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A Categorification of Subword Complexes and Its Hall Algebra

Mikhail Gorsky, Zijun Li

Abstract

Bergeron and Ceballos defined a Hopf algebra structure on equivalence classes of subword complexes. We introduce a category of subword complexes, endow it with a proto-exact-like structure, and show that the corresponding dual Hall Hopf algebra is isomorphic to the algebra of Bergeron-Ceballos. We prove that the full subcategory of root-independent objects is proto-abelian in the sense of Dyckerhoff. We give a categorical lift of flips in subword complexes. We consider a version of a category of formal direct sums of subobjects for a root-independent subword complex and interpret it in terms of quivers. If the corresponding quiver is a tree, the category is endowed with a proto-exact structure. We show that its Hall algebra is isomorphic to the Hall algebra of the category of representations of this quiver over $\mathbb{F}_1$. Under certain conditions, a flip corresponds to changing a proto-exact structure while keeping the category the same up to isomorphism, which corresponds to a non-trivial automorphism of the Hall algebra. In type $A$, this leads to a realization of the nilpotent part of the universal enveloping algebra and its automorphisms.

A Categorification of Subword Complexes and Its Hall Algebra

Abstract

Bergeron and Ceballos defined a Hopf algebra structure on equivalence classes of subword complexes. We introduce a category of subword complexes, endow it with a proto-exact-like structure, and show that the corresponding dual Hall Hopf algebra is isomorphic to the algebra of Bergeron-Ceballos. We prove that the full subcategory of root-independent objects is proto-abelian in the sense of Dyckerhoff. We give a categorical lift of flips in subword complexes. We consider a version of a category of formal direct sums of subobjects for a root-independent subword complex and interpret it in terms of quivers. If the corresponding quiver is a tree, the category is endowed with a proto-exact structure. We show that its Hall algebra is isomorphic to the Hall algebra of the category of representations of this quiver over . Under certain conditions, a flip corresponds to changing a proto-exact structure while keeping the category the same up to isomorphism, which corresponds to a non-trivial automorphism of the Hall algebra. In type , this leads to a realization of the nilpotent part of the universal enveloping algebra and its automorphisms.

Paper Structure

This paper contains 13 sections, 29 theorems, 88 equations.

Table of Contents

  1. Categorification of subword complexes
  2. Subword complexes and flats
  3. The construction of the category of subword complexes C
  4. Categorical description of flips
  5. A proto-abelian subcategory
  6. root configuration quiver and proto-exact structure on the category SX
  7. Root configuration quiver for objects in D
  8. Subquiver category
  9. Proto-exact structure on SX
  10. Isomorphic Hall algebras
  11. Flips and Reflections
  12. Type A
  13. Examples of non-linear orientations

Key Result

Lemma 1.4

BC Any flat $F$ of an quadruple $X\in \tilde{\mathcal{C}}$ induces a quadruple $X_F\in \tilde{\mathcal{C}}$, such that $V_{X_F}=\operatorname{V}(F)$ and $n_{X_F}=|F|$. Simple roots of $\Phi_{X_F}$ are given by $\beta_F$. We have $\operatorname{r}_X(i)=\operatorname{r}_{X_F}(\operatorname{b}_F(i))$ f

Theorems & Definitions (101)

  • Definition 1.1
  • Remark 1.2
  • Definition 1.3
  • Lemma 1.4
  • Definition 1.5
  • Remark 1.6
  • Definition 1.7
  • Definition 1.8
  • Theorem 1.9
  • proof
  • ...and 91 more