Abstract

The Dolbeault complex of a quantized compact Hermitian symmetric space is expressed in terms of the Koszul complex of a braided symmetric algebra of Berenstein and Zwicknagl. This defines a spectral triple quantizing the Dolbeault‐Dirac operator associated to the canonical spin c structure.

Highlights

  • The aim of this paper is to connect the noncommutative geometry of the quantized compact Hermitian symmetric spaces and the theory of the braided symmetric and exterior algebras of Berenstein and Zwicknagl [6, 56]

  • Connes’ spectral triples [16, 17] provide a framework for quantizing the Kähler metric. This requires one to deform, in addition to the algebra of functions, the spinor fields with respect to the canonical spinc structure as well as the Dolbeault–Dirac operator ∂ + ∂∗ acting on them

  • The typical fiber of the canonical spinc structure on G/P is the exterior algebra Λ(g/p), so the classical Clifford algebra can be identified with EndC(Λ(g/p)); see Section 5.1

Read more

Summary

Introduction

The aim of this paper is to connect the noncommutative geometry of the quantized compact Hermitian symmetric spaces (see, for example, [18,19,20, 31, 32, 42, 44, 46, 51]) and the theory of the braided symmetric and exterior algebras of Berenstein and Zwicknagl [6, 56]. Connes’ spectral triples [16, 17] provide a framework for quantizing the Kähler metric This requires one to deform, in addition to the algebra of functions, the spinor fields with respect to the canonical spinc structure as well as the Dolbeault–Dirac operator ∂ + ∂∗ acting on them. The connection to the work of Berenstein and Zwicknagl arises in the construction of Clq. Classically, the typical fiber of the canonical spinc structure on G/P is the exterior algebra Λ(g/p), so the classical (complex) Clifford algebra can be identified with EndC(Λ(g/p)); see Section 5.1. The paper is organized as follows: Section 2 contains background material and notation on semisimple Lie algebras, parabolic Lie subalgebras, their quantizations and representations.

Simple Lie algebras g
Parabolic subalgebras p
The cominuscule case
Type 1 representations
Braidings
Coboundary structure
Quantum symmetric and exterior algebras
Quantization of the nilradical
Quadratic duality
Linear duality
The quantum exterior algebras are Frobenius algebras
The classical Clifford algebra
The quantum Clifford algebra
The Dolbeault operator ð
An example
Lie algebras and root data
The quantized enveloping algebra
The coboundary structure
The quantum symmetric and exterior algebras
The twisted quantum Schubert cell
A motivation
Findings
The classical geometric picture
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call