Killing Form Research Articles

Theorems of existence and of vanishing of conformally killing forms

On an n-dimensional compact, orientable, connected Riemannian manifold, we consider the curvature operator acting on the space of covariant traceless symmetric 2-tensors. We prove that, if the curvature operator is negative, then the manifold admits no nonzero conformally Killing p-forms for p = 1, 2, …, n − 1. On the other hand, we prove that the dimension of the vector space of conformally Killing p-forms on an n-dimensional compact simply-connected conformally flat Riemannian manifold (M,g) is not zero.

On compact semisimple Lie groups as 2-plectic manifolds

In this paper a compact and semisimple Lie group G is considered endowed with a 2-plectic structure ω, induced by the Killing form. We show that the Lie group of 2-plectomorphisms of G is finite dimensional and compact, and hence the Darboux’s theorem fails to be true for this 2-plectic structure. Also it is shown that ω induces a left-invariant \({\mathfrak{g}^{*}}\) valued 2-form which is proportional to dΘ, where Θ is the Cartan–Maurer 1-form on G. At last we consider the action of G on its tangent bundle which is furnished with the 2-plectic structure ωc, the complete lift of ω, and calculate covariant momentum map of this action.

ONEOptimal: A Maple Package for Generating One-Dimensional Optimal System of Finite Dimensional Lie Algebra

We present a Maple computer algebra package, ONEOptimal, which can calculate one-dimensional optimal system of finite dimensional Lie algebra for nonlinear equations automatically based on Olver's theory. The core of this theory is viewing the Killing form of the Lie algebra as an invariant for the adjoint representation. Some examples are given to demonstrate the validity and efficiency of the program.

Invariant scalar product on extended Poincaré algebra

Two methods can be used to calculate explicitly the Killing form on a Lie algebra. The first one is a direct calculation of the traces of the generators in a matrix representation of the algebra, and the second one is the usage of the group invariance of the scalar product. We use both methods in our calculation of the scalar product on the extended Poincaré algebra in order to have a cross check of our results. The algebra is infinite-dimensional and requires careful treatment of the infinities. The scalar product on the extended algebra found by both methods coincides and the important conclusion which follows is that Poincaré generators are orthogonal to the gauge generators.

Killing forms on toric Sasaki-Einstein spaces

We summarize recent results on the construction of Killing forms on Sasaki- Einstein manifolds. The complete set of special Killing forms of the Sasaki-Einstein spaces are presented. It is pointed out the existence of two additional Killing forms associated with the complex holomorphic volume form of Calabi-Yau cone manifold. In the case of toric Sasaki-Einstein manifolds the Killing forms are expressed in terms of toric data.

The classification of homogeneous Einstein metrics on flag manifolds with [formula omitted

Consider a compact connected simple Lie group G. We study homogeneous Einstein metrics for a class of compact homogeneous spaces, namely generalized flag manifolds G/H with second Betti number b2(G/H)=1. There are 8 infinite families G/H corresponding to a classical simple Lie group G and 25 exceptional flag manifolds, which all have some common geometric features; for example they admit a unique invariant complex structure which gives rise to unique invariant Kähler–Einstein metric. The most typical examples are the compact isotropy irreducible Hermitian symmetric spaces for which the Killing form is the unique homogeneous Einstein metric (which is Kähler). For non-isotropy irreducible spaces the classification of homogeneous Einstein metrics has been recently completed for 24 of the 26 cases. In this paper we construct the Einstein equation for the two unexamined cases, namely the flag manifolds E8/U(1)×SU(4)×SU(5) and E8/U(1)×SU(2)×SU(3)×SU(5). In order to determine explicitly the Ricci tensors of an E8-invariant metric we use a method based on the Riemannian submersions. For both spaces we classify all homogeneous Einstein metrics and thus we conclude that any flag manifold G/H with b2(M)=1 admits a finite number of non-isometric non-Kähler invariant Einstein metrics. The precise number of these metrics is given in Table 1.

Some properties of the family Γ of modular Lie superalgebras

In this paper, we continue to investigate some properties of the family Γ of finite-dimensional simple modular Lie superalgebras which were constructed by X. N. Xu, Y. Z. Zhang, L. Y. Chen (2010). For each algebra in the family, a filtration is defined and proved to be invariant under the automorphism group. Then an intrinsic property is proved by the invariance of the filtration; that is, the integer parameters in the definition of Lie superalgebras Γ are intrinsic. Thereby, we classify these Lie superalgebras in the sense of isomorphism. Finally, we study the associative forms and Killing forms of these Lie superalgebras and determine which superalgebras in the family are restrictable.

