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Posets of finite GK-dimensional graded pre-Nichols algebras of diagonal type

Iván Angiono, Emiliano Campagnolo

Abstract

We classify graded pre-Nichols algebras of diagonal type with finite Gelfand-Kirillov dimension. The characterization is made through an isomorphism of posets with the family of appropriate subsets of the set of positive roots coming from central extensions of Nichols algebras of diagonal type, generalizing the corresponding extensions for small quantum groups in de Concini-Kac-Procesi forms of quantum groups. On the way to achieving this result, we also classify graded quotients of algebras of functions of unipotent algebraic groups attached to semisimple Lie algebras.

Posets of finite GK-dimensional graded pre-Nichols algebras of diagonal type

Abstract

We classify graded pre-Nichols algebras of diagonal type with finite Gelfand-Kirillov dimension. The characterization is made through an isomorphism of posets with the family of appropriate subsets of the set of positive roots coming from central extensions of Nichols algebras of diagonal type, generalizing the corresponding extensions for small quantum groups in de Concini-Kac-Procesi forms of quantum groups. On the way to achieving this result, we also classify graded quotients of algebras of functions of unipotent algebraic groups attached to semisimple Lie algebras.
Paper Structure (11 sections, 19 theorems, 56 equations)

This paper contains 11 sections, 19 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. On pre-Nichols algebras of diagonal type
  3. Nichols algebras and pre-Nichols algebras
  4. Central extensions of braided Hopf algebras
  5. Distinguished pre-Nichols algebras
  6. Twist equivalence and pre-Nichols algebras
  7. Hopf ideals of algebras of functions of unipotent groups
  8. Posets of pre-Nichols algebras of diagonal type
  9. Eminent pre-Nichols algebras which are not distinguished
  10. The connected case
  11. The non-connected case

Key Result

Theorem 1.1

Let $\mathfrak{q}$ be a braiding matrix whose connected components are not of Cartan types $A_{\theta}$, $D_{\theta}$ with label $q=-1$ neither one dimensional with label $q=\pm 1$. For each $\underline{\beta}\in\underline{\widehat{\mathfrak O}^{\mathfrak{q}}_+}$ let $z_{\beta}$ be a generator of $\ Then each $\mathscr{B}(\mathfrak{q},B)$ is an $\mathbb N_0^{\theta}$-graded pre-Nichols algebra suc

Theorems & Definitions (45)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Example 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • Proposition 2.7
  • ...and 35 more