Prime Roots Of Unity Research Articles

Categorifying Hecke algebras at prime roots of unity, part I

We equip the type A A diagrammatic Hecke category with a special derivation, so that after specialization to characteristic p p it becomes a p p -dg category. We prove that the defining relations of the Hecke algebra are satisfied in the p p -dg Grothendieck group. We conjecture that the p p -dg Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztig basis and the p p -canonical basis. More precise conjectures will be found in the sequel. Here are some other results contained in this paper. We provide an incomplete proof of the classification of all degree + 2 +2 derivations on the diagrammatic Hecke category, and a complete proof of the classification of those derivations for which the defining relations of the Hecke algebra are satisfied in the p p -dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence. We prove that no such derivation exists in simply-laced types outside of finite and affine type A A . We also examine a particular Bott-Samelson bimodule in type A 7 A_7 , which is indecomposable in characteristic 2 2 but decomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any characteristic, which is the analogue in the p p -dg setting of being indecomposable.

Cell 2-representations and categorification at prime roots of unity

Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2-categories, enriched with a p-differential, which satisfy finiteness conditions analogous to those of finitary or fiat 2-categories. We construct cell 2-representations in this setup, and consider a class of 2-categories stemming from bimodules over a p-dg category in detail. This class is of particular importance in the categorification of quantum groups, which allows us to apply our results to cyclotomic quotients of the categorifications of small quantum group of type sl2 at prime roots of unity by Elias–Qi (2016) [3]. Passing to stable 2-representations gives a way to construct triangulated 2-representations, but our main focus is on working with p-dg enriched 2-representations that should be seen as a p-dg enhancement of these triangulated ones.

A CATEGORIFICATION OF A QUANTUM FROBENIUS MAP

A quantum Frobenius map a la Lusztig for $\mathfrak{s}\mathfrak{l}_{2}$ is categorified at a prime root of unity.

A categorification of quantum [formula omitted] at prime roots of unity

We categorify the Beilinson–Lusztig–MacPherson integral form of quantum sl(2) specialized at a prime root of unity.

A categorification of the Burau representation at prime roots of unity

We construct a p-DG structure on an algebra Koszul dual to a zigzag algebra used by Khovanov and Seidel to construct a categorical braid group action. We show there is a braid group action in this p-DG setting.

Equiangular lines and covers of the complete graph

The relation between equiangular sets of lines in the real space and distance-regular double covers of the complete graph is well known and studied since the work of Seidel and others in the 70s. The main topic of this paper is to continue the study on how complex equiangular lines relate to distance-regular covers of the complete graph with larger index. Given a set of equiangular lines meeting the relative (or Welch) bound, we show that if the entries of the corresponding Gram matrix are prime roots of unity, then these lines can be used to construct an antipodal distance-regular graph of diameter three. We also study in detail how the absolute (or Gerzon) bound for a set of equiangular lines can be used to derive bounds of the parameters of abelian distance-regular covers of the complete graph.

An approach to categorification of some small quantum groups II

We categorify an idempotented form of quantum sl2 and some of its simple representations at a prime root of unity.

An approach to categorification of some small quantum groups

We categorify one half of the small quantum sl(2) at a prime root of unity. An extension of this construction to an arbitrary simply-laced case is proposed.

Quantum invariants of 3-manifolds: Integrality, splitting, and perturbative expansion

We consider quantum invariants of 3-manifolds associated with arbitrary simple Lie algebras. Using the symmetry principle we show how to decompose the quantum invariant as the product of two invariants, one of them is the invariant corresponding to the projective group. We then show that the projective quantum invariant is always an algebraic integer, if the quantum parameter is a prime root of unity. We also show that the projective quantum invariant of rational homology 3-spheres has a perturbative expansion a la Ohtsuki. The presentation of the theory of quantum 3-manifold invariants is self-contained.

A simple proof of integrality of quantum invariants at prime roots of unity

Recently, Hitoshi Murakami has shown that the quantum SU(2)- and SO(3)-invariants of 3-manifolds at roots of unity of prime order are algebraic integers. Unfortunately, his proof is by a very complicated computation. Here, a quite different and very simple proof is presented, based on the second author's method to show that the Turaev–Viro invariant is the square of the modulus of the Reshetikhin–Turaev invariant.

Analog diversity coding to provide transparent self-healing communication networks

pg 100 diversity coding, as introduced in Ayanoglu et al. (1990), is a method of protection against failures in a communication network or a storage system, which is based on introducing a digital error-correcting code across independent links. This technique makes efficient use of the extra network capacity needed for coding and has the additional advantages of being nearly instantaneous, not requiring a feedback channel, rerouting, or resynchronization. In high-speed (multi Gbps) networks, digital coding will be difficult to implement, and the purpose of the present paper is to demonstrate how diversity coding may be implemented in the analog domain using the discrete fourier transform (DFT). In particular, the authors show that the DFT is a continuous-amplitude maximum-distance separable code over the field of complex numbers when the transform kernel is a prime root of unity. This code can be used to generate self-healing or fault-tolerant communication networks for continuous- or discrete-amplitude signals, as long as continuous-amplitude parity channels are available. The authors describe electrical and optoelectronic implementations, and a signal estimation approach to combat channel noise and thereby improve the performance of the analog diversity coding system. The most important advantage of this technique is in greatly simplifying the encoders and decoders of diversity coding systems for high-speed networks, such as fiber-optic wavelength division multiplexed networks. Application of analog diversity coding to systems with analog sources, such as telemetry systems is also possible.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>