THE LAPLACIAN ON HOMOGENEOUS SPECIAL UNITARY SPACES AND THE CORRESPONDING INVERSE BRANCHING RULE

The solution of the eigenvalue problem of the Laplacian with the normal metric induced by the Killing form on homogeneous special unitary spaces SU(n+1)/SU(n) = S2n+1 is given. The corresponding inverse branching rule from SU(n) to SU(n+ 1) is obtained. AMS Subject Classification: 17B20, 22E46, 43A85, 81R05

Nash-type equilibria on Riemannian manifolds: A variational approach

Motivated by Nash equilibrium problems on ‘curved’ strategy sets, the concept of Nash–Stampacchia equilibrium points is introduced via variational inequalities on Riemannian manifolds. Characterizations, existence, and stability of Nash–Stampacchia equilibria are studied when the strategy sets are compact/noncompact geodesic convex subsets of Hadamard manifolds, exploiting two well-known geometrical features of these spaces both involving the metric projection map. These properties actually characterize the non-positivity of the sectional curvature of complete and simply connected Riemannian spaces, delimiting the Hadamard manifolds as the optimal geometrical framework of Nash–Stampacchia equilibrium problems. Our analytical approach exploits various elements from set-valued and variational analysis, dynamical systems, and non-smooth calculus on Riemannian manifolds. Examples are presented on the Poincaré upper-plane model and on the open convex cone of symmetric positive definite matrices endowed with the trace-type Killing form.

THE LAPLACIAN ON HOMOGENEOUS SPECIAL ORTHOGONAL SPACES AND THE INVERSE BRANCHING RULE

The solution of the eigenvalue problem of the Laplacian with the normal metric induced by the Killing form on homogeneous special orthogonal spaces SO(2n + 1)/SO(2n) is given. The corresponding inverse branching rule from SO(2n) to SO(2n + 1) is obtained.

Curvature spectra of simple Lie groups

The Killing form \beta\ of a real (or complex) semisimple Lie group G is a left-invariant pseudo-Riemannian (or, respectively, holomorphic) Einstein metric. Let \Omega\ denote the multiple of its curvature operator, acting on symmetric 2-tensors, with the factor chosen so that \Omega\beta=2\beta. The result of Meyberg [8], describing the spectrum of \Omega\ in complex simple Lie groups G, easily implies that 1 is not an eigenvalue of \Omega\ in any real or complex simple Lie group G except those locally isomorphic to SU(p,q), or SL(n,R), or SL(n,C) or, for even n only, SL(n/2,H), where p\ge q\ge0 and p+q=n>2. Due to the last conclusion, on simple Lie groups G other the ones just listed, nonzero multiples of the Killing form \beta\ are isolated among left-invariant Einstein metrics. Meyberg's theorem also allows us to understand the kernel of \Lambda, which is another natural operator. This in turn leads to a proof of a known, yet unpublished, fact: namely, that a semisimple real or complex Lie algebra with no simple ideals of dimension 3 is essentially determined by its Cartan three-form.

Discovering real lie subalgebras of $\mathfrak {e}_6$e6 using Cartan decompositions

The process of complexification is used to classify Lie algebras and identify their Cartan subalgebras. However, this method does not distinguish between real forms of a complex Lie algebra, which can differ in signature. In this paper, we show how Cartan decompositions of a complexified Lie algebra can be combined with information from the Killing form to identify real forms of a given Lie algebra. We apply this technique to \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(3,\mathbb {O})$\end{document}sl(3,O), a real form of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}_6$\end{document}e6 with signature (52, 26), thereby identifying chains of real subalgebras and their corresponding Cartan subalgebras within \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}_6$\end{document}e6. Motivated by an explicit construction of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(3,\mathbb {O})$\end{document}sl(3,O), we then construct an Abelian group of order 8 which acts on the real forms of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}_6$\end{document}e6, leading to the identification of 8 particular copies of the 5 real forms of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {e}_6$\end{document}e6, which can be distinguished by their relationship to the original copy of \documentclass[12pt]{minimal}\begin{document}$\mathfrak {sl}(3,\mathbb {O})$\end{document}sl(3,O).

Gradient flows for the minimum distance to the sum of adjoint orbits

Let G be a connected semisimple Lie group and its Lie algebra. Let be the Cartan decomposition corresponding to a Cartan involution θ of . The Killing form B induces a positive definite symmetric bilinear form B θ on defined by B θ(X, Y) = − B(X, θY). Given , we consider the optimization problem where the norm ‖ · ‖ is induced by B θ and K is the connected subgroup of the G with the Lie algebra . We obtain the gradient flow of the corresponding smooth function on the manifold K × ··· × K. Our results give unified extensions of several results of Li, Poon and Schulte-Herbrüggen [C.K. Li, Y.T. Poon, and T. Schulte-Herbruggen, Least-squares approximation by elements from matrix orbits achieved by gradient flows on compact Lie groups, Math. Comp. 80 (2011), 1601–1621]. They are also true for reductive Lie groups.

