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Register a new userIndefinite Einstein metrics on nice Lie groups
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We introduce a systematic method to produce left-invariant, non-Ricci-flat Einstein metrics of indefinite signature on nice nilpotent Lie groups. On a nice nilpotent Lie group, we give a simple algebraic characterization of non-Ricci-flat left-invariant Einstein metrics in both the class of metrics for which the nice basis is orthogonal and a more general class associated to order two permutations of the nice basis. We obtain classifications in dimension 8 and, under the assumption that the root matrix is surjective, dimension 9; moreover, we prove that Einstein nilpotent Lie groups of nonzero scalar curvature exist in every dimension $geq 8$.
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We call a metric $m$-quasi-Einstein if $Ric_X^m$, which replaces a gradient of a smooth function $f$ by a vector field $X$ in $m$-Bakry-Emery Ricci tensor, is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant metrics on simple Lie groups. First, we prove that $X$ is a left-invariant Killing vector field if the metric on a compact simple Lie group is $m$-quasi-Einstein. Then we show that every compact simple Lie group admits non-trivial $m$-quasi-Einstein metrics except $SU(3)$, $E_8$ and $G_2$, and most of them admit infinitely many metrics. Naturally, the study on $m$-quasi-Einstein metrics can be extended to pseudo-Riemannian case. And we prove that every compact simple Lie group admits non-trivial $m$-quasi-Einstein Lorentzian metrics and most of them admit infinitely many metrics. Finally, we prove that some non-compact simple Lie groups admit infinitely many non-trivial $m$-quasi-Einstein Lorentzian metrics.
We call a metric $m$-quasi-Einstein if $Ric_X^m$ (a modification of the $m$-Bakry-Emery Ricci tensor in terms of a suitable vector field $X$) is a constant multiple of the metric tensor. It is a generalization of Einstein metrics which contains Ricci solitons. In this paper, we focus on left-invariant vector fields and left-invariant Riemannian metrics on quadratic Lie groups. First we prove that any left-invariant vector field $X$ such that the left-invariant Riemannian metric on a quadratic Lie group is $m$-quasi-Einstein is a Killing field. Then we construct infinitely many non-trivial $m$-quasi-Einstein metrics on solvable quadratic Lie groups $G(n)$ for $m$ finite.
We illustrate an algorithm to classify nice nilpotent Lie algebras of dimension $n$ up to a suitable notion of equivalence; applying the algorithm, we obtain complete listings for $nleq9$. On every nilpotent Lie algebra of dimension $leq 7$, we determine the number of inequivalent nice bases, which can be $0$, $1$, or $2$. We show that any nilpotent Lie algebra of dimension $n$ has at most countably many inequivalent nice bases.
In the paper Einstein metrics on compact simple Lie groups attached to standard triples, the authors introduced the definition of standard triples and proved that every compact simple Lie group $G$ attached to a standard triple $(G,K,H)$ admits a left-invariant Einstein metric which is not naturally reductive except the standard triple $(Sp(4),2Sp(2),4Sp(1))$. For the triple $(Sp(4),2Sp(2),4Sp(1))$, we find there exists an involution pair of $sp(4)$ such that $4sp(1)$ is the fixed point of the pair, and then give the decomposition of $sp(4)$ as a direct sum of irreducible $ad(4sp(1))$-modules. But $Sp(4)/4Sp(1)$ is not a generalized Wallach space. Furthermore we give left-invariant Einstein metrics on $Sp(4)$ which are non-naturally reductive and $Ad(4Sp(1))$-invariant. For the general case $(Sp(2n_1n_2),2Sp(n_1n_2),2n_2Sp(n_1))$, there exist $2n_2-1$ involutions of $sp(2n_1n_2)$ such that $2n_2sp(n_1))$ is the fixed point of these $2n_2-1$ involutions, and it follows the decomposition of $sp(2n_1n_2)$ as a direct sum of irreducible $ad(2n_2sp(n_1))$-modules. In order to give new non-naturally reductive and $Ad(2n_2Sp(n_1)))$-invariant Einstein metrics on $Sp(2n_1n_2)$, we prove a general result, i.e. $Sp(2k+l)$ admits at least two non-naturally reductive Einstein metrics which are $Ad(Sp(k)timesSp(k)timesSp(l))$-invariant if $k<l$. It implies that every compact simple Lie group $Sp(n)$ for $ngeq 4$ admits at least $2[frac{n-1}{3}]$ non-naturally reductive left-invariant Einstein metrics.
We study the Ricci tensor of left-invariant pseudoriemannian metrics on Lie groups. For an appropriate class of Lie groups that contains nilpotent Lie groups, we introduce a variety with a natural $mathrm{GL}(n,mathbb{R})$ action, whose orbits parametrize Lie groups with a left-invariant metric; we show that the Ricci operator can be identified with the moment map relative to a natural symplectic structure. From this description we deduce that the Ricci operator is the derivative of the scalar curvature $s$ under gauge transformations of the metric, and show that Lie algebra derivations with nonzero trace obstruct the existence of Einstein metrics with $s eq0$. Using the notion of nice Lie algebra, we give the first example of a left-invariant Einstein metric with $s eq0$ on a nilpotent Lie group. We show that nilpotent Lie groups of dimension $leq 6$ do not admit such a metric, and a similar result holds in dimension $7$ with the extra assumption that the Lie algebra is nice.
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