论文标题
Rota-baxter的非亚伯延长谎言代数和自动形态的诱导性
Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms
论文作者
论文摘要
一个rota-baxter lie代数$ \ mathfrak {g} _t $是配备了rota-baxter operator $ t：\ mathfrak {g} \ rightArrow \ rightarrow \ mathfrak {g} $。在本文中，我们考虑了一个rota-baxter的非阿布尔延伸，lie代数$ \ mathfrak {g} _t $ by另一个rota-baxter lie代数$ \ mathfrak {h} _s。 \ Mathfrak {h} _s）$分类了{等价类的}。 Given a non-abelian extension $$ 0 \rightarrow \mathfrak{h}_S \xrightarrow{i} \mathfrak{e}_U \xrightarrow{p} \mathfrak{g}_T \rightarrow 0$$ of Rota-Baxter Lie algebras, we also show that the obstruction for a pair of Rota-Baxter $ \ mathrm {aut}（\ Mathfrak {h} _s）\ times \ times \ times \ mathrm {aut}（\ mathfrak {g} _t）$中的自动晶体是由$ \ mathrm {autfrak}（autfrak}（\ mathfrak}（\ mathfrak）组成的$ h^2_ {nab}（\ mathfrak {g} _t，\ mathfrak {h} _s）$。作为副产品，我们在rota-baxter lie代数的背景下获得了井短序列。
A Rota-Baxter Lie algebra $\mathfrak{g}_T$ is a Lie algebra $\mathfrak{g}$ equipped with a Rota-Baxter operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. In this paper, we consider non-abelian extensions of a Rota-Baxter Lie algebra $\mathfrak{g}_T$ by another Rota-Baxter Lie algebra $\mathfrak{h}_S.$ We define the non-abelian cohomology $H^2_{nab} (\mathfrak{g}_T, \mathfrak{h}_S)$ which classifies {equivalence classes of} such extensions. Given a non-abelian extension $$ 0 \rightarrow \mathfrak{h}_S \xrightarrow{i} \mathfrak{e}_U \xrightarrow{p} \mathfrak{g}_T \rightarrow 0$$ of Rota-Baxter Lie algebras, we also show that the obstruction for a pair of Rota-Baxter automorphisms in $\mathrm{Aut}(\mathfrak{h}_S ) \times \mathrm{Aut}(\mathfrak{g}_T)$ to be induced by an automorphism in $\mathrm{Aut}(\mathfrak{e}_U)$ lies in the cohomology group $H^2_{nab} (\mathfrak{g}_T, \mathfrak{h}_S)$. As a byproduct, we obtain the Wells short-exact sequence in the context of Rota-Baxter Lie algebras.
