论文标题
vector Field for vector field的不可总结模块lie代数为$ \ mathbb {s}^1 \ times \ times \ mathbb {c} $
Irreducible Jet modules for the vector field Lie algebra on $\mathbb{S}^1\times \mathbb{C}$
论文作者
论文摘要
对于$ \ mathbb {c} $上的交换代数$ a $ a $ a $，表示$ \ mathfrak {g} = \ text {der}（a）$。 Smash产品$ A \＃U（\ Mathfrak {G}）$的模块称为JET $ \ Mathfrak {G} $ - 模块，其中$ u（\ Mathfrak {G}）$是通用的通用代数，$ \ m athfrak {g} $。 $ a = \ mathbb {c} [t_1^{\ pm 1}，t_2] $。我们表明$ a \ #u（\ m athfrak {g} {g}）\ cong \ cong \ mathcal {d} \ otimes u（l） $ \ mathbb {c} [t_1^{\ pm 1}，t_2，\ frac {\ partial} {\ partial t_1}，\ frac {\ partial} {\ partial} {\ partial t_2}}] $，$ l $是$ a \ a \ a \ a \ a \ a \ a \ a \ a \ a \ a \ n op a \ a \ u（\ a \ a）代数对应于$ \ Mathfrak {g} $。使用lie代数同构$θ：l \ rightarrow \ Mathfrak \ Mathfrak {m} _ {1,0}Δ$，$ \ m athfrak {m m athfrak {m} _ {1,0} _ {1,0}Δ$ $不可还原的有限尺寸$ L $ -MODULE是不可约$ \ Mathfrak {gl} _2 $ -MODULE的同构。作为应用程序，我们在$ \ mathfrak {g} $上具有均匀界限的重量空间的张量产品实现。
For a commutative algebra $A$ over $\mathbb{C}$,denote $\mathfrak{g}=\text{Der}(A)$. A module over the smash product $A\# U(\mathfrak{g})$ is called a jet $\mathfrak{g}$-module, where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$.In the present paper, we study jet modules in the case of $A=\mathbb{C}[t_1^{\pm 1},t_2]$.We show that $A\#U(\mathfrak{g})\cong\mathcal{D}\otimes U(L)$, where $\mathcal{D}$ is the Weyl algebra $\mathbb{C}[t_1^{\pm 1},t_2, \frac{\partial}{\partial t_1},\frac{\partial}{\partial t_2}]$, and $L$ is a Lie subalgebra of $A\# U(\mathfrak{g})$ called the jet Lie algebra corresponding to $\mathfrak{g}$.Using a Lie algebra isomorphism $θ:L \rightarrow \mathfrak{m}_{1,0}Δ$, where $\mathfrak{m}_{1,0}Δ$ is the subalgebra of vector fields vanishing at the point $(1,0)$, we show that any irreducible finite dimensional $L$-module is isomorphic to an irreducible $\mathfrak{gl}_2$-module. As an application, we give tensor product realizations of irreducible jet modules over $\mathfrak{g}$ with uniformly bounded weight spaces.
