Weak Generators of W-algebras for type A = A형 W-대수의 약생성자

다국어 초록 (Multilingual Abstract)

We study the structure of both the affine and classical affine W-algebras associated with Lie algebras and Lie superalgebras of type A.
After reviewing the definitions of vertex algebras and Poisson vertex algebras and recalling the correspondence between them, we introduce the two equivalent formulations of the classical affine W-algebra and explain how they correspond to the affine W-algebra.
We then analyze the structural properties of affine W-algebras, focusing in particular on the Hamiltonian operators and strong generating sets that appear in both the quantum and classical settings.
Furthermore, we examine the relations between the generators of these two algebras and clarify how the conformal structures are preserved under the quasiclassical limit.

In the latter part of the paper, we construct explicit weak generating sets for affine W-algebras with both large and small conformal weights.
For the large weight cases, we provide construction for classical affine W-algebras $\mathcal{W}^k(\mathfrak{sl}_N, f)$, $\mathcal{W}^k(\mathfrak{sl}_{N_1\mid N_2}, f)$ and affine W-algebras $W^k(\mathfrak{sl}_N, f)$, $W^k(\mathfrak{sl}_{N_1\mid N_2}, f)$.
Analogous results are obtained for the small weight cases, where we describe their inductive generation mechanisms.
Finally, several examples are given, including the principal, rectangular and minimal nilpotent cases, as well as examples associated with the Lie superalgebras $\mathfrak{sl}_{n+1\mid n}$ and $\mathfrak{sl}_{m\mid n}$ with $m\geq n+2$, which demonstrate the general theory developed in this thesis.

국문 초록 (Abstract)

본 논문에서는 A형 리 대수 및 리 초대수에 대응하는 아핀 W-대수와 고전 아핀 W-대수의 구조를 연구한다.
우선 꼭짓점 대수와 푸아송 꼭짓점 대수의 정의를 정리하고 이들 사이의 대응관계를 설명한다.
그 후 고전 아핀 W-대수의 두가지 동치인 정의를 제시하고 이들이 아핀 W-대수와 어떻게 대응하는지를 설명한다.
다음으로 양자 및 고전 아핀 W-대수의 구조적 성질을 분석하며, 특히 해밀턴 작용소와 강생성자의 성질을 중심으로 고찰한다.
또한 두 대수의 생성자들 사이의 관계를 규명하고 준고전적 극한에서 등각 구조가 어떻게 보존되는 지를 명확히 한다.

논문의 후반부에서는 큰 등각 차수와 작은 등각 차수를 갖는 경우에 대해 아핀 W-대수의 구체적인 약생성자를 구성한다.
큰 등각 차수의 경우, $\mathfrak{sl}_N$과 $\mathfrak{sl}_{N_1\mid N_2}$에 해당하는 아핀 W-대수에 대한 구성을 제시하고, 작은 등각 차수의 경우에도 비슷하게 구체적인 생성을 제시하며 이들이 귀납적으로 생성되는 과정을 설명한다.
마지막으로, 주요 멱영원의 경우, 직사각형 멱영원의 경우, 최소 멱영원의 경우뿐 아니라 $\mathfrak{sl}_{n+1\mid n}$ 및 $\mathfrak{sl}_{m\mid n}$ $(m\geq n+2)$와 관련된 멱영원의 예시들을 통해 본 논문에서 전개된 일반 이론을 구체적으로 확인한다.

목차 (Table of Contents)

• 1 Introduction 1
• 1.1 W-algebras 2
• 1.2 Main results 3
• 1.3 Organization 5
• 2 Preliminaries 6
• 2.1 Vertex algebras 6
• 2.2 Poisson vertex algebras 15
• 2.3 Relations between vertex algebras and Poisson vertex algebras 16
• 3 Affine W-algebras 19
• 3.1 Affine W-algebras $W^k(\mathfrak{g},f)$ 20
• 3.2 Classical affine W-algebras $\mathcal{W}^k(\mathfrak{g},f)$ 25
• 3.2.1 The first definition of classical affine W-algebras 25
• 3.2.2 The second definition of classical affine W-algebras 29
• 3.2.3 Equivalence of the two definitions of classical affine W-algebras 31
• 3.3 Relations between affine and classical affine W-algebras 33
• 4 Structures of affine W-algebras 38
• 4.1 Properties of affine W-algebras 38
• 4.1.1 Hamiltonian operator of affine W-algebras 39
• 4.1.2 Strong generating set for affine W-algebras 41
• 4.2 Properties of classical affine W-algebras 47
• 4.2.1 Hamiltonian operator of classical affine W-algebras 48
• 4.2.2 Strong generating set for classical affine W-algebras 49
• 4.3 Relations between generators of affine and classical affine W-algebras 58
• 5 Weak Generating Set with Large Conformal Weights 61
• 5.1 $\mathcal{W}^k(\mathfrak{sl}_N,f)$ 63
• 5.1.1 Basis of centralizer of $f$ 63
• 5.1.2 Weak generating set of $\mathcal{W}^k(\mathfrak{sl}_N,f)$ 65
• 5.2 Weak generating set of $\mathcal{W}^k(\mathfrak{sl}_{N_1\mid N_2} ,f)$ 75
• 5.3 Weak generating set of $W^k(\mathfrak{sl}_N,f)$ and $W^k(\mathfrak{sl}_{N_1\mid N_2} ,f)$ 77
• 6 Weak Generating Set with Small Conformal Weights 79
• 6.1 Weak generating set of $\mathcal{W}^k(\mathfrak{sl}_N,f)$ 80
• 6.2 Weak generating set of $\mathcal{W}^k(\mathfrak{sl}_{N_1\mid N_2},f)$ 87
• 6.3 Weak generating set of $W^k(\mathfrak{sl}_N,f)$ and $W^k(\mathfrak{sl}_{N_1\mid N_2} ,f)$ 89
• 7 Examples of weak generating sets in classical affine W-algebras 91
• 7.1 $\mathcal{W}^k(\mathfrak{sl}_N,f_{prin})$ with a principal nilpotent $f_{prin}$ 92
• 7.2 $\mathcal{W}^k(\mathfrak{sl}_{n+1\mid n},(n+ 1,n))$ 94
• 7.3 $\mathcal{W}^k(\mathfrak{sl}_{m
• n},(m,n))$ with $m\geq n+ 2$ 95
• 7.4 $\mathcal{W}^k(\mathfrak{g},f)$ with a rectangular nilpotent $f$ 97
• 7.5 $\mathcal{W}^k(\mathfrak{g},f)$ with a minimal nilpotent $f$ 99
• 7.6 Example 100
• Abstract (in Korean) 105