By Shahn Majid
Here's a self-contained creation to quantum teams as algebraic items. in accordance with the author's lecture notes for the half III natural arithmetic path at Cambridge collage, the publication is acceptable as a chief textual content for graduate classes in quantum teams or supplementary interpreting for contemporary classes in complex algebra. the cloth assumes wisdom of simple and linear algebra. a few familiarity with semisimple Lie algebras may even be worthwhile. the quantity is a primer for mathematicians however it can also be beneficial for mathematical physicists.
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Extra resources for A Quantum Groups Primer
It is in this algebra that we require R-1 : H ® H -* k to exist. Explicitly, it means a linear map such that 7 R-1(h(1) = E(h)e(9) ®9(1))R(h(2) ®9(2)) R(h(1) (9 9(1))R-1(h(Z) ®g(2)) for all h, g E H. Keeping such considerations in mind, it is easy to dualise the remainder of Drinfeld's axioms to obtain the following definition. 4 A dual quasitriangular (or coquasitriangular) bialgebra or Hopf algebra is a pair (H, R) where 1. H is a bialgebra or Hopf algebra. 2. R is a convolution-invertible map R : H ® H -+ k such that E 9(1)h(1)R(h(2) ® 9(2)) _ E R(h(1) ® 9(1))h(2)9(2) for all h, g E H (quasicommutativity axiom).
1 - qm 1-q where we suppose (for convenience) that the q-integers [m], are nonzero for 0 < m < n. By convention, they are defined as the usual integers when q = 1. Also by convention, [, ] q = 1 when m = 0, n. If A, B are elements of an algebra obeying BA = qAB then (A+B) = n ll m=0 AmBn-m [ mI Proof We proceed by induction. 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).
The idea this time is to describe Hopf algebras H which are noncommutative but for which the noncommutativity is under control by a map H ® H -> k. We think of a quasitriangular structure as a map k -> H ® H and reverse the arrows. The first step is the to find the correct notion of R-1. 3 Let C be a coalgebra and A an algebra. Then the set of linear maps Hom(C, A) has an associative product and unit 6' =mo(0®')o0, I=qoe. This is called the `convolution algebra'. Explicitly, (/')(c) = 1:0(c(1))(c(2)), I(c) = le(c), be E C.