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Negative Ricci curvature on some non-solvable Lie groups II
Mathematische Zeitschrift ( IF 0.8 ) Pub Date : 2019-05-13 , DOI: 10.1007/s00209-019-02310-z
Cynthia Will

We construct many examples of Lie groups admitting a left-invariant metric of negative Ricci curvature. We study Lie algebras which are semidirect products $${\mathfrak {l}}= ({\mathfrak {a}} \oplus {\mathfrak {u}} ) < imes {\mathfrak {n}}$$ l = ( a ⊕ u ) ⋉ n and we obtain examples where $${\mathfrak {u}} $$ u is any semisimple compact real Lie algebra, $${\mathfrak {a}} $$ a is one-dimensional and $${\mathfrak {n}}$$ n is a representation of $${\mathfrak {u}} $$ u which satisfies some conditions. In particular, when $${\mathfrak {u}} = {{\mathfrak {s}}}{{\mathfrak {u}}}(m)$$ u = s u ( m ) , $${{\mathfrak {s}}}{{\mathfrak {o}}} (m)$$ s o ( m ) or $${{\mathfrak {s}}}{{\mathfrak {p}}} (m)$$ s p ( m ) and $${\mathfrak {n}}$$ n is a representation of $${\mathfrak {u}} $$ u in some space of homogeneous polynomials, we show that these conditions are indeed satisfied. In the case $${\mathfrak {u}} = {{\mathfrak {s}}}{{\mathfrak {u}}}(2)$$ u = s u ( 2 ) we get a more general construction where $${\mathfrak {n}}$$ n can be any nilpotent Lie algebra where $${{\mathfrak {s}}}{{\mathfrak {u}}}(2)$$ s u ( 2 ) acts by derivations. We also prove a general result in the case when $${\mathfrak {u}} $$ u is a semisimple Lie algebra of non-compact type.

中文翻译:

一些不可解李群上的负 Ricci 曲率 II

我们构造了许多承认负 Ricci 曲率的左不变度量的李群的例子。我们研究李代数,它是半直积 $${\mathfrak {l}}= ({\mathfrak {a}} \oplus {\mathfrak {u}} ) < imes {\mathfrak {n}}$$ l = ( a ⊕ u ) ⋉ n 并且我们得到这样的例子,其中 $${\mathfrak {u}} $$ u 是任何半单紧实李代数,$${\mathfrak {a}} $$ a 是一维的,$$ {\mathfrak {n}}$$ n 是满足某些条件的 $${\mathfrak {u}} $$ u 的表示。特别地,当 $${\mathfrak {u}} = {{\mathfrak {s}}}{{\mathfrak {u}}}(m)$$ u = su ( m ) 时, $${{\mathfrak {s}}}{{\mathfrak {o}}} (m)$$ so ( m ) 或 $${{\mathfrak {s}}}{{\mathfrak {p}}} (m)$$ sp ( m ) 和 $${\mathfrak {n}}$$ n 是 $${\mathfrak {u}} $$ u 在一些齐次多项式空间中的表示,我们表明这些条件确实得到满足。在 $${\mathfrak {u}} = {{\mathfrak {s}}}{{\mathfrak {u}}}(2)$$ u = su ( 2 ) 的情况下,我们得到一个更一般的构造,其中 $ ${\mathfrak {n}}$$ n 可以是任何幂零李代数,其中 $${{\mathfrak {s}}}{{\mathfrak {u}}}(2)$$ su ( 2 ) 通过推导起作用. 我们还证明了在 $${\mathfrak {u}} $$ u 是非紧型半单李代数的情况下的一般结果。
更新日期:2019-05-13
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