By Larry A. Lambe, David E. Radford (auth.)

ISBN-10: 1461368421

ISBN-13: 9781461368427

ISBN-10: 1461541093

ISBN-13: 9781461541097

Chapter 1 The algebraic necessities for the ebook are lined the following and within the appendix. This bankruptcy will be used as reference fabric and may be consulted as wanted. a scientific therapy of algebras, coalgebras, bialgebras, Hopf algebras, and represen tations of those items to the level wanted for the booklet is given. the fabric the following now not particularly stated are available for the main half in [Sweedler, 1969] in a single shape or one other, with a number of exceptions. loads of emphasis is put on the coalgebra that's the twin of n x n matrices over a box. this is often the main uncomplicated instance of a coalgebra for our reasons and is on the middle of such a lot algebraic buildings defined during this booklet. we've got discovered pointed bialgebras invaluable in reference to fixing the quantum Yang-Baxter equation. accordingly we boost their conception in a few aspect. the category of examples defined in bankruptcy 6 in reference to the quantum double contains pointed Hopf algebras. We notice the quantized enveloping algebras defined Hopf algebras. therefore for plenty of purposes pointed bialgebras are in other places are pointed of primary curiosity within the learn of the quantum Yang-Baxter equation and items quantum groups.

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**Example text**

The definition of Hopf algebra is made in terms of this structure. 1 INTRODUCTION TO THE QYBE The Convolution Algebra Let (C,~,c:) be a coalgebra and (A,m,1J) be an algebra over the field k. 16) for all J, g E Hom(C, A). The map 7JE is the multiplicative identity for this algebra. 16) is expressed by the equations for all J, g E Hom(C, A) and c E C. When A= k the convolution algebra Hom(C, k) = C* is the dual algebra of C. Let rr : C D be a coalgebra map and t : A easy exercise in definitions shows that ---+- B be an algebra map.

Then f(Mr) ~ Nr. Thus the restriction fr =JIM~ is a module map fr : Mr- Nr. Proof: The fact that Mr is a subspace of M is a straightforward exercise. Let m E Mr and assume that m =f. 0. Write Pm = I:;=l mi ® Ci where r = Rankpm. 1. As c*·m = I:;=l c* (ci)mi for c* E C*, by definition it follows that C* ·m is contained in the span of { m 1, ... , mr}. Since { c 1, ... , Cr} is linearly independent, for given 1 :::; i :::; r there is a c* E C* such that c* (Cj) = 8i,j for all 1 :::; j :::=; r. For such a functional c* we have c*·m =mi.

Show that C* is isomorphic to the algebra of functions from S to k under point-wise multiplication. 4 Let V be a vector space over k and g be a symbol. (v) =g®v+v®g for all v E V. Show that C* = kt: EB I as a vector space, where / 2 = (0). 1 which has linear basis {co, c1, ... (cn) = L:~=o Cn-i ®c; and t:(cn) = ho,n for all n :2: 0. c· a) Show that 7r : k[[X]] - defined by 7r(L:::"=o anXn)(cm) = Om is an algebra isomorphism of the algebra of formal power series in X over k and the dual algebra C*.

### Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach by Larry A. Lambe, David E. Radford (auth.)

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