[Paper Review] Connections over twisted tensor products of algebras

Javier López Peña|arXiv (Cornell University)|Oct 31, 2006

Algebraic structures and combinatorial models13 references3 citations

TL;DR

This paper introduces a constructive method to build connections on twisted tensor products of algebras from connections on the individual factors, proving the resulting product connection is flat if both input connections are flat. The curvature of the product connection is shown to be independent of the twisting map and module twisting map, ensuring consistency and generalizing classical product connections to noncommutative settings, with explicit computation of all product connections on the quantum plane $k_q[x,y]$.

ABSTRACT

Motivated by some results in classical differential geometry, we give a constructive procedure for building up a connection over a (twisted) tensor product of two algebras, starting from connections defined on the factors. The curvature for the product connection is explicitly calculated, and shown to be independent of the choice of the twisting map and the module twisting map used to define the product connection. As a consequence, we obtain that a product of two flat connections is again a flat connection. We show that our constructions also behaves well with respect to bimodule structures, namely being the product of two bimodule connections again a bimodule connection. As an application of our theory, all the product connections on the quantum plane are computed.

Motivation & Objective

• To develop a systematic method for constructing connections on twisted tensor products of algebras using connections on the component algebras.
• To generalize classical product connections in differential geometry to noncommutative settings via twisted tensor products.
• To prove that the curvature of the product connection is independent of the choice of twisting map and module twisting map.
• To show that the product of two flat connections remains flat, extending a key property from classical geometry.
• To compute all possible product connections on the quantum plane $k_q[x,y]$ using the proposed construction.

Proposed method

• Define a product connection on the twisted tensor product $A times_R B$ using connections on $A$-module $E$ and $B$-module $F$, via a formula involving the individual connections and twisting maps.
• Use the twisting map $R: B igotimes A o A igotimes B$ and its induced module twisting map $\tau_{F,A}: F igotimes A o A igotimes F$ to define the action on tensor products of elements.
• Construct the product connection $\nabla$ as a sum of terms: $\nabla(e \otimes b, a \otimes f) = \nabla^{gr}(e \otimes b, a \otimes f) + \sum \varphi_i(a_j) \otimes 1 \otimes \omega_i \otimes b + \sum a \otimes \psi_k(b_l) \otimes 1 \otimes \eta_k$, where $\varphi_i, \psi_k$ are components of the gauge potentials.
• Verify that the resulting operator satisfies the Leibniz rule and defines a well-behaved connection on the product module $E \otimes F$.
• Compute the curvature of the product connection and show it vanishes if both input connections are flat, regardless of the twisting map used.
• Apply the construction explicitly to the quantum plane $k_q[x,y] = k[x] \rtimes_R k[y]$ with $R(y \otimes x) = qx \otimes y$, deriving the full family of product connections in terms of gauge potentials.

Experimental results

Research questions

• RQ1 How can connections on two algebras be combined to define a connection on their twisted tensor product?
• RQ2 Does the curvature of the resulting product connection depend on the choice of twisting map or module twisting map?
• RQ3 Is the product of two flat connections on the factors necessarily flat on the twisted product?
• RQ4 Can the construction be extended to bimodule connections, preserving the bimodule structure?
• RQ5 What are all possible product connections on the quantum plane $k_q[x,y]$?

Key findings

• The product connection constructed on a twisted tensor product of algebras is independent of the choice of twisting map and module twisting map, ensuring consistency across different realizations of the product.
• The curvature of the product connection is explicitly computed and shown to vanish if both input connections are flat, proving that flatness is preserved under the product construction.
• The product of two flat connections yields a flat connection on the twisted tensor product, generalizing a classical result to the noncommutative setting.
• The construction preserves the bimodule structure, meaning the product of two bimodule connections is again a bimodule connection.
• All product connections on the quantum plane $k_q[x,y]$ are fully computed, expressed in terms of the gauge potentials of the input connections on $k[x]$ and $k[y]$, with explicit formulas involving $q$-deformed actions.
• The Grassmann connection on free modules over $k[x]$ and $k[y]$ is extended to the quantum plane via the product construction, yielding a canonical flat product connection.