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Quantum Groups and Quantum Cohomology
 
Davesh Maulik Massachusetts Institute of Technology, Cambridge, MA
Andrei Okounkov Columbia University, New York, NY
A publication of the Société Mathématique de France
Quantum Groups and Quantum Cohomology
Softcover ISBN:  978-2-85629-900-5
Product Code:  AST/408
List Price: $67.00
AMS Member Price: $53.60
Please note AMS points can not be used for this product
Quantum Groups and Quantum Cohomology
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Quantum Groups and Quantum Cohomology
Davesh Maulik Massachusetts Institute of Technology, Cambridge, MA
Andrei Okounkov Columbia University, New York, NY
A publication of the Société Mathématique de France
Softcover ISBN:  978-2-85629-900-5
Product Code:  AST/408
List Price: $67.00
AMS Member Price: $53.60
Please note AMS points can not be used for this product
  • Book Details
     
     
    Astérisque
    Volume: 4082019; 212 pp
    MSC: Primary 17; 14;

    In this paper, the authors study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver \(Q\). Using a geometric \(R\)-matrix formalism, they construct a Hopf algebra \(\mathbf{Y}_Q\), the Yangian of \(Q\), acting on the cohomology of these varieties, and show several results about their basic structure theory. The authors prove a formula for quantum multiplication by divisors in terms of this Yangian action. The quantum connection can be identified with the trigonometric Casimir connection for \(\mathbf {Y}_Q\); equivalently, the divisor operators correspond to certain elements of Baxter subalgebras of \(\mathbf{Y}_Q\). A key role is played by geometric shift operators which can be identified with the quantum KZ difference connection.

    In the second part, the authors give an extended example of the general theory for moduli spaces of sheaves on \(\mathbb{C}^2\), framed at infinity. Here, the Yangian action is analyzed explicitly in terms of a free field realization; the corresponding \(R\)-matrix is closely related to the reflection operator in Liouville field theory. The authors show that divisor operators generate the quantum ring, which is identified with the full Baxter subalgebras. As a corollary of our construction, the authors obtain an action of the W-algebra \(\mathcal{W}(\mathfrak{gl}(r))\) on the equivariant cohomology of rank \(r\) moduli spaces, which implies certain conjectures of Alday, Gaiotto, and Tachikawa.

    Readership

    Graduate students and research mathematicians.

  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Accessibility – to request an alternate format of an AMS title
Volume: 4082019; 212 pp
MSC: Primary 17; 14;

In this paper, the authors study the classical and quantum equivariant cohomology of Nakajima quiver varieties for a general quiver \(Q\). Using a geometric \(R\)-matrix formalism, they construct a Hopf algebra \(\mathbf{Y}_Q\), the Yangian of \(Q\), acting on the cohomology of these varieties, and show several results about their basic structure theory. The authors prove a formula for quantum multiplication by divisors in terms of this Yangian action. The quantum connection can be identified with the trigonometric Casimir connection for \(\mathbf {Y}_Q\); equivalently, the divisor operators correspond to certain elements of Baxter subalgebras of \(\mathbf{Y}_Q\). A key role is played by geometric shift operators which can be identified with the quantum KZ difference connection.

In the second part, the authors give an extended example of the general theory for moduli spaces of sheaves on \(\mathbb{C}^2\), framed at infinity. Here, the Yangian action is analyzed explicitly in terms of a free field realization; the corresponding \(R\)-matrix is closely related to the reflection operator in Liouville field theory. The authors show that divisor operators generate the quantum ring, which is identified with the full Baxter subalgebras. As a corollary of our construction, the authors obtain an action of the W-algebra \(\mathcal{W}(\mathfrak{gl}(r))\) on the equivariant cohomology of rank \(r\) moduli spaces, which implies certain conjectures of Alday, Gaiotto, and Tachikawa.

Readership

Graduate students and research mathematicians.

Review Copy – for publishers of book reviews
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.