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.     

Central Fourier analysis for Lorentz spaces on compact Lie groups

We determine the critical indexes for theL ^p,q uniform boundedness of the characters of a compact connected semisimple Lie group. These results show a basic difference between the simple and the semisimple case, a situation which does not appear when dealing withL ^p spaces. We provide applications to the general theory of central functions on compact Lie groups.     

Twistor quantization of loop spaces of compact Lie groups

We consider the problem of twistor quantization for the loop space Ω_TG of a compact Lie group G. We show that this problem is solvable in the critical dimension. Dedicated to Vasilii Sergeevich Vladimirov for his 85th birthday __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 3, pp. 450–467, December, 2008.     

Harmonic maps into loop spaces of compact Lie groups

We study harmonic maps from Riemann surfaces M to the loop spaces ΩG of compact Lie groups G, using the twistor approach. Harmonic maps into loop spaces are of special interest because of their relation to the Yang-Mills equations on ℝ⁴. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 16, Differential and Functional Differential Equations. Part 2, 2006.     

Harmonic maps into loop spaces of compact Lie groups

We study harmonic maps from Riemann surfaces M to the loop spaces ΩG of compact Lie groups G, using the twistor approach. We conjecture that harmonic maps of the Riemann sphere ℂℙ¹ into ΩG are related to Yang-Mills G-fields on ℝ⁴. This work was partly supported by the RFBR (Grant Nos. 04-01-00236, 06-02-04012), by the program of Support of Scientific Schools (Grant No. 1542.2003.1), and by the Scientific Program of RAS “Nonlinear Dynamics”     

Homology of iterated loop spaces of homogeneous spaces associated with exceptional Lie groups

We study the mod 2 homology of the double and triple loop spaces of homogeneous spaces associated with exceptional Lie groups. The main computational tools are the Serre spectral sequence for fibrations Ωⁿ⁺¹G→Ωⁿ⁺¹(G/H)→ΩⁿH for n=1,2, and the Eilenberg-Moore spectral sequence associated with related fiber squares, which both converge to the same destination space H_∗(Ωⁿ(G/H);F₂). We also develop the generalized Bockstein lemma to determine the higher Bockstein actions.     

Homology decompositions for p-compact groups

We construct a homotopy theoretic setup for homology decompositions of classifying spaces of p-compact groups. This setup is then used to obtain a subgroup decomposition for p-compact groups which generalizes the subgroup decomposition with respect to p-stubborn subgroups for a compact Lie group constructed by Jackowski, McClure and Oliver.     

Harmonic maps into loop spaces eliminate of compact Lie groups

We study harmonic maps from Riemann surfaces M to the loop spacesΩG of compact Lie groups G,using the twistor approach.We conjecture that harmonic maps of the Riemann sphere CP~1 intoΩG are related to Yang-Mills G-fields on R~4.     

Product decompositions of solvable Lie groups

Summary The article investigate the structure of real solvable connected Lie groups. It is described how one can decompose a solvable Lie group in direct and semidirect products of closed connected subgroups. In particular, the commutator group, Cartan subgroups, the center, maximal compactly embedded subgroups and tori are considered. Furthermore, one can find special solvable Lie groups and their product decompositions, namely compactly generated solvable Lie groups and those Lie groups which are generated by maximal compactly embedded subgroups. This article was processed by the author using the LATEX style filecljour1 from Springer-Verlag.     

The homotopy Lie algebra of classifying spaces

Let X be a 1-connected CW-complex of finite type and L_x its rational homotopy Lie algebra. In this work, we show that there is a spectral sequence whose E² term is the Lie algebra Ext_ULx(Q, L_x), and which converges to the homotopy Lie algebra of the classifying space B autX. Moreover, some terms of this spectral sequence are related to derivations of L_x and to the Gottlieb group of X.     

Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces

In this paper we give global characterisations of Gevrey–Roumieu and Gevrey–Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace–Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their α-duals in the sense of Köthe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.     

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.     

The rational homotopy Lie algebra of classifying spaces for formal two-stage spaces

We compute the center and nilpotency of the graded Lie algebra for a large class of formal spaces X. The latter calculation determines the rational homotopical nilpotency of the space of self-equivalences aut₁(X) for these X. Our results apply, in particular, when X is a complex or symplectic flag manifold.