We construct cocycles on the Lie algebra of pseudoand q-pseudodifferential symbols of one variable and on their close relatives: the sine-algebra and the Poisson algebra on two-torus. A ``quantum'' Godbillon Vey cocycle on (pseudo)differential operators appears in this construction as a natural generalization of the Gelfand Fuchs 3-cocycle on periodic vector fields. A nontrivial embedding of th...

In [2, 3] we gave a number of formulas for qKdV equations derived in a q-Virasoro context. The constructions were meaningful even though the cocycle terms did not visibly satisfy a Jacobi identity. In the present note we modify the framework slightly and produce a cocycle term (morally equivalent to one of those used in [2, 3]) and thereby exhibit a more convincing candidate for a canonical qKd...

Meyer [9] defined a cocycle g 2Z.MgIZ/ (g 0) called the Meyer cocycle which represents four times generator of the second cohomology class when g 3. Let D1 , D2 , and D3 be mutually disjoint disks in S , and Int Di the interior of Di for i D 1; 2; 3. We denote by P WDS q iD1 Int Di the pair of pants, and ̨; ˇ; 2 1.P / be the homotopy classes as shown in Figure 1. We consider a †g;r –bundle E '; ...

We study the lattice (grid) generated by the incidence vectors of cocycles of a binary matroid and its dual lattice. We characterize those binary matroids for which the obvious necessary conditions for a vector to belong to the cocycle lattice are also sufficient. This characterization yields a polynomial time algorithm to check whether a matroid has this property, and also to construct a basis...

Suppose L is a semisimple Levi subgroup of a connected Lie group G, X is a Borel G-space with finite invariant measure, and α : X × G → GLn(R) is a Borel cocycle. Assume L has finite center, and that the real rank of every simple factor of L is at least two. We show that if L is ergodic on X, and the restriction of α to X × L is cohomologous to a homomorphism (modulo a compact group), then, aft...

Let $A$ be an abelian topological group and $B$ a trivial topological $A$-module. In this paper we define the second bilinear cohomology with a trivial coefficient. We show that every abelian group can be embedded in a central extension of abelian groups with bilinear cocycle. Also we show that in the category of locally compact abelian groups a central extension with a continuous section can b...

In this paper, we prove the generic version of Cantor spectrum property for quasi-periodic Schrödinger operators with finitely smooth and small potentials, also show pure point a class multi-frequency \(C^k\) long-range on \(\ell ^2({\mathbb Z}^d)\). These results are based reducibility properties differentiable \(SL(2,{\mathbb R})\) cocycles. More precisely, that if base frequency is Diophanti...

We show that transgression of the Dixmier-Douady class coming from the natural gerbe arising in the discussion of Faddeev's anomaly is the Faddeev-Mickelsson two cocycle.

In this brief note, it is shown that the random hyperbolicity of a random linear cocycle is equivalent to having the Lipschitz shadowing property.

In this brief note, it is shown that the random hyperbolicity of a random linear cocycle is equivalent to having the Lipschitz shadowing property.