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Minimal nilpotent finite $W$-algebra and cuspidal module category of $\mathfrak{sp}_{2n}$

Genqiang Liu, Mingjie Li

Abstract

Let $U_S$ be the localization of $U(\mathfrak{sp}_{2n})$ with respect to the Ore subset $S$ generated by the root vectors $X_{ε_1-ε_2},\dots,X_{ε_1-ε_n}, X_{2ε_1}$. We show that the minimal nilpotent finite $W$-algebra $W(\mathfrak{sp}_{2n}, e)$ is isomorphic to the centralizer $C_{U_S}(B)$ of some subalgebra $B$ in $U_S$, and it can be identified with a tensor product factor of $U_S$. As an application, we show that the category of weight $\mathfrak{sp}_{2n}$-modules with injective actions of all root vectors and finite-dimensional weight spaces is equivalent to the category of finite-dimensional modules over $W(\mathfrak{sp}_{2n}, e)$, explaining the coincidence that both of them are semi-simple.

Minimal nilpotent finite $W$-algebra and cuspidal module category of $\mathfrak{sp}_{2n}$

Abstract

Let be the localization of with respect to the Ore subset generated by the root vectors . We show that the minimal nilpotent finite -algebra is isomorphic to the centralizer of some subalgebra in , and it can be identified with a tensor product factor of . As an application, we show that the category of weight -modules with injective actions of all root vectors and finite-dimensional weight spaces is equivalent to the category of finite-dimensional modules over , explaining the coincidence that both of them are semi-simple.

Paper Structure

This paper contains 11 sections, 13 theorems, 80 equations.

Table of Contents

  1. introduction
  2. Preliminaries on the minimal nilpotent finite $W$-algebra $W(\mathfrak{sp}_{2n},e)$
  3. Lie algebra $\mathfrak{sp}_{2n}$
  4. Minimal nilpotent finite $W$-algebra $W(\mathfrak{sp}_{2n},e)$ and its twisted form
  5. Centralizer realization of the finite $W$-algebra $W(\mathfrak{sp}_{2n},e)$
  6. Centralizer of $\mathfrak{h}_n\oplus \mathfrak{m}_n$
  7. Isomorphism between $W(\mathfrak{sp}_{2n}, e)$ and $C_{U_S}(B)$
  8. One dimensional representations of $W(\mathfrak{sp}_{2n},e)$
  9. Equivalence between the category of cuspidal modules over $\mathfrak{sp}_{2n}$ and $\mathbf{fmod}$ over $W(\mathfrak{sp}_{2n},e)$
  10. Block decomposition
  11. The equivalence between $W\text{-fmod}$ and $\mathcal{C}_{\mathbf{a}}(\chi_\lambda)$

Key Result

Theorem 1.1

There is an algebra isomorphism $U_S\cong D^\partial_n\otimes W(\mathfrak{sp}_{2n},e)$, where $D_n^{\partial}$ is the localization of the Weyl algebra $D_n$ with respect to the Ore subset $\{\partial_1^{i_1}\dots\partial_n^{i_n}\mid i_1,\dots,i_n\in \mathbb{Z}_{\geq 0}\}$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • ...and 17 more