By Shahn Majid
Here's a self-contained creation to quantum teams as algebraic items. in response to the author's lecture notes for the half III natural arithmetic direction at Cambridge collage, the e-book is acceptable as a major textual content for graduate classes in quantum teams or supplementary interpreting for contemporary classes in complex algebra. the fabric assumes wisdom of uncomplicated and linear algebra. a few familiarity with semisimple Lie algebras might even be worthwhile. the quantity is a primer for mathematicians however it can also be worthwhile for mathematical physicists.
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For example, if we take uq(b+) the sub-Hopf algebra generated by g, E in uq(sl2) then we can identify uq(b+)*OP = uq(b_), the sub-Hopf algebra generated by g, F. This is a version of the self-duality pairing of Uq(b+) with itself which we have discussed in Lecture 2. Denoting the generators of uq(b+)*OP by g', F say, we can recover uq(sl2) as D(uq(b+)) 8 Quantum double. Dual quasitriangular structures 47 modulo (the ideal generated by) the relation g = g'. Its quasitriangular structure is then the quotient of that of D(uq(b+)).
G is Abelian. 2. R E k(G x G) is a bicharacter. e. we need R a function of two variables on the group. e. nowhere vanishing). e. G is Abelian. The remaining axiom 3 is clearly R(xy, z) = R(x, z)R(y, z), R(x, yz) = R(x, z)R(x, y), x, y, z E G. e. R(x, e) = 1 = R(e, x) for all x E G (here e is the group identity). These equations are precisely the definition of a bicharacter on G. Actually, our present motivation comes not from this example k(G) but from generalising the group algebra kG and Lie theory.
That 8211 is another quasitriangular structure is an elementary exercise from the definitions. e. R21 (Oh)R21 = T o Ah, which is the same axiom for 8211 in the role of R. For the last part of the lemma we compute (id (D T o A)R in two ways: using the third axiom directly, or using the second axiom and then the third axiom. Thus, (id (&T o 0)R = (id ®T) (id ®A)R = (id ®T)R13R12 = R12R13 and (id ®T o 0)R = R23((id ®0)R)R23 = R23R13R12R23 The Yang-Baxter equation arises in physics and also in knot theory as we will see later in the course.
A Quantum Groups Primer by Shahn Majid