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Sugawara Operators for Classical Lie Algebras
 
Alexander Molev University of Sydney, Sydney, Australia
Sugawara Operators for Classical Lie Algebras
Hardcover ISBN:  978-1-4704-3659-9
Product Code:  SURV/229
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-4391-7
Product Code:  SURV/229.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN:  978-1-4704-3659-9
eBook: ISBN:  978-1-4704-4391-7
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
Sugawara Operators for Classical Lie Algebras
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Sugawara Operators for Classical Lie Algebras
Alexander Molev University of Sydney, Sydney, Australia
Hardcover ISBN:  978-1-4704-3659-9
Product Code:  SURV/229
List Price: $129.00
MAA Member Price: $116.10
AMS Member Price: $103.20
eBook ISBN:  978-1-4704-4391-7
Product Code:  SURV/229.E
List Price: $125.00
MAA Member Price: $112.50
AMS Member Price: $100.00
Hardcover ISBN:  978-1-4704-3659-9
eBook ISBN:  978-1-4704-4391-7
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 Details
     
     
    Mathematical Surveys and Monographs
    Volume: 2292018; 304 pp
    MSC: Primary 17; 16

    The celebrated Schur-Weyl 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 finite-dimensional Lie algebras. Sugawara operators are analogs of Casimir elements for the affine Kac-Moody 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 Harish-Chandra isomorphism connects the Sugawara operators with the classical \(\mathcal{W}\)-algebras, which play the role of the Weyl group invariants in the finite-dimensional theory.

    Readership

    Graduate 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 $\mathfrak {sp}_N$
    • Feigin-Frenkel 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 Harish-Chandra isomorphism
    • Higher Hamiltonians in the Gaudin model
    • Wakimoto modules
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 2292018; 304 pp
MSC: Primary 17; 16

The celebrated Schur-Weyl 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 finite-dimensional Lie algebras. Sugawara operators are analogs of Casimir elements for the affine Kac-Moody 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 Harish-Chandra isomorphism connects the Sugawara operators with the classical \(\mathcal{W}\)-algebras, which play the role of the Weyl group invariants in the finite-dimensional theory.

Readership

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 $\mathfrak {sp}_N$
  • Feigin-Frenkel 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 Harish-Chandra isomorphism
  • Higher Hamiltonians in the Gaudin model
  • Wakimoto modules
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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