Download A Quantum Groups Primer by Shahn Majid PDF

By Shahn Majid

Here's a self-contained creation to quantum teams as algebraic items. in keeping with the author's lecture notes for the half III natural arithmetic direction at Cambridge collage, the publication is acceptable as a major textual content for graduate classes in quantum teams or supplementary examining for contemporary classes in complicated algebra. the cloth assumes wisdom of uncomplicated and linear algebra. a few familiarity with semisimple Lie algebras might even be beneficial. the quantity is a primer for mathematicians however it may also be priceless for mathematical physicists.

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The specific meaning of such a quasitriangular structure R will emerge during the next several lectures. However, there are many examples and indeed, most self-respecting quantum groups are either quasitriangular or dual to a quasitriangular one. 2 Let (H, R) be a quasitriangular bialgebra. Then 29 30 5 Quasitriangular structures 1. (e ®id)R = (id (9 e)R = 1. 2. (H,R211) is also a quasitriangular bialgebra (8211 = T(R-1) is called the `conjugate' quasitriangular structure). 3. R12R13R23 = R23R13R12 holds in H ® H ® H (the Yang-Baxter equation).

This leads to the following convenient notation for working with such braidings. We write morphisms pointing generally downwards (say) and denote tensor product by horizontal juxtaposition. Instead of a usual arrow for ', W-1 we use the shorthand VW VW %PV,W = /\ W V , (IF W,V)-1 /y W V to distinguish them. We denote any other morphisms as nodes on a string with the appropriate number of input and output legs. 3, where the doubled lines in part (a) refer to the composite objects V ® W and W ® Z in a convenient extension of the notation.

M+1Bn-1-m+ E rn - 11 AmBn-m I m q m=0 m=0 IL m J q E n E qn-m n-1 Ln-11 m-1 q m=0 39 [n- -11 AmBn-m m Jq 7 q-Binomials 40 =An+Bn+ n-1(qn_m n-1 [n_i] + [m-1]q m=1 m Am Bn-m q/ Th e expression in parentheses combines to [ m ] q, as required, after an elementary computation using the identity qn-m[m]q + [n - m]q = [n]q (the addition rule for q-integers). We used this in the expression for AE' in uq(s12). Similarly, it implies the formula for the coproduct of the version of Uq (b+) in Lecture 1. Also, one does not actually need the q-integers here to be invertible.

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