本文選題：motivic　切入點(diǎn)：Hall代數(shù)　出處：《清華大學(xué)》2016年博士論文

【摘要】：本學(xué)位論文的主要內(nèi)容是研究幾何化實(shí)現(xiàn)范疇上的李代數(shù)和結(jié)合代數(shù)。令(?)bi_A是阿貝爾范疇A上全體對(duì)象的stack,令CF((?)bi_A)表示全體(?)bi_A上的可構(gòu)函數(shù)組成的Q上的向量空間。在[1]中,Joyce證明了 CF((?)bi_A)在卷積乘法下是一個(gè)結(jié)合代數(shù),并且由不可分解可構(gòu)函數(shù)全體組成的子空間CFind((?)bi_A)是CF((?)bi_A)的李子代數(shù),其中不可分解可構(gòu)函數(shù)是指支撐點(diǎn)都對(duì)應(yīng)于A的不可分解對(duì)象的可構(gòu)函數(shù)。令CFfin((?)bi_A)是由支持有限的可構(gòu)函數(shù)組成的向量空間,CFfi芫((?)bi_A)是CFfin((?)bi_A)和CFind((?)bi_A)的交。假設(shè)A是有限表示型的,即對(duì)A中任意兩個(gè)對(duì)象,它們擴(kuò)張出來(lái)的中間項(xiàng)的同構(gòu)類(lèi)是有限多個(gè)的,那么Joyce證明了 CFfin((?)bi_A)是李代數(shù)CFfining((?)bi_A)的包絡(luò)代數(shù)。進(jìn)一步的,Joyce將可構(gòu)函數(shù)推廣到stack函數(shù),將CF((?)bi_A)推廣到了 SF((?)bi_A),其中SF((?)bi_A)表示stack函數(shù)生成的Q上的向量空間。Joyce證明了 SF((?)bi_A)是結(jié)合的(Ringel-Hall代數(shù))。Kontsevich和Soibelman[2],Bridgeland[3]考慮了其它范疇上的stack函數(shù)所構(gòu)成的代數(shù),這一類(lèi)代數(shù)被統(tǒng)稱(chēng)為motivic Hall代數(shù)。本文的主要結(jié)果是:(1)對(duì)于一個(gè)Krull-Schmidt正合K-范疇A,我們證明CF((?)bi_A)和CFind((?)bi_A)分別是一個(gè)結(jié)合代數(shù)和CF((?)bi_A)的李子代數(shù)。(2)存在CF((?)bi_A)的一個(gè)子代數(shù)CFKS((?)bi_A),使得CFKS((?)bi_A)同構(gòu)于CFind((?)bi_A)的包絡(luò)代數(shù)。(3)在代數(shù)CFKS((?)bi_A)上存在一個(gè)余乘結(jié)構(gòu),它和乘法結(jié)構(gòu)是相容的,即退化形式的格林公式成立。這些結(jié)論改進(jìn)了 Joyce的結(jié)果,同時(shí)也是[4]的推廣。(4)給定一個(gè)遺傳阿貝爾范疇A和它的motivic Hall代數(shù)MH(A),我們定義了MH(A)上的余乘和余單位,并證明了MH(A)上的格林公式,即MH(A)上乘法和余乘的相容性,由此MH(A)具有雙代數(shù)(bialgebra)結(jié)構(gòu)。
[Abstract]:The main content of this dissertation is to study lie algebras and associative algebras in the category of geometric realization.So that Mr A is the stack-set of all objects on Abel category A, and that CFT / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C means the vector space on Q which is composed of constructable functions on the whole.In [1], Joyce proved that the CFT / T _ T _ A) is a associative algebra under convolution multiplication, and that the subspace C _ Find _ T _ A, which is composed of all the indecomposable constructable functions, is the plum subalgebra of the CFS / C _ T _ 2 / _ _ _ (_ _ _.The indecomposable constructors refer to the structurable functions of indecomposable objects with support points corresponding to A.The CFFFI / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C / C /Assuming that A is a finite representation type, that is, for any two objects in A, the isomorphic classes of the intermediate terms extended by them are finite, then Joyce proves that the CFFFINPIN A) is the enveloping algebra of the lie algebra CFFFINING A / T A).Further, Joyce extends the constructable function to the stack function.灝咰F((?)bi_A)鎺ㄥ箍鍒頒簡(jiǎn) SF((?)bi_A),鍏朵腑SF((?)bi_A)琛ㄧずstack鍑芥暟鐢熸垚鐨凲涓婄殑鍚戦噺絀洪棿.Joyce璇佹槑浜，

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