By Shahn Majid

ISBN-10: 0521010411

ISBN-13: 9780521010412

Here's a self-contained advent to quantum teams as algebraic gadgets. in line with the author's lecture notes for the half III natural arithmetic path at Cambridge college, the ebook is appropriate as a chief textual content for graduate classes in quantum teams or supplementary interpreting for contemporary classes in complicated algebra. the cloth assumes wisdom of simple and linear algebra. a few familiarity with semisimple Lie algebras might even be necessary. the amount is a primer for mathematicians however it can also be priceless for mathematical physicists.

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This says that (S (D id) (Q) is Ad-invariant. 32 5 Quasitriangular structures Proof We write T R'") ® R'(2) as an explicit notation for a second copy of R, with the prime used to distinguish it from the first copy. Then Adh((S (Sid)(Q)) = h(1)(SQ('))Sh(2) ® h(3) Q(2) Sh(4) = h(,)(SR('))(SR' 2 )Sh(2)® h(3) R'«'R(2)Sh(4, h(1)(SR"))Sh(3)(S)Z/(2)) ®R'("h(2)R(2)Sh(4) = = h(1)Sh(Z)(SR('))(SR'(2)) ®R'(')R(2)h(3) Sh(4) = e(h)(S(9 id)(Q), using that S is an antialgebra map, and axiom 2 of a quasitriangular structure for the third and fourth equalities.

Hence 1 ` q-ab+cd6c 2agb = 1 [ q-a(b-2d)gb = g2d qcd fc(Q(1))Q(2) = l c aL,b a,,bb,,c n for all d, where { f c} is a dual basis to the basis {ga} of CZ/n. If n is odd, then 2d is a permutation of {0, . . , n - 11 as d runs through this set, and otherwise not. Hence Z/ is factorisable if n is odd. 7. One should also check that R. is invertible. Indeed, n-1 -1 = 1 gabga ®gb 1: a,b=O as one may easily check by a computation similar to that for Q. This example generates as its category of modules a very interesting braided category, as we will see later in the course.

For a quasitriangular Hopf algebra we will find that they are isomorphic, but by a nontrivial isomorphism called the `braiding' 1P. 3 A braided monoidal category (C, 0, 1, 4), 1, r, T) is 1. A monoidal category (C, ®,1, c,1, r). 2. e. 2. The model here is the usual twist or transposition map V ® W --W 0 V for vector spaces. If we suppress 4, then the hexagon conditions are T V ®W,Z ='PV,z o T W,Z, W V,W ®z = ,VVZ o T V,W 9 Braided categories V®(W®Z) MOT 55 (V®W)®Z AY®id V®(Z®W) (V®W)®Z I 41-1 j* (V®Z)®W AF Z®(Vow) V®(W®Z) 'P (W®®V)®Z I W®/(V®Z) (W®Z)®V / id ®qY (Z®V)®W W ®(Z®V) Fig.

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