Biderivations of the planar Galilean conformal algebra and their applications

In this paper, the biderivations without skew-symmetric condition of the planar Galilean conformal algebra are presented. As applications, the characterizations of the forms of linear commuting maps and the commutative post-Lie algebra structures on the planar Galilean conformal algebra are given.     

Exterior algebra structure on relative invariants of reflection groups

Let G be a reflection group acting on a vector space V (over a field with zero characteristic). We denote by S(V ^*) the coordinate ring of V, by M a finite dimensional G-module and by ?? a one-dimensional character of G. In this article, we define an algebra structure on the isotypic component associated to ?? of the algebra ${S(V^*) \otimes \Lambda(M^*)}$ . This structure is then used to obtain various generalizations of usual criterions on regularity of integers.     

Sewing homeomorphism and conformal invariants

This paper is devoted to the study of some fundamental properties of the sewing homeomorphism induced by a Jordan domain. In particular, using conformal invariants such as harmonic measure, extremal distance, and reduced extremal distance, we give several necessary and sufficient conditions for the sewing homeomorphism to be bi-Lipschitz or bi-Hlder. Furthermore, equivalent conditions for a Jordan curve to be a quasicircle are also obtained.     

Differential invariants of a Euclidean algebra

Functional bases of second-order differential invariants of a Euclidean algebra and a conformal algebra are found for a set of scalar functions depending on n variables.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 767–772, June, 1990.     

On the decomposition of global conformal invariants II

This paper is a continuation of [S. Alexakis, The decomposition of global conformal invariants I, submitted for publication, see also math.DG/0509571], where we complete our partial proof of the Deser-Schwimmer conjecture on the structure of “global conformal invariants.” Our theorem deals with such invariants P(gn) that locally depend only on the curvature tensor Rijkl (without covariant derivatives).In [S. Alexakis, The decomposition of global conformal invariants I, Ann. of Math., in press] we developed a powerful tool, the “super divergence formula” which applies to any Riemannian operator that always integrates to zero on compact manifolds. In particular, it applies to the operator Ign(?) that measures the “non-conformally invariant part” of P(gn). This paper resolves the problem of using this information we have obtained on the structure of Ign(?) to understand the structure of P(gn).     

Homological algebra of Novikov-Shubin invariants and morse inequalities

It is shown in this paper that the topological phenomenonzero in the continuous spectrum, discovered by S.P. Novikov and M.A. Shubin, can be explained in terms of a homology theory on the category of finite polyhedra with values in a certain abelian category. This approach implies homotopy invariance of the Novikov-Shubin invariants. Its main advantage is that it allows the use of the standard homological techniques, such as spectral sequences, derived functors, universal coefficients etc., while studying the Novikov-Shubin invariants. It also leads to some new quantitative invariants, measuring the Novikov-Shubin phenomenon in a different way, which are used in the present paper in order to strengthen the Morse type inequalities of Novikov and Shubin [NSh2].The research was supported by a grant from US-Israel Binational Science Foundation.     

On conjugation invariants in the dual Steenrod algebra

We investigate the canonical conjugation, , of the mod dual Steenrod algebra, , with a view to determining the subspace, , of elements invariant under . We give bounds on the dimension of this subspace for each degree and show that, after inverting , it becomes polynomial on a natural set of generators. Finally we note that, without inverting , is far from being polynomial.

     

The planar algebra of a semisimple and cosemisimple Hopf algebra

To a semisimple and cosemisimple Hopf algebra over an algebraically closed field, we associate a planar algebra defined by generators and relations and show that it is a connected, irreducible, spherical, non-degenerate planar algebra with non-zero modulus and of depth two. This association is shown to yield a bijection between (the isomorphism classes, on both sides, of) such objects.     

q-Deformation of the Virasoro algebra and lattice conformal theories

A natural definition of the q-deformation of the Virasoro and superconformal algebras is suggesteded. New Liealgebraic symmetries describe a lattice version of the continual theory. A close link between the deformation constructed in this paper and the lattice version of the Faddeev-Takhtadjian-Volkov Virasaro algebra is shown. 18 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 235, 1900, pp. 217–227.     

Temperley-Lieb planar algebra modules arising from the ADE planar algebras

A Hilbert module over a planar algebra P is essentially a Hilbert module over a canonically defined algebra spanned by the annular tangles in P. It follows that any planar algebra Q containing P is a module over P, and in particular, any subfactor planar algebra is a module over the Temperley-Lieb planar algebra with the same modulus. We describe a positivity result that allows us to describe irreducible Temperley-Lieb planar algebra modules, and apply the result to decompose the planar algebras determined by the Coxeter graphs A_n (n?3), D_n (n?4), E₆, E₇, and E₈.     

Some conformal and projective scalar invariants of Riemannian manifolds

It is proved that, on any closed oriented Riemannian n-manifold, the vector spaces of conformal Killing, Killing, and closed conformal Killing r-forms, where 1 ≤ r ≤ n ? 1, have finite dimensions t _r , k _r , and p _r , respectively. The numbers t _r are conformal scalar invariants of the manifold, and the numbers k _r and p _r are projective scalar invariants; they are dual in the sense that t _r = t _n?r and k _r = p _n?r . Moreover, an explicit expression for a conformal Killing r-form on a conformally flat Riemannian n-manifold is given.