[Paper Review] Reductions of piecewise-trivial principal comodule algebras

Piotr M. Hajac, Jan Rudnik+1 more|arXiv (Cornell University)|Dec 31, 2010

Advanced Operator Algebra Research22 references3 citations

TL;DR

This paper establishes a noncommutative-geometric criterion for reducing piecewise-trivial principal comodule algebras, generalizing classical bundle reduction theory. It proves that a principal comodule algebra reduces to a quotient Hopf algebra if and only if there exists a piecewise trivialization where transition functions annihilate the Hopf ideal and the associated actions of the ideal on invariants are trivial—verified via a quantum deformation of $\mathbb{R}P^2$ and a nontrivial $U(1)$-prolongation of a $\mathbb{Z}/2\mathbb{Z}$-bundle.

ABSTRACT

Let $G'$ be a closed subgroup of a topological group $G$. A principal $G$-bundle $X$ is reducible to a locally trivial principal $G'$-bundle $X'$ if and only if there exists a local trivialisation of $X$ such that all transition functions take values in $G'$. We prove a noncommutative-geometric counterpart of this theorem. To this end, we employ the concept of a piecewise-trivial principal comodule algebra as a replacement of a locally trivial compact principal bundle. To illustrate our theorem, first we define a new noncommutative deformation of the $\mathbb{Z}/2\mathbb{Z}$-principal bundle $S^2 ightarrow \mathbb{R}P^2$ that yields a piecewise-trivial principal comodule algebra. It is the C*-algebra of a quantum cube whose each face is given by the Toeplitz algebra. The $\mathbb{Z}/2\mathbb{Z}$-invariant subalgebra defines the C*-algebra of a quantum $\mathbb{R}P^2$. It is given as a triple-pullback of Toeplitz algebras. Next, we prolongate this noncommutative $\mathbb{Z}/2\mathbb{Z}$-principal bundle to a noncommutative $U(1)$-principal bundle, so that the former becomes a reduction of the latter thus instantiating our theorem. Moreover, using K-theory results, we prove that the prolongated noncommutative bundle is not trivial.

Motivation & Objective

• To generalize classical reduction theory of principal bundles to the noncommutative setting using piecewise-trivial principal comodule algebras.
• To provide a criterion for reducibility of such comodule algebras to quotient Hopf algebras.
• To construct and analyze a noncommutative deformation of the $\mathbb{Z}/2\mathbb{Z}$-principal bundle $S^2 \to \mathbb{R}P^2$ via the quantum cube and Toeplitz algebras.
• To demonstrate that a quantum $U(1)$-bundle prolongating this $\mathbb{Z}/2\mathbb{Z}$-bundle is nontrivial using K-theory.

Proposed method

• Define piecewise-triviality via finite closed coverings of the comodule algebra, replacing local triviality in the non-Lie setting.
• Use the Hopf–Galois Reduction Theorem to relate reduction ideals to equivariant algebra homomorphisms.
• Construct a quantum $\mathbb{R}P^2$ as a triple-pullback of Toeplitz algebras, realizing the $\mathbb{Z}/2\mathbb{Z}$-invariant subalgebra of a quantum cube.
• Prolongate the $\mathbb{Z}/2\mathbb{Z}$-bundle to a $U(1)$-bundle by extending the structure algebra to a $U(1)$-comodule algebra.
• Apply K-theory to prove that the prolonged bundle is nontrivial, showing it does not admit a global section.
• Use the Miyashita–Ulbrich action and coaction invariance to characterize the correspondence between algebra homomorphisms and maps on coinvariants.

Experimental results

Research questions

• RQ1 Under what conditions can a piecewise-trivial principal comodule algebra be reduced to a principal comodule algebra over a quotient Hopf algebra?
• RQ2 How can the classical notion of bundle reduction via transition functions be generalized to noncommutative geometry using piecewise trivializations?
• RQ3 Can a noncommutative deformation of $\mathbb{R}P^2$ be constructed as a quantum bundle with a $\mathbb{Z}/2\mathbb{Z}$-action and Toeplitz algebraic structure?
• RQ4 Is the $U(1)$-prolongation of this quantum bundle trivial, and how can K-theory detect its nontriviality?
• RQ5 What role do the actions of the Hopf ideal on coinvariants play in determining reducibility?

Key findings

• A principal comodule algebra $P$ over a Hopf algebra $H$ admits a reduction to a principal $H/J$-comodule algebra if and only if there exists a piecewise trivialization where all transition functions annihilate the Hopf ideal $J$.
• The associated action of $J$ on the algebra of $H$-coaction invariants must be trivial for such a reduction to exist.
• A new noncommutative deformation of $S^2 \to \mathbb{R}P^2$ is constructed as the C*-algebra of a quantum cube, with each face isomorphic to the Toeplitz algebra.
• The $\mathbb{Z}/2\mathbb{Z}$-invariant subalgebra of this quantum cube yields a C*-algebra for a quantum $\mathbb{R}P^2$, realized as a triple-pullback of Toeplitz algebras.
• The $\mathbb{Z}/2\mathbb{Z}$-principal bundle is prolongated to a $U(1)$-principal bundle, which is proven to be nontrivial via K-theory.
• The $U(1)$-bundle is not trivial because the existence of a $\operatorname{Ker} \pi$-reduction would require $q^3 = 1$, which fails for generic $q$.