Quantum Algebras Associated to Irreducible Generalized Flag Manifolds

TL;DR

This thesis explores quantum symmetric and exterior algebras, proving their universal properties, flatness conditions, and developing quantum Clifford algebras, with applications to noncommutative geometry of quantum flag manifolds.

Contribution

It establishes the universal property of quantum symmetric algebras, proves a conjecture on their collapsing behavior, and constructs quantum Clifford algebras for geometric applications.

Findings

Quantum symmetric algebra is the universal commutative algebra generated by a module.

Quantum exterior algebras that are flat deformations are Frobenius algebras.

Construction of a quantum Clifford algebra and a Dirac operator on quantum flag manifolds.

Abstract

The first part of this thesis deals with certain properties of the quantum symmetric and exterior algebras of Type 1 representations of $U_q(g)$ defined by Berenstein and Zwicknagl. We define a notion of a commutative algebra object in a coboundary category, and we prove that the quantum symmetric algebra of a module is the universal commutative algebra generated by that module. That is, the functor assigning to a module its quantum symmetric algebra is left adjoint to a forgetful functor. We also prove a conjecture of Berenstein and Zwicknagl, stating that the quantum symmetric and exterior cubes exhibit the same amount of "collapsing" relative to their classical counterparts. We prove that those quantum exterior algebras that are flat deformations of their classical analogues are Frobenius algebras. We also develop a rigorous framework for discussing continuity and limits of the structures involved as the deformation parameter q varies along the positive real line. The second part deals with quantum analogues of Clifford algebras and their application to the noncommutative geometry of certain quantum homogeneous spaces. We introduce the quantum Clifford algebra through its spinor representation via creation and annihilation operators on one of the flat quantum exterior algebras discussed in the first part. The proof that the spinor representation is irreducible relies on the Frobenius property discussed previously. We use this quantum Clifford algebra to revisit Krahmer's construction of a Dolbeault-Dirac-type operator on a quantized irreducible flag manifold. This operator is of the form $d+d^*$, and we relate $d$ to the boundary operator for the Koszul complex of a certain quantum symmetric algebra, which shows that $d^2=0$. This is a first step toward a Parthasarathy-type formula for the spectrum of the square of the Dirac operator.