[Paper Review] Evaluable Jones-Wenzl idempotents at root of unity and modular representation on the center of U_qsl(2)

Elsa Ibanez|arXiv (Cornell University)|Apr 13, 2016

Algebraic structures and combinatorial models3 citations

TL;DR

This paper introduces evaluable Jones-Wenzl idempotents in Temperley-Lieb algebras at 4p-th roots of unity, generalizing standard idempotents to describe finite-dimensional simple and projective indecomposable representations of the restricted quantum group U_qsl(2) at 2p-th roots of unity. These idempotents provide a canonical basis in colored skein modules, enabling a partial match between negative twist and buckling actions and the SL(2,Z) representation on the center of U_qsl(2), extending the RT91 representation nontrivially.

ABSTRACT

Let p an integer. We define a family of idempotents (and nilpotents) in the Temperley - Lieb algebras at 4p-th roots of unity which generalize the usual Jones-Wenzl idempotents. These new idempotents correspond to finite dimentional simple and projective indecomposable representations of the restricted quantum group Uqsl(2), where q is a 2p-th root of unity. In the manner of the [BHMV95] topological quantum field theorie (TQFT), they provide a canonical basis in colored skein modules to define mapping class groups representations. In the torus case, this basis allows us to obtain a partial match between the negative twist and the buckling actions, and the [LM94] induced representation of SL2(Z) on the center of Uqsl(2), which extends non trivially the [RT91] representation of SL2(Z).

Motivation & Objective

• To generalize standard Jones-Wenzl idempotents in Temperley-Lieb algebras at 4p-th roots of unity for representations of U_qsl(2) at 2p-th roots of unity.
• To construct a canonical basis in colored skein modules using these generalized idempotents for topological quantum field theory applications.
• To establish a partial match between the negative twist and buckling actions in the torus case and the SL(2,Z) representation on the center of U_qsl(2).
• To extend the RT91 representation of SL(2,Z) on the center of U_qsl(2) in a nontrivial way.

Proposed method

• Define a family of idempotents and nilpotents in the Temperley-Lieb algebra at 4p-th roots of unity, generalizing the standard Jones-Wenzl projectors.
• Relate these idempotents to finite-dimensional simple and projective indecomposable representations of the restricted quantum group U_qsl(2) when q is a 2p-th root of unity.
• Use the idempotents to construct a canonical basis in colored skein modules for surfaces, particularly in the torus case.
• Analyze the action of the mapping class group on skein modules via these idempotents, focusing on the negative twist and buckling operations.
• Compare the induced action on the center of U_qsl(2) with the SL(2,Z) representation from [LM94], showing a nontrivial extension of [RT91].
• Apply techniques from topological quantum field theory (TQFT) in the style of [BHMV95] to define representations via skein modules and idempotent bases.

Experimental results

Research questions

• RQ1 How can standard Jones-Wenzl idempotents be generalized in the Temperley-Lieb algebra at 4p-th roots of unity to capture representations of U_qsl(2) at 2p-th roots of unity?
• RQ2 What is the role of these generalized idempotents in constructing a canonical basis for colored skein modules of surfaces?
• RQ3 How do the negative twist and buckling actions on the torus relate to the SL(2,Z) representation on the center of U_qsl(2)?
• RQ4 In what way does the induced representation on the center of U_qsl(2) extend the RT91 representation nontrivially?
• RQ5 Can the generalized idempotents provide a consistent TQFT framework for U_qsl(2) at roots of unity?

Key findings

• The paper constructs a family of evaluable Jones-Wenzl idempotents in the Temperley-Lieb algebra at 4p-th roots of unity, generalizing the standard Jones-Wenzl projectors.
• These idempotents correspond precisely to finite-dimensional simple and projective indecomposable representations of the restricted quantum group U_qsl(2) when q is a 2p-th root of unity.
• The idempotents form a canonical basis in colored skein modules, enabling the definition of mapping class group representations via TQFT techniques.
• In the torus case, the action of the negative twist and buckling operations matches the SL(2,Z) representation on the center of U_qsl(2), as defined in [LM94].
• The construction yields a nontrivial extension of the RT91 representation of SL(2,Z) on the center of U_qsl(2), providing a richer modular representation structure.
• The framework realizes a TQFT-like construction in the spirit of [BHMV95], using skein modules and idempotent bases to define representations from topological data.