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Hardcover ISBN:  9781470436599 
Product Code:  SURV/229 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9781470443917 
Product Code:  SURV/229.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9781470436599 
eBook ISBN:  9781470443917 
Product Code:  SURV/229.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 229; 2018; 304 ppMSC: Primary 17; 16
The celebrated SchurWeyl duality gives rise to effective ways of constructing invariant polynomials on the classical Lie algebras. The emergence of the theory of quantum groups in the 1980s brought up special matrix techniques which allowed one to extend these constructions beyond polynomial invariants and produce new families of Casimir elements for finitedimensional Lie algebras. Sugawara operators are analogs of Casimir elements for the affine KacMoody algebras.
The goal of this book is to describe algebraic structures associated with the affine Lie algebras, including affine vertex algebras, Yangians, and classical \(\mathcal{W}\)algebras, which have numerous ties with many areas of mathematics and mathematical physics, including modular forms, conformal field theory, and soliton equations. An affine version of the matrix technique is developed and used to explain the elegant constructions of Sugawara operators, which appeared in the last decade. An affine analogue of the HarishChandra isomorphism connects the Sugawara operators with the classical \(\mathcal{W}\)algebras, which play the role of the Weyl group invariants in the finitedimensional theory.
ReadershipGraduate students and researchers interested in algebraic aspects of representation theory and applications to mathematical physics.

Table of Contents

Chapters

Idempotents and traces

Invariants of symmetric algebras

Manin matrices

Casimir elements for $\mathfrak {gl}_N$

Casimir elements for $\mathfrak {o}_N$ and $\mathfrak {sp}_N$

FeiginFrenkel center

Generators in type $A$

Generators in types $B, C$ and $D$

Commutative subalgebras of $\textrm {U}(\mathfrak {g})$

Yangian characters in type $A$

Yangian characters in types $B, C$ and $D$

Classical $\mathcal {W}$algebras

Affine HarishChandra isomorphism

Higher Hamiltonians in the Gaudin model

Wakimoto modules


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The celebrated SchurWeyl duality gives rise to effective ways of constructing invariant polynomials on the classical Lie algebras. The emergence of the theory of quantum groups in the 1980s brought up special matrix techniques which allowed one to extend these constructions beyond polynomial invariants and produce new families of Casimir elements for finitedimensional Lie algebras. Sugawara operators are analogs of Casimir elements for the affine KacMoody algebras.
The goal of this book is to describe algebraic structures associated with the affine Lie algebras, including affine vertex algebras, Yangians, and classical \(\mathcal{W}\)algebras, which have numerous ties with many areas of mathematics and mathematical physics, including modular forms, conformal field theory, and soliton equations. An affine version of the matrix technique is developed and used to explain the elegant constructions of Sugawara operators, which appeared in the last decade. An affine analogue of the HarishChandra isomorphism connects the Sugawara operators with the classical \(\mathcal{W}\)algebras, which play the role of the Weyl group invariants in the finitedimensional theory.
Graduate students and researchers interested in algebraic aspects of representation theory and applications to mathematical physics.

Chapters

Idempotents and traces

Invariants of symmetric algebras

Manin matrices

Casimir elements for $\mathfrak {gl}_N$

Casimir elements for $\mathfrak {o}_N$ and $\mathfrak {sp}_N$

FeiginFrenkel center

Generators in type $A$

Generators in types $B, C$ and $D$

Commutative subalgebras of $\textrm {U}(\mathfrak {g})$

Yangian characters in type $A$

Yangian characters in types $B, C$ and $D$

Classical $\mathcal {W}$algebras

Affine HarishChandra isomorphism

Higher Hamiltonians in the Gaudin model

Wakimoto modules