General Volume-Preserving Mechanical Systems

We present the general form of equations that generate a volume-preserving flow on a symplectic manifold M, ) via the highest Euler–Lagrange cohomology. It is shown that for every volume-preserving flow there are some 2-forms that play a similar role to the Hamiltonian in Hamilton mechanics. The ordinary canonical equations are included as a special case with a 2-form 1/(n - 1)H, where H is the corresponding Hamiltonian.     

Cohomology of Quantized Function Algebras at Roots of Unity

Let G be a simply-connected, semisimple algebraic group overk, an algebraically closed field of characteristic zero. Let O[G] be the quantized function algebra of G at a primitivelth root of unity , and let be the ‘restricted’ quantized function algebra at, a finite-dimensional k-algebra obtained from O[G] by factoringout a centrally generated ideal. It is known that is a Hopf algebra. We study the cohomology ring, a graded commutative algebra, and, for any finite-dimensional -module M, the -module . We prove that for sufficiently large l there isan isomorphism of graded algebras where each X_i is homogeneous of degree $2$, and $2N$ equalsthe number of roots associated to G. Moreover we show that inthis case is a finitely generated -module. We also show under much less restrictive conditions on l that continues to be a finitely generated graded commutativealgebra over which is a finitely generated module. 1991 Mathematics Subject Classification: 16W30,17B37, 17B56.     

Bundles over Quantum Sphere and Noncommutative Index Theorem

The Noncommutative Index Theorem is used to prove that the Chern numbers of quantum Hopf line bundles over the standard Podle quantum sphere equal the winding numbers of the representations defining these bundles. This result gives an estimate of the positive cone of the algebraic K₀ of the standard quantum sphere.     

Group actions and group extensions

In this paper we study finite group extensions represented by special cohomology classes. As an application, we obtain some restrictions on finite groups which can act freely on a product of spheres or on a product of real projective spaces. In particular, we prove that if acts freely on , then .

     

Novikov–Shubin Signatures, I

Torsion objects of von Neumann categories describe thephenomenon spectrum near zero discovered by Novikov and Shubin. Inthis paper we classify Hermitian forms on torsion objects of a finitevon Neumann category. We prove that any such Hermitian form can berepresented as the discriminant form of a degenerate Hermitian form on aprojective module. We also find the relation between the Hermitian formson projective modules which holds if and only if their discriminantforms are congruent. A notion of superfinite von Neumann category isintroduced. It is proven that the classification of torsion Hermitianforms in a superfinite category can be completely reduced to theisomorphism types of their positive and the negative parts.     

A Generalization of the Bott–Borel–Weil Theorem and Euler Number Multiplets of Representations

The Bott–Borel–Weil theorem (BBW) is a statement about a complex homogeneous space X=G/R where G is a compact semisimple Lie group and R is the centralizer of a torus in G. One knows that BBW is equivalent to the determination of how R operates on the cohomology of a certain nilpotent Lie algebra of antiholomorphic tangent vectors operating on an arbitrary irreducible G-module. Upon replacing the complex structure with a space S of spinors BBW is equivalent to a statement about the eigenvalues of the Casimir operator of R in the module S V where V is the irreducible G-module with highest weight . But then the complex structure is eliminated and the problem makes sense for any compact subgroup R G. We solve the problem in the case where R and G have the same rank. The very special case where R is the centralizer of a torus then yields BBW. Involved in the general case (i.e. arbitrary R) is a new (at least to mathematicians) Dirac operator with a cubic term whose square is expressed in terms of the Casimir operator. Actually the spectral resolution of the Dirac operator is determined in the general case. The kernel of the Dirac operator is only given in the equal rank case. A key feature in that case is that, associated to , there is a multiplet of representations of R, having rather striking properties, and cardinality equal to the Euler number of G/R.     

Tangential Star Products

We etablish a necessary and sufficient condition under which there exists a tangential and well graded star product, differential or not, on the dual of a nilpotent Lie algebra . We also give enlightening examples with explicit computations.     

Derivations and Automorphisms of the Positive Part of the Two-parameter Quantum Group Ur,s(B3)

We compute the derivations of the positive part of the two-parameter quantum group U_(r,s)(B_3) and show that the Hochschild cohomology group of degree 1 of this algebra is a threedimensional vector space over the base field C. We also compute the groups of(Hopf) algebra automorphisms of the augmented two-parameter quantized enveloping algebra ?_(r,s)~(≥0)(B_3).     

On the quotient class of non-archimedean fields

The quotient class of a non-archimedean field is the set of cosets with respect to all of its additive convex subgroups. The algebraic operations on the quotient class are the Minkowski sum and product. We study the algebraic laws of these operations. Addition and multiplication have a common structure in terms of regular ordered semigroups. The two algebraic operations are related by an adapted distributivity law.