ®: C x C -> C is a functor. e. a collection of 3. 1. 4. e. 1. The pentagon condition equates the two ways to go ((V ® W) (D Z) ® U V ®(W ®(Z ® U)) by applying 4) repeatedly.

Factorisable Hopf algebras are at the opposite extreme and are motivated in a different way from the theory of finite-dimensional complex semisimple Lie algebras, where the Killing form can be viewed as an isomorphism g - g*. For this reason the element Q is sometimes called the `inverse quantum Killing form'. 6 For any quasitriangular Hopf algebra H, Adh((S(&id)(Q)) = e(h)(S®id)(Q), Vh E H, where Ad is the quantum group adjoint action extended to H ® H. This says that (S (D id) (Q) is Ad-invariant.

E. t = 0). ] In this variant the generators are H, X+, X_, with the relations and structure [H, Xt] = ±2Xf, qH q- [X+, X-] = q_q-1 OH=H®1+1®H, OX±=Xf®q2 +q 2®X+ R=q H OH °O (1 - q-2)n [n]! (n_1) 2 q q [n] = qn - q q-q 1 Here q = e2 should be substituted in all expressions. We also use here some other popular conventions where the q-integers are more symmet- 7 q-Binomials 43 rical between q and q-'. One can check that if g=qH, E=X+q2, F=q 2Xthen these obey the relations of Uq(sl2) given before. Moreover, and most importantly, we have the quasitriangular structure once again, as a formal powerseries in t.