本文選題：Witt代數(shù) + Virasoro-like代數(shù)�。� 參考：《鄭州大學(xué)》2016年碩士論文

【摘要】：設(shè)d是正整數(shù).d維洛朗多項式環(huán)Ad=C[t1±1,t2±1…,td±1]是交換結(jié)合代數(shù),稱它的導(dǎo)子李代數(shù)為Witt代數(shù),記作Wd.量子環(huán)面代數(shù)Cq是類似于Ad的非交換結(jié)合代數(shù),它的導(dǎo)子李代數(shù)我們記為Der(Cq)當(dāng)d=2時,Witt代數(shù)W2有自然子代數(shù)L=L2以及去掉Cartan子代數(shù)后的L=L2,我們統(tǒng)稱為Virasoro-like代數(shù)；此時稱Der(Cq)的自然子代數(shù)G(q)＝L2(q),L(q)＝L2(q)為q-Virasoro-like代數(shù).1986年,沈光宇在Wd上定義了一類權(quán)模：設(shè)a∈Cd, b∈C,V是g[d模,且單位矩陣在其上的作用是常數(shù)b,令Fbα(V)= V(?)d,則Fbα(V)成為一個自然的Wd模.函子Fbα也稱為Larsson函子,后來Rao刻畫了這類權(quán)模的結(jié)構(gòu).當(dāng)d=2時,Fbα(V)可以看作Virasoro-like代數(shù)上的模,類似地,我們也可以定義q-Virasoro-like代數(shù)上的模Fqα(V).本文中,我們用統(tǒng)一的方法研究了這些模的不可約性,主要結(jié)果有：(1)我們確定了Fbα(V)的不可約性.特別地,當(dāng)權(quán)空間V是無限維時,我們得到Virasoro-like代數(shù)上一類新的權(quán)空間無限維的不可約模；(2)無論V是有限還是無限維不可約s[2模,模Fqα(V)都可以分解成一個不可約子模與另一個子模的直和,同時我們也確定了另一個子模的不可約性.此外,我們還刻畫了代數(shù)L上C[(?)1,(?)2]-作用自由的模的結(jié)構(gòu),它同構(gòu)于Q(Λ2,α).
[Abstract]:Let d be a positive integer.D dimensional Laurent polynomial ring Ad=C[t1 + 1, T2 + 1... TD + 1] is a commutative associative algebra, which is called the derivation Lie algebra of its derivative as Witt algebra, and it is recorded as a Wd. quantum torus algebra Cq is a noncommutative associative algebra similar to Ad, and its derivate Lie algebra is recorded as Der (Cq) when d=2, Witt algebra W2 has a natural subalgebra L=L2 and after removing the subalgebra of subalgebra. The natural subalgebra G (q) of Der (q) = L2 (q), L (q) = L2 (q) is q-Virasoro-like algebra.1986 year. Shen Guangyu defines a class of weights, and the function of the unit matrix is constant. The structure of this class of weights is depicted. When d=2, Fb alpha (V) can be considered as a model on Virasoro-like algebra. Similarly, we can define the modulus Fq alpha (V) on q-Virasoro-like algebra. In this paper, we study the irreducibility of these modules in a unified way. The main fruit is: (1) we have determined the irreducibility of Fb a (V). When the weight space V is infinite dimension, we get a new class of Irreducible Modules on the infinite dimension of the weight space on Virasoro-like algebra; (2) no matter whether V is finite or infinitely irreducible s[2 module, the mode Fq alpha (V) can be decomposed into a direct sum of an irreducible submode and another submode, and we also determine the irreducibility of another submode. Besides, We also characterize the structure of the modulus of freedom of C[(?) 1, (?) 2]- on algebraic L, which is isomorphic to Q (2, alpha).

【學(xué)位授予單位】：鄭州大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2016
【分類號】：O152.5
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本文編號：1800789