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Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach (Hardcover, 1997 ed.)
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Introduction to the Quantum Yang-Baxter Equation and Quantum Groups: An Algebraic Approach (Hardcover, 1997 ed.)
Series: Mathematics and Its Applications, 423
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Chapter 1 The algebraic prerequisites for the book are covered here
and in the appendix. This chapter should be used as reference
material and should be consulted as needed. A systematic treatment
of algebras, coalgebras, bialgebras, Hopf algebras, and represen
tations of these objects to the extent needed for the book is
given. The material here not specifically cited can be found for
the most part in Sweedler, 1969] in one form or another, with a few
exceptions. A great deal of emphasis is placed on the coalgebra
which is the dual of n x n matrices over a field. This is the most
basic example of a coalgebra for our purposes and is at the heart
of most algebraic constructions described in this book. We have
found pointed bialgebras useful in connection with solving the
quantum Yang-Baxter equation. For this reason we develop their
theory in some detail. The class of examples described in Chapter 6
in connection with the quantum double consists of pointed Hopf
algebras. We note the quantized enveloping algebras described Hopf
algebras. Thus for many reasons pointed bialgebras are elsewhere
are pointed of fundamental interest in the study of the quantum
Yang-Baxter equation and objects quantum groups."
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