Hamiltonian reductions of free particles under polar actions of compact Lie groups

We investigate classical and quantum Hamiltonian reductions of free geodesic systems of complete Riemannian manifolds. We describe the reduced systems under the assumption that the underlying compact symmetry group acts in a polar manner in the sense that there exist regularly embedded, closed, connected submanifolds intersecting all orbits orthogonally in the configuration space. Hyperpolar actions on Lie groups and on symmetric spaces lead to families of integrable systems of the spin Calogero-Sutherland type. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 1, pp. 161–176, April, 2008.     

Symplectically asystatic actions of compact Lie groups

In this paper we introduce the notion of symplectically asystatic Hamiltonian action on a Kahler manifold. In the algebraic setting we prove that if a complex linear group G acts on a Kahler manifold in a symplectically asystatic fashion, then the G-orbits are spherical. Finally, we give the complete classification of symplectically asystatic irreducible representations.     

Topologically conjugate classifications of the translation actions on low-dimensional compact connected Lie groups

In this article, we focus on the left translation actions on noncommutative compact connected Lie groups with topological dimension 3 or 4, consisting of SU(2), U(2), SO(3), SO(3)×S~1 and Spin~C(3). We define the rotation vectors(numbers) of the left actions induced by the elements in the maximal tori of these groups, and utilize rotation vectors(numbers) to give the topologically conjugate classification of the left actions. Algebraic conjugacy and smooth conjugacy are also considered. As a by-product, we show that for any homeomorphism f : L(p,-1) × S~1→ L(p,-1) × S~1, the induced isomorphism(π?f?i)_* maps each element in the fundamental group of L(p,-1) to itself or its inverse, where i : L(p,-1) → L(p,-1) × S~1 is the natural inclusion and π : L(p,-1) × S~1→ L(p,-1) is the projection.     

Definable groups and compact p-adic Lie groups

We formulate p-adic analogues of the o-minimal group conjecturesfrom the works of Hrushovski, Peterzil and Pillay [J. Amer.Math. Soc., to appear] and Pillay [J. Math. Log. 4 (2004) 147–162];that is, we formulate versions that are appropriate for groupsG definable in (saturated) P-minimal fields. We then restrictour attention to saturated models K of Th(_p) and Th(_{p, an}),record some elementary observations when G is defined over thestandard model _p, and then make a detailed analysis of the casewhere G = E(K) for E an elliptic curve over K. Essentially,our P-minimal conjectures hold in these contexts and, moreover,our case study of elliptic curves yields counterexamples toa more naive direct translation of the o-minimal conjectures.     

Lipschitz spaces on compact Lie groups

We prove the equivalence between Lipschitz spaces on a compact Lie group defined in terms of Weierstrass integrals and by means of higher order difference operators.     

On multipliers on compact Lie groups

In this note we announce L ^p multiplier theorems for invariant and noninvariant operators on compact Lie groups in the spirit of the well-known Hörmander-Mikhlin theorem on ? ⁿ and its versions on the torus $\mathbb{T}^n$ . Applications to mapping properties of pseudo-differential operators on L ^p -spaces and to a priori estimates for nonhypoelliptic operators are given.     

Representations at fixed points of smooth actions of compact connected Lie groups

Let be a compact connected Lie group acting smoothly on a connected closed manifold with nonempty fixed point set . In this paper, we study the relation between the cohomology of or and the equivalent representations of at fixed points.

     

Smooth actions of Lie groups and Lie algebras on manifolds

Necessary or sufficient conditions are presented for the existence of various types of actions of Lie groups and Lie algebras on manifolds.     

Central oscillating multipliers on compact Lie groups

Fefferman and Stein and Wainger proved optimal L ^p boundedness for certain oscillating multipliers on ${\mathbb{R}^{d}}$ . In this article, we prove analogues of their results for a compact Lie group.     

Subelliptic estimates on compact semisimple Lie groups

In this paper, we consider a natural subelliptic structure in semisimple, compact and connected Lie groups, and estimate the constant in the so-called subelliptic Friedrichs-Knapp-Stein inequality, which has implications in the regularity theory of p-energy minimizers.     

Time optimal factorizations on compact Lie groups

In this paper we study the relationship between factorization problems on SU(2ⁿ) or more generally on compact Lie groups G and time optimal control problems. Both types of problems naturally arise in physics, such as in quantum computing and in controlling coupled spin systems (NMR‐spectroscopy). In the first part we show that certain factorization problems can be reformulated as time optimal control problems on G. In the second part a necessary condition for the existence of finite optimal factorizations is discussed. At the end we illustrate our results by an example on Euler angle factorizations. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonhomogeneous G-spaces of compact Lie groups

A duality theorem for a compact G-manifold M of a compact Lie group G is proved. For the case when all the orbits in M are principal, it is proved that the quotient space of the complexification of M by the action of the associated algebraic group G^c is isomorphic to the quotient space of M by the action of G.Translated from Matematicheskie Zametki, Vol, 13, No. 4, pp. 523–529, April, 1973.This paper was completed under the guidance of A. L. Onishchik, to whom the author expresses his thanks.     

Spherical resolutions for compact Lie groups

Let G be a compact Lie group and A a G-space. When does there exist a relative G-CW-complex (X,A) with free G-action on X\A, such that X has the homology of a sphere? This paper gives sufficient conditions, which can be used for the construction of homotopy representations.