New Einstein metrics on Sp(n) which are non-naturally reductive

Czechoslovak Mathematical Journal - We prove that there are at least two new non-naturally reductive Ad(Sp(l) × Sp(k) × Sp(k) × Sp(k)) invariant Einstein metrics on Sp(l + 3k) (k...     

Space-time locality in theSp(2)-symmetric Lagrangian formalism

The existence of a local solution to the Sp(2) master equation for the gauge field theory is proved in the perturbation theory under standard regularity assumptions for the action. The arbitrariness of solutions to the Sp(2) master equation is described, provided they are proper. The effective action can be chosen to be Sp(2) invariant and (under the additional assumption that the gauge transformation generators are Lorentz tensors) Lorentz invariant. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 115. No. 3, pp. 373–388, June, 1998.     

关于广义超特殊p-群的自同构群

用如下的方式确定了广义超特殊p-群G的自同构群.设|G|=p2n+m,|ζG|=pm,|N|=pl并且G'≤N≤ζG,其中n≥1且m≥2.AutnG表示AutG中平凡地作用在N上的所有自同构形成的正规子群.则(1)当p是奇素数时,AutG/AunG≌Z(p-1)pl-1.进一步地,(i)如果G的幂指数是pm,则Autn...     

Geodesics and Curvatures of Special Sub-Riemannian Metrics on Lie Groups

Let G be a full connected semisimple isometry Lie group of a connected Riemannian symmetric space M = G/K with the stabilizer K; p : G → G/K = M the canonical projection which is a Riemannian submersion for some G-left invariant and K-right invariant Riemannian metric on G, and d is a (unique) sub-Riemannian metric on G defined by this metric and the horizontal distribution of the Riemannian submersion p. It is proved that each geodesic in (G, d) is normal and presents an orbit of some one-parameter isometry group. By the Solov'ev method, using the Cartan decomposition for M = G/K, the author found the curvatures of the homogeneous sub-Riemannian manifold (G, d). In the case G = Sp(1) × Sp(1) with the Riemannian symmetric space S³ = Sp(1) = G/ diag(Sp(1) × Sp(1)) the curvatures and torsions are calculated of images in S³ of all geodesics on (G, d) with respect to p.     

LANGLANDS DUALITY AND G2 SPECTRAL CURVES

We first demonstrate how duality for the fibres of the so-calledHitchin fibration works for the Langlands dual groups Sp(2m)and SO(2m + 1). We then show that duality for G₂ is implementedby an involution on the base space which takes one fibre toits dual. A formula for the natural cubic form is given andshown to be invariant under the involution.     

The Cubic Symplectic Theta Function

It is proved that the cubic symplectic theta function is invariant with respect to the group Sp(4,). Bibliography: 10 titles.     

Optimal pinching constants of odd dimensional homogeneous spaces

In this article we compute the pinching constants of all invariant Riemannian metrics on the Berger space B ¹³=SU(5)/(Sp(2)×_ℤ2S¹) and of all invariant U(2)-biinvariant Riemannian metrics on the Aloff–Wallach space W ⁷ _1,1=SU(3)/S¹ _1,1. We prove that the optimal pinching constants are precisely in both cases. So far B ¹³ and W ⁷ _1,1 were only known to admit Riemannian metrics with pinching constants .?We also investigate the optimal pinching constants for the invariant metrics on the other Aloff–Wallach spaces W ⁷ _k,l =SU(3)/S¹ _k,l . Our computations cover the cone of invariant T²-biinvariant Riemannian metrics. This cone contains all invariant Riemannian metrics unless k/l=1. It turns out that the optimal pinching constants are given by a strictly increasing function in k/l∈[0,1]. Thus all the optimal pinching constants are ≤.?In order to determine the extremal values of the sectional curvature of an invariant Riemannian metric on W ⁷ _k,l we employ a systematic technique, which can be applied to other spaces as well. The computation of the pinching constants for B ¹³ is reduced to the curvature computation for two proper totally geodesic submanifolds. One of them is diffeomorphic to ℂℙ³/ℤ₂ and inherits an Sp(2)-invariant Riemannian metric, and the other is W ⁷ _1,1 embedded as recently found by Taimanov. This approach explains in particular the coincidence of the optimal pinching constants for W ⁷ _1,1 and the Berger space B ¹³. Oblatum 9-XI-1998 & 3-VI-1999 / Published online: 20 August 1999     

Sul classificante del gruppoSp(n)·Sp(1)

Sunto Si studia la coomologia a coefficienti inZ ₂ del gruppo Sp(n)·Sp(1) e del suo spazio classificante B_Sp(n)·Sp(1), stabilendo in particolare che H*(Sp(n)·Sp(1),Z ₂) è generato da un sistema semplice di elementi universalmente trasgressivi e H*(B_Sp(n)·Sp(1),Z ₂) è un anello di polinomi suZ ₂ in classi di Stiefel-Whitney di opportuni fibrati. Ne vengono tratte alcune conseguenze geometriche. Entrata in Redazione il 4 agosto 1975. Lavoro eseguito con contributo del CNR, nell'ambito del Gruppo Nazionale per le Strutture Algebriche e Geometriche e loro Applicazioni.     

