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Enhanced nilHecke algebras and baby Verma modules

Elia Rizzo, Pedro Vaz

TL;DR

This work builds a $p$-differential (p-dg) enhancement of the nilHecke framework to categorify baby Verma modules for the small quantum group $u_q(\mathfrak{sl}_2)$ at a root of unity. It constructs a derivation on the enhanced nilHecke algebra $A_n$, extends it to a $p$-dg structure, and shows that the Grothendieck group $\mathbf{K}_0(A)$ realizes a baby Verma module $M(\lambda)$ under a categorical $\mathfrak{sl}_2$-action via functors $\mathcal{E},\mathcal{F}$. The authors further lift a Witt-type $\mathfrak{sl}_2$-action to the extended ring $R_n$, yielding a non-diagonalizable $\mathfrak{sl}_2$-module structure on $R_n$ and filtrations on $A_n$ by weight submodules; these results connect Verma-module categorification with root-of-unity phenomena. The framework offers a coherent path between $p$-dg categorification, Verma module theory at roots of unity, and structured $\mathfrak{sl}_2$-actions, with potential implications for link homology and cyclotomic phenomena.

Abstract

We define a derivation on the enhanced nilHecke algebra yielding a p-dg algebra when working over a field of characteristic p. We define functors on the category of p-dg modules resulting in an action of small quantum sl2 on the Grothendieck group, which is isomorphic to a baby Verma module. We upgrade the derivation into an action of the Lie algebra sl(2).

Enhanced nilHecke algebras and baby Verma modules

TL;DR

This work builds a -differential (p-dg) enhancement of the nilHecke framework to categorify baby Verma modules for the small quantum group at a root of unity. It constructs a derivation on the enhanced nilHecke algebra , extends it to a -dg structure, and shows that the Grothendieck group realizes a baby Verma module under a categorical -action via functors . The authors further lift a Witt-type -action to the extended ring , yielding a non-diagonalizable -module structure on and filtrations on by weight submodules; these results connect Verma-module categorification with root-of-unity phenomena. The framework offers a coherent path between -dg categorification, Verma module theory at roots of unity, and structured -actions, with potential implications for link homology and cyclotomic phenomena.

Abstract

We define a derivation on the enhanced nilHecke algebra yielding a p-dg algebra when working over a field of characteristic p. We define functors on the category of p-dg modules resulting in an action of small quantum sl2 on the Grothendieck group, which is isomorphic to a baby Verma module. We upgrade the derivation into an action of the Lie algebra sl(2).
Paper Structure (17 sections, 39 theorems, 214 equations)

This paper contains 17 sections, 39 theorems, 214 equations.

Table of Contents

  1. Reminders
  2. p-dg algebras and modules
  3. p-dg nilHecke algebras
  4. The enhanced nilHecke algebra
  5. The p-dg enhanced nilHecke algebra
  6. A derivation on Rn
  7. Partial derivatives on the extended polynomial ring
  8. Equivariant homomorphisms
  9. The Grothendieck group of (A,d) and baby Verma modules
  10. Baby Verma modules
  11. The Grothendieck group of (A,d)
  12. A categorical action
  13. An action of sl2
  14. Witt actions
  15. The sl2-module Rn
  16. ...and 2 more sections

Key Result

Proposition 1.9

The algebra $A_n$ is generated by the elements $x_1,\dotsc,x_n$, $T_1,\dotsc,T_{n-1}$, and $\omega_1,\dotsc,\omega_n$, modulo the nilHecke relations and the relations involving the $\omega$'s It is a $\mathbb{Z}^2$-graded super algebra, where the $x_i$ and the $T_i$ are even, while the $\omega_i$ are odd and the $\mathbb{Z}^2$ grading is given by $\deg(x_i)=(2,0)$, $\deg(T_j)=(-2,0)$, $\deg(w_i)

Theorems & Definitions (100)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Proposition 1.9
  • proof
  • ...and 90 more