论文标题
Kontsevich的符号衍生物的交换案例的第二个同源组。
The second homology group of the commutative case of Kontsevich's symplectic derivation Lie algebra
论文作者
论文摘要
由Kontsevich定义的符号衍生物谎言代数与各种几何对象有关，包括图形的模量空间和Riemann表面，图形同源物，Hamiltonian矢量场等。它们及其Chevalley-Eilenberg链具有$ \ Mathbb的chevalley-eilenberg链，称为Mathbb a prange。我们考虑其中之一$ \ mathfrak {c} _g $，称为“交换案例”，其正权重零件$ \ mathfrak {c} _g^{+} \ subset \ subset \ mathfrak {c} _g $。 $ \ mathfrak {c} _g^{+} $的符号不变同源性与交换图同源性密切相关，因此从图形同源性理论的角度来看，有一些计算结果。但是，整个同源组$ h_ \ bullet（\ mathfrak {c} _g^{+}）$尚不清楚。我们通过使用$ \ mathrm {sp}（2G; \ m m iartbb {q}）$的经典表示理论来确定$ h_2（\ mathfrak {c} _g^{+}）$和重量分解。
The symplectic derivation Lie algebras defined by Kontsevich are related to various geometric objects including moduli spaces of graphs and of Riemann surfaces, graph homologies, Hamiltonian vector fields, etc. Each of them and its Chevalley-Eilenberg chain complex have a $\mathbb{Z}_{\geq 0}$-grading called weight. We consider one of them $\mathfrak{c}_g$, called the "commutative case", and its positive weight part $\mathfrak{c}_g^{+} \subset \mathfrak{c}_g$. The symplectic invariant homology of $\mathfrak{c}_g^{+}$ is closely related to the commutative graph homology, hence there are some computational results from the viewpoint of graph homology theory. However, the entire homology group $H_\bullet (\mathfrak{c}_g^{+})$ is not known well. We determined $H_2 (\mathfrak{c}_g^{+})$ by using classical representation theory of $\mathrm{Sp}(2g; \mathbb{Q})$ and the decomposition by weight.