The Yagita Invariant of Some Symplectic Groups

If a group G of finite virtual cohomological dimension has p-periodic Farrell cohomology, then the Yagita invariant of this group equals its p-period. The group of symplectic 2(p + 1) × 2(p + 1) matrices over ${\mathbb{Z}}$ has elementary abelian p-subgroups of rank at least 2 and hence ${{\rm Sp}(2(p+1), \mathbb{Z})}$ and ${{\rm Sp}(2(p+1), \mathbb{Q})}$ do not have p-periodic Farrell cohomology. We compute the Yagita invariants of these groups for any odd prime p.     

广义超特殊p-群的自同构群(Ⅱ)

重新确定了广义超特殊p-群G的自同构群的结构.设|G|=p~(2n+m),|ζG|=p~m,其中n≥1,m≥2,Aut_cG是AutG中平凡地作用在ζG上的元素形成的正规子群,则(i)若p是奇素数,则AutG=〈θ〉×Aut_cG,其中θ的阶是(p-1)p~(m-1);若p=2,则AutG=〈θ_1,θ_2〉×Aut_cG,其中〈θ_1,θ_2〉=〈θ_1〉×〈θ_2〉≌Z_(2m-2)×Z_2.(ii)如果G的幂指数是p~m,那么Aut_cG/InnG≌Sp(2n,p).(iii)如果G的幂指数是p~(m+1),那么Aut_cG/InnG≌K×Sp(2n-2,p),其中K是p~(2n-1)阶超特殊p-群(若p是奇素数)或者初等Abel 2-群.特别地,当n=1时,Aut_cG/InnG≌Z_p.     

A new class of quaternionic manifolds

Riemannian manifolds with structure group Sp(n)·Sp(1) are studied. Moreover for the Riemannian manifolds with structure group Sp(n) a new class is defined, and example of compact nilmanifolds in this class are constructed.     

Invariants for an adjoint action of classical groups

Let H=Sp(n) or H=O(n); and char K≠2 in the orthogonal case. We prove that an invariant algebra K[M(n)^m]^H is generated by elements σ_i(Y_j1..._j2, where every matrix Y_i either is X_i or the (symplectic) transpose of X_i. Supported by RFFR grant No. 98-01-00932. Translated fromAlgebra i Logika, Vol. 38, No. 5, pp. 549–584, September–October, 1999.     

Sections of U(2n)/Sp(s) over S4n−1

We describe the sections of U(2n)/Sp(s) over S⁴ⁿ⁻¹ in terms of the sections of the symplectic Stiefel manifold Sp(n)/Sp(s) and we express the orders of obstructions to sectioning U(2n)/Sp(s) in terms of orders of obstructions to sectioning Sp(n)/Sp(s). In certain cases we give the exact values of these orders.     

A branching law from Sp(n) TO Sp(q) × Sp(n-q) and an application to laplace operator spectra

In this paper, we give a branching law from the group Sp(n) to the subgroup Sp(q) × Sp(n-q). We propose an application of this result to compute the Laplace spectrum on the forms of the manifold Sp(n)/Sp(q)×Sp(n-q), using the “identification” of the Laplace operator with the Casimir operator in symmetric spaces.     

Classification of invariant valuations on the quaternionic plane

We describe the orbit space of the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$ on the real Grassmann manifolds _Grk(H²)${\mathrm{Gr}}_k({\mathbb{H}}^2)$ in terms of certain quaternionic matrices of Moore rank not larger than 2. We then give a complete classification of valuations on the quaternionic plane H²${\mathbb{H}}^2$ which are invariant under the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$.     

Non-naturally reductive Einstein metrics on Sp(n)

We prove that Sp(2k+l)admits at least two non-naturally reductive Einstein metrics which are Ad(Sp(k)×Sp(k)×Sp(l))-invariant ifk<1.It implies that every compact simple Lie group Sp(n)for n≥4 admits at least 2[(n-1)/3]non-naturally reductive left-invariant Einstein metrics.     

The heat kernels on some classical manifolds and classical domains

In this paper, by the method used in [7, 9, 10], we construct the heat kernels (i.e. the fundamental solutions of the heat equation) of the following spaces: the quaternion hyperbolic spaceSp (1,n)/Sp(1)×Sp (n), the quaternion classical domain of type IIISp (n, ℂ)/Sp (n), the complex Grassmannian manifoldU (m+n)/U(m)×U(n). Project supported by the National Natural Science Foundation of China.     

矩阵空间上保弱伴随矩阵的线性映射

为了刻画矩阵空间上保弱伴随矩阵的线性映射f,引入了保弱伴随矩阵的概念,以矩阵的弱伴随矩阵为不变量,得到了当n≥3时数域F上从线性矩阵空间Mn×n（F）到Mm×m（F）的保弱伴随矩阵的线性映射f的形式．     

Alcune osservazioni sui sottogruppi abeliani del gruppoSp(n)·Sp(1)

Sunto Si studiano i sottogruppi abeliani diSp(n)·Sp(1). Viene data una caratterizzazione dei sottogruppi abeliani G diSp(n)·Sp(1) che sono immagini di sottogruppi abeliani del rivestimento universaleSp(n)·Sp(1). In relazione all'azione diSp(n)·Sp(1) su R⁴ⁿ ≡ Qⁿ si determinano i sottogruppi abeliani G per i quali Qⁿ si decompone nella somma diretta di di n sottospazi quaternionali1-dimensionali invarianti per G. Si caratterizzano i sottogruppi abeliani massimali tra quelli del tipo Z_p × … × Z_p (p numero primo). Entrata in Redazione il 21 Luglio 1976.