On the Nonexistence of Left-Invariant Ricci Solitons — A Conjecture and Examples

Taketomi, Yuichiro; Tamaru, Hiroshi

doi:10.1007/s00031-017-9439-4

Cited by 3 publications

(2 citation statements)

References 27 publications

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“…For example, the multiplication relation of µ := (d 6 , •, • ) is given by: e 1 e 1 = e 1 , e 1 e 2 = e 2 , e 2 e 1 = e 2 . With respect to the given orthonormal basis {e 1 , e 2 }, the left and right multiplications of µ are represented by In the sequel, we follow a similar procedure as in [6,16] to classify all Hermitian inner products on By (3.7), Lemma 3.6 and a straightforward calculation, it follows that the critical type is of (3a 4 + 6a 2 + 8, 5a 4 + 10a 2 + 8, 2(3a 4 + 8a 2 + 8))…”

Section: Examplesmentioning

confidence: 99%

The moment map for the variety of associative algebras

Zhang¹,

Yan²

2023

Preprint

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We consider the moment map m : PV n → iu(n) for the action of GL(n) on V n = ⊗ 2 (C n ) * ⊗ C n , and study the critical points of the functional F n = m 2 : PV n → R. Firstly, we prove that [µ] ∈ PV n is a critical point if and only if M µ = c µ I + D µ for some c µ ∈ R and D µ ∈ Der(µ), where m([µ]) = Mµ µ 2 . Then we show that any algebra µ admits a Nikolayevsky derivation φ µ which is unique up to automorphism, and if moreover, [µ] is a critical point of F n , then φ µ = − 1 cµ D µ . Secondly, we characterize the maxima and minima of the functional F n : A n → R, where A n denotes the projectivization of the algebraic varieties of all ndimensional associative algebras. Furthermore, for an arbitrary critical point [µ] of F n : A n → R, we also obtain a description of the algebraic structure of [µ]. Finally, we classify the critical points ofwhere (•, •) is an Ad(U(n))-invariant real inner product on iu(n), and ρ µ : GL(n) → R is defined by ρ µ (g) = g.µ, g.µ . The function m is the moment map from symplectic geometry, corresponding to the Hamiltonian action U(n) of V n on the symplectic manifold PV n (see [4,12]). In this paper, we study the critical points of the functional F n = m 2 : PV n → R, with an emphasis on the critical points that lie in the projectivization of the algebraic variety of all n-dimensional associative algebras A n .

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Section: Examplesmentioning

confidence: 99%

The moment map for the variety of associative algebras

Zhang¹,

Yan²

2023

Preprint

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“…Both of the Riemannian and pseudo-Riemannian cases are interesting, but sometimes the properties and methodologies are different. Recently many results on the Riemannian case have been obtained (just as examples, see [5,6,8,19] and references therein). However, the studies on the pseudo-Riemannian case seem to be still developing.…”

Section: Introductionmentioning

confidence: 99%

On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups

Kubo¹,

Onda²,

Taketomi³

et al. 2016

Hiroshima Math. J.

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In this paper, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics, and is based on the moduli space of left-invariant pseudo-Riemannian metrics. As one of applications, we show that any left-invariant pseudo-Riemannian metrics of arbitrary signature on the Lie groups of real hyperbolic spaces have constant sectional curvatures.

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The moment map for the variety of associative algebras

Zhang,

Yan

2024

Sci. China Math.

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On the Nonexistence of Left-Invariant Ricci Solitons — A Conjecture and Examples

Cited by 3 publications

References 27 publications

The moment map for the variety of associative algebras

The moment map for the variety of associative algebras

On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups

The moment map for the variety of associative algebras

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