Hardcover ISBN:  9780821843741 
Product Code:  SURV/143 
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Electronic ISBN:  9781470413705 
Product Code:  SURV/143.E 
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Book DetailsMathematical Surveys and MonographsVolume: 143; 2007; 400 ppMSC: Primary 17; 81; Secondary 05;
The Yangians and twisted Yangians are remarkable associative algebras taking their origins from the work of St. Petersburg's school of mathematical physics in the 1980s. The general definitions were given in subsequent work of Drinfeld and Olshansky, and these algebras have since found numerous applications in and connections with mathematical physics, geometry, representation theory, and combinatorics.
The book is an introduction to the theory of Yangians and twisted Yangians, with a particular emphasis on the relationship with the classical matrix Lie algebras. A special algebraic technique, the \(R\)matrix formalism, is developed and used as the main instrument for describing the structure of Yangians. A detailed exposition of the highest weight theory and the classification theorems for finitedimensional irreducible representations of these algebras is given.
The Yangian perspective provides a unifying picture of several families of Casimir elements for the classical Lie algebras and relations between these families. The Yangian symmetries play a key role in explicit constructions of all finitedimensional irreducible representations of the orthogonal and symplectic Lie algebras via weight bases of GelfandTsetlin type.ReadershipGraduate students and research mathematicians interested in representation theory and quantum groups.

Table of Contents

Chapters

1. Yangian for $\mathfrak {gl}_N$

2. Twisted Yangians

3. Irreducible representations of $\mathrm {Y}(\mathfrak {gl}_N)$

4. Irreducible representations of $\mathrm {Y}(\mathfrak {g}_N)$

5. GelfandTsetlin bases for representations of $\mathrm {Y}(\mathfrak {gl}_N)$

6. Tensor products of evaluation modules for $\mathrm {Y}(\mathfrak {gl}_N)$

7. Casimir elements and Capelli identities

8. Centralizer construction

9. Weight bases for representations of $\mathfrak {g}_N$


Additional Material

Reviews

This book is well written and will be indispensable for anyone working on Yangians. It will also be of interest for the new point of view it brings to branching rules. Some versions of the theorems proved hold for other algebras, in particular quantized enveloping algebras of finite and affine type. Each chapter concludes with examples of these versions, and bibliographic notes linking the chapter to the extensive bibliography at the end of the book.
Mathematical Reviews


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The Yangians and twisted Yangians are remarkable associative algebras taking their origins from the work of St. Petersburg's school of mathematical physics in the 1980s. The general definitions were given in subsequent work of Drinfeld and Olshansky, and these algebras have since found numerous applications in and connections with mathematical physics, geometry, representation theory, and combinatorics.
The book is an introduction to the theory of Yangians and twisted Yangians, with a particular emphasis on the relationship with the classical matrix Lie algebras. A special algebraic technique, the \(R\)matrix formalism, is developed and used as the main instrument for describing the structure of Yangians. A detailed exposition of the highest weight theory and the classification theorems for finitedimensional irreducible representations of these algebras is given.
The Yangian perspective provides a unifying picture of several families of Casimir elements for the classical Lie algebras and relations between these families. The Yangian symmetries play a key role in explicit constructions of all finitedimensional irreducible representations of the orthogonal and symplectic Lie algebras via weight bases of GelfandTsetlin type.
Graduate students and research mathematicians interested in representation theory and quantum groups.

Chapters

1. Yangian for $\mathfrak {gl}_N$

2. Twisted Yangians

3. Irreducible representations of $\mathrm {Y}(\mathfrak {gl}_N)$

4. Irreducible representations of $\mathrm {Y}(\mathfrak {g}_N)$

5. GelfandTsetlin bases for representations of $\mathrm {Y}(\mathfrak {gl}_N)$

6. Tensor products of evaluation modules for $\mathrm {Y}(\mathfrak {gl}_N)$

7. Casimir elements and Capelli identities

8. Centralizer construction

9. Weight bases for representations of $\mathfrak {g}_N$

This book is well written and will be indispensable for anyone working on Yangians. It will also be of interest for the new point of view it brings to branching rules. Some versions of the theorems proved hold for other algebras, in particular quantized enveloping algebras of finite and affine type. Each chapter concludes with examples of these versions, and bibliographic notes linking the chapter to the extensive bibliography at the end of the book.
Mathematical Reviews