Indefinite Einstein metrics on simple Lie groups

The set E of Levi-Civita connections of left-invariant pseudo-Riemannian Einstein metrics on a given semisimple Lie group always includes D, the Levi-Civita connection of the Killing form. For the groups SU(l,j) (or SL(n,R), or SL(n,C) or, if n is even, SL(n/2,IH)), with 0<=j<=l and j+l>2 (or, n>2), we explicitly describe the connected component C of E, containing D. It turns out that C, a relatively-open subset of E, is also an algebraic variety of real dimension 2lj (or, real/complex dimension [n^2/2] or, respectively, real dimension 4[n^2/8]), forming a union of (j + 1)(j + 2)/2 (or, [n^2]+1 or, respectively, [n/4] + 1) orbits of the adjoint action. In the case of SU(n) one has 2lj=0, so that a positive-definite multiple of the Killing form is isolated among suitably normalized left-invariant Riemannian Einstein metrics on SU(n).

Symplectic Dolbeault operators on Kähler manifolds

For a K\"ahler Manifold $M$, the "symplectic Dolbeault operators" are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar\partial$ and $\bar\partial^*$, arise from Dirac operators on the canonical complex spinors on $M$. We give special attention to two special classes of K\"ahler manifolds: Riemann surfaces and flag manifolds ($G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). In the case of flag manifolds, we work with the Hermitian structure induced by the Killing form and a choice of positive roots (this is actually not a K\"ahler structure but is a K\"ahler with torsion (KT) structure). For Riemann surfaces the symplectic Dolbeault operators are elliptic and we compute their indices. In the case of flag manifolds, we will see that the representation theory of $G$ plays a role and that these operators can be used to distinguish (as Hermitian manifolds) between the flag manifolds corresponding to the Lie algebras $B_n$ and $C_n$. We give a thorough analysis of these operators on $\C P^1$ (the intersection of these classes of spaces), where the symplectic Dolbeault operators have an especially interesting structure.

Hidden symmetries on Kerr-NUT-(A)dS metrics of Einstein-Sasaki type

The hidden symmetries of higher dimensional Euclideanised Kerr-NUT-(A)dS metrics are investigated. In certain scaling limits these metrics are related to the Einstein-Sasaki ones. The complete set of Killing-Yano tensors of the Einstein-Sasaki spaces are presented. For this purpose the Killing forms of the Calabi-Yau cone over the Einstein-Sasaki manifold are constructed. Two new Killing forms on Einstein-Sasaki manifolds are identified associated with the complex volume form of the cone manifolds. As a concrete example we present the complete set of Killing-Yano tensors on the five-dimensional Einstein-Sasaki Y(p, q) spaces. The corresponding hidden symmetries are not anomalous and the geodesic equations are superintegrable.

On Kropina metrics

In this paper we study the curvature properties of Kropina metric. We find expressions for Riemann curvature and Ricci curvature of a Kropina metric when the 1-form \(\beta \) is a Killing form of constant length. We give a characterization of projectively flat Kropina metric and Kropina metric with isotropic \(S\)-curvature.

A note on semidirect sum of Lie algebras

In the paper there are investigated some properties of Lie algebras, the construction which has a wide range of applications like computer sciences (especially to computer visions), geometry or physics, for example. We concentrate on the semidirect sum of algebras and there are extended some theoretic designs as conditions to be a center, a homomorphism or a derivative. The Killing form of the semidirect sum where the second component is an ideal of the first one is considered as well.

Extrinsic hyperspheres in manifolds with special holonomy

We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kähler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.

On Einstein–Kropina metrics

In this paper, a characteristic condition of Einstein–Kropina metrics is given. By the characteristic condition, we prove that a non-Riemannian Kropina metric F=α2β with constant Killing form β on an n-dimensional manifold M, n⩾2, is an Einstein metric if and only if α is also an Einstein metric. By using the navigation data (h,W), it is proved that an n-dimensional (n⩾2) Kropina metric F=α2β is Einstein if and only if the Riemannian metric h is Einstein and W is a unit Killing vector field with respect to h. Moreover, we show that every Einstein–Kropina metric must have vanishing S-curvature, and any conformal map between Einstein–Kropina metrics must be homothetic.