[Paper Review] Circle actions on a quantum Seifert manifold

Tomasz Brzeziński|arXiv (Cornell University)|Mar 30, 2012

Algebraic structures and combinatorial models11 references8 citations

TL;DR

This paper constructs quantum weighted real projective spaces as quotients of a non-orientable quantum Seifert 3-manifold via circle actions, generalizing the quantum disc and quantum real projective plane. It computes the $K$-groups of their continuous function algebras using Toeplitz algebra extensions and symbol maps, showing $K_0 \cong \mathbb{Z}_2 \oplus \mathbb{Z}^l$ and $K_1 = 0$ for the negative twist case, with explicit isomorphisms to iterated pullbacks of Toeplitz algebras.

ABSTRACT

The quotients of a (non-orientable) quantum Seifert manifold by circle actions are described. In this way quantum weighted real projective spaces that include the quantum disc and the quantum real projective space as special cases are obtained. Bounded irreducible representations of the coordinate algebras and the K-groups of the algebras of continuous functions on quantum weighted real projective spaces are presented.

Motivation & Objective

• To generalize quantum orbifolds by constructing quantum weighted real projective spaces as quotients of a non-orientable quantum Seifert 3-manifold under circle actions.
• To extend methods from Hajac, Matthes, and Szymański on the quantum disc and quantum real projective plane to this new class of quantum spaces.
• To compute the $K$-groups of the $C^*$-algebras of continuous functions on quantum weighted real projective spaces.
• To analyze bounded irreducible representations of the coordinate algebras of these quantum spaces.
• To establish isomorphisms between the function algebras and iterated pullbacks of Toeplitz algebras, linking them to the quantum disc and classical circle boundary.

Proposed method

• Define the coordinate $*$-algebra $\mathcal{O}(\Sigma^3_q)$ of the quantum Seifert 3-manifold via generators $\zeta_0, \zeta_1, \xi$ with $q$-deformed commutation relations and a central unitary $\xi$.
• Introduce a circle action via a $\mathbb{Z}_2$-coaction on the quantum 2-sphere, extended to a $U(1)$-coaction on $\mathcal{O}(\Sigma^3_q)$ with weights $k,l$ coprime integers.
• Identify the coinvariant subalgebra $\mathcal{O}(\mathbb{RP}_q(l;+))$ as the fixed point algebra under the coaction, parameterized by coprime weights $k,l$.
• Use the symbol map and Toeplitz algebra structure to realize $C(\mathbb{RP}_q(l;+))$ as an $l$-fold pullback of $C(D_q)$ algebras over $C(S^1)$, with $C(D_q) \cong \mathcal{T}$.
• Construct a short exact sequence involving compact operators and $C(\mathbb{RP}_q(l;+))$, enabling $K$-group computation via the six-term exact sequence.
• Compute $K$-groups using the index map and cokernel of the homomorphism $\delta_\pm$, yielding $K_0 \cong \mathbb{Z}_2 \oplus \mathbb{Z}^l$ and $K_1 = 0$ for the negative twist case.

Experimental results

Research questions

• RQ1 How can circle actions on a non-orientable quantum Seifert 3-manifold be classified and used to construct quantum weighted real projective spaces?
• RQ2 What is the structure of the $C^*$-algebra of continuous functions on these quantum weighted projective spaces?
• RQ3 How do the $K$-groups of these algebras relate to those of the quantum disc and quantum real projective plane?
• RQ4 Can the methods of Hajac, Matthes, and Szymański for the quantum disc be generalized to this new class of quantum orbifolds?
• RQ5 What is the role of the symbol map and Toeplitz algebra extensions in realizing the function algebras as pullbacks of $C(D_q)$?

Key findings

• The algebra $C(\mathbb{RP}_q(l;+))$ is isomorphic to the $l$-fold pullback $C(D_q) \oplus_\sigma \cdots \oplus_\sigma C(D_q)$, where $\sigma$ is the symbol map from the boundary inclusion of the classical circle into the quantum disc.
• For the positive twist case, $\pi(C(\mathbb{RP}_q(l;+))) \cong \mathcal{T}_1 \oplus_\sigma \cdots \oplus_\sigma \mathcal{T}_l$, with each $\mathcal{T}_r \cong \mathcal{T}$, the Toeplitz algebra.
• The $K$-groups of $C(\mathbb{RP}_q(l;+))$ are $K_0 \cong \mathbb{Z}^l$ and $K_1 \cong \mathbb{Z}^l$, arising from the cokernel of the index map $\delta_+$.
• For the negative twist case, $K_1(C(\mathbb{RP}_q(l;-))) = 0$ and $K_0(C(\mathbb{RP}_q(l;-))) \cong \mathbb{Z}_2 \oplus \mathbb{Z}^l$, with the isomorphism induced by the cokernel of $\delta_-$.
• The computation recovers the known $K$-groups of $C(\mathbb{RP}_q^2)$ when $l=1$, confirming consistency with prior results.
• The function algebra $C(\mathbb{RP}_q(l;-))$ is isomorphic to $C(\mathbb{RP}_q^2) \oplus_{\bar{\sigma}} \cdots \oplus_{\bar{\sigma}} C(\mathbb{RP}_q^2)$, $l$ times, with $\bar{\sigma}$ induced by the equatorial inclusion of $S^1$ in $S_q^2$.