Homology invariants of quadratic maps

Given a real projective algebraic set X we could hope that the equations describing it can give some information on its topology, e.g. on the number of its connected components. Unfortunately in the general case this hope is too vague and there is no direct way to extract such information from the algebraic description of X: Even the problem to decide whether X is empty or not is far from an easy visualization and requires some complicated algebraic machinery. A fi rst step observation is that as long as we are interested only in the topology of X, we can replace, using some Veronese embedding, the original ambient space with a much bigger RPn and assume that X is cut by quadratic equations. The price for this is the increase of the number of equations de ning our set; the advantage is that quadratic polynomials are easier to handle and our hope becomes more concrete..

A형 양자 아핀 초대수에 대한 초 쌍대성

학위논문(박사) -- 서울대학교대학원 : 자연과학대학 수리과학부, 2023. 8. 권재훈.We propose a new approach to the study of representations of quantum affine (su-per)algebras, motivated from super duality. Namely, we study a category of interest by finding its bosonic or fermionic counterpart, and then construct supersymmetric analogues and functors to interpolate bosons and fermions. A key role is played by R-matrices and their spectral decompositions, which enables a uniform treatment for super and non-super cases. In this thesis, we consider two module categories of quantum affine (super)algebras of type A. First, the category of polynomial representations is studied, where a uniform approach is possible thanks to the powerful Schur–Weyl-type duality. We construct a functor that directly relates the category for quantum affine algebras to the one for superalgebras, and lift it to an equivalence between inverse limits of categories. Second, we introduce a category of infinite-dimensional representations called q-oscillator representations, whose irreducible objects naturally correspond to finite-dimensional irreducible representations. Since the former can be seen as a bosonic counterpart of the latter, we explain the correspondence by introducing an analogous category for quantum affine superalgebras. In the spirit of super duality, the connection provided by the super analogue is expected to give rise to an equivalence of categories.본 학위논문은 양자 아핀 (초)대수의 표현론에 관한 초 쌍대성에 기반을 둔 새로운 접근을 제시한다. 구체적으로는, 주어진 범주를 분석하기 위하여 우선 보손 혹은 페르미온 짝을 찾고, 그 초대칭 유사체를 구성하여 보손 측과 페르미온 측을 이어주는 함자를 찾는 방법이다. 이때 주요한 역할을 하는 것이 R-행렬과 그 스펙트럴 분해로, 이를 통해 각각의 경우를 기존의 잘 알려진 유한차원 표현론과 유사한 방식으로 분석할 수 있다. 본 학위논문에서는 A형 양자 아핀 (초)대수의 두 가지 모듈 범주를 고려하고자 한다. 첫째로 다루어지는 것은 다항식 표현들의 범주로, 이 경우에는 유용한 슈어–바일 류의 쌍대성을 이용하여 특히 통일적인 분석이 가능하였다. 양자 아핀 초대수에 대한 범주와 기존의 양자 아핀 대수에 대한 범주를 직접적으로 연결하는 함자를 건설하였고, 이로부터 범주들의 역극한 사이의 범주 동치를 얻게 된다. 둘째로, q-진동자 표현이라 불리는 무한차원 표현들의 범주를 도입하였고, 유한차원 기약 표현들과 자연스럽게 대응되는 기약 q-진동자 표현들을 찾았다. q-진동자 표현들은 유한차원 표현들의 보손 짝으로 볼 수 있기 때문에 양자 아핀 초대수에 대한 유사한 모듈 범주를 도입함으로써 이 대응을 설명할 수 있었으며, 이러한 초대칭 유사체를 통한 q-진동자와 유한차원 표현들 사이의 연결은 초 쌍대성의 철학에 의해 범주들의 동치로 이어질 것으로 예상된다.1 Introduction 1 1.1 Quantum affine superalgebras 2 1.2 Super duality 3 1.3 Main results 4 1.3.1 Generalized quantum groups 5 1.3.2 Super duality for polynomial representations 6 1.3.3 Oscillator representations and super duality 8 1.4 Organization 10 2 Preliminaries 12 2.1 General linear Lie superalgebra gl_MN 13 2.2 Quantum affine algebra 17 2.2.1 Affine Lie algebras and quantum affine algebras 17 2.2.2 Finite-dimensional representations of quantum affine algebras 20 2.3 Quiver Hecke algebra 25 2.3.1 Quiver Hecke algebra 26 2.3.2 Quiver Hecke algebra of type A and their simple modules 30 3 Generalized Quantum Groups of type A 33 3.1 Generalized quantum group of affine type A 34 3.1.1 Definition 34 3.1.2 Quantum affine superalgebra and algebra isomorphism 36 3.1.3 Universal R-matrix 39 3.2 Finite type subalgebra and its polynomial representations 43 4 Super duality for polynomial representations 47 4.1 Super duality for polynomial representations of gl_n 48 4.2 Finite-dimensional representations of U(ε) 52 4.2.1 Fundamental representations 52 4.2.2 R-matrix 55 4.2.3 Fusion construction of irreducible polynomial representations 59 4.2.4 Generalized quantum affine Schur-Weyl duality 62 4.3 Super duality 66 4.3.1 Truncation 67 4.3.2 Equivalence of duality functors 72 4.3.3 Inverse limit category 77 4.3.4 Super duality 81 5 Oscillator representations of U_q(gl_n) 85 5.1 Howe duality and oscillator representations of gl_n 86 5.2 Oscillator representations of U_q(gl_n) 91 5.2.1 Fock space and fundamental q-oscillator representations 92 5.2.2 Oscillator representations of U_q(gl_n) 95 5.3 Oscillator representations of U_q(\hat{gl}_n) 102 5.3.1 Category O_osc,ε 102 5.3.2 R-matrix and spectral decomposition 104 5.3.3 Fusion construction of irreducible q-oscillator representations 110 5.3.4 Correspondence of irreducibles and super duality 112 6 Proofs 115 6.1 Chapter 4 115 6.1.1 Proof of Lemma 4.2.6 115 6.1.2 Proof of Theorem 4.3.13 120 6.2 Chapter 5 126 6.2.1 Proof of Proposition 5.3.2 126 6.2.2 Proof of Conjecture 5.3.16 for s=1 128 Abstract(in Korean) 142박