Commutator length of symplectomorphisms

Each element $x$ of the commutator subgroup $[G, G]$ of a group $G$ can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of $x$. The commutator length of $G$ is defined as the supremum of commutator lengths of elements of $[G, G]$. We show that for certain closed symplectic manifolds $(M,\omega)$, including complex projective spaces and Grassmannians, the universal cover $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ of the group of Hamiltonian symplectomorphisms of $(M,\omega)$ has infinite commutator length. In particular, we present explicit examples of elements in $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ that have arbitrarily large commutator length -- the estimate on their commutator length depends on the multiplicative structure of the quantum cohomology of $(M,\omega)$. By a different method we also show that in the case $c_1 (M) = 0$ the group $\widetilde{\hbox{\rm Ham}\, (M,\omega)$ and the universal cover ${\widetilde{\Symp}}_0\, (M,\omega)$ of the identity component of the group of symplectomorphisms of $(M,\omega)$ have infinite commutator length.     

D—AKNS族的换位表示

本文根据曹策问教授的想法,求得了与D-AKNS族发展方程相联系的特征值之泛函梯度与Lenard算子对;并由此得到了D-AKNS族非线性发展方程的换位表示。文末还讨论了换位表示与定态D-AKNS方程之间的关系.     

Commutator Leavitt Path Algebras

For any field K and directed graph E, we completely describe the elements of the Leavitt path algebra L _K (E) which lie in the commutator subspace [L _K (E), L _K (E)]. We then use this result to classify all Leavitt path algebras L _K (E) that satisfy L _K (E)?=?[L _K (E),L _K (E)]. We also show that these Leavitt path algebras have the additional (unusual) property that all their Lie ideals are (ring-theoretic) ideals, and construct examples of such rings with various ideal structures.     

K-Theory of the Commutator Ideal

We consider for unital c ^*-algebras the short exact sequence 0 I A * _C B A _max B 0and give an explicit formula for the boundary map for separable C ^*-algebras for which the Künneth formula for tensor products holds without Tor-terms. In the course of the article, we study the similar case of an ideal concerning one-relator groups.     

Commutator Properties for Periodic Splines

Commutator properties are established for periodic splines with distinct uniformly spaced knots (on uniform meshes) operated on by certain pseudo-differential operators. The commutation involves the operations of multiplication by a smooth function and application of a discrete version of orthogonal projection obtained by using a quadrature rule (which need integrate only constants exactly) to approximate the inner product. The results mirror a well-known super-approximation property of splines multiplied by smooth functions.     

Commutator identities and integrable hierarchies

Theoretical and Mathematical Physics - The approach based on commutator identities for elements of associative algebras was previously effectively used to investigate $$(2{+}1)$$ -dimensional...     

Commutator equations in free groups

Letf ₁, …,f _n be free generators of a free groupF. We consider the equation [z ₁, …,z _n]_ω. where ω and ω′ indicate the disposition of brackets in the higher commutators [z ₁, …,z _n]_ω and [f ₁, …,f _n]_ω. We give a necessary and sufficient condition on ω and ω′ for the existence of solutions of this equation. It is also shown that for any solutionz ₁=r₁, …,z _z=r _n we have <r ₁, …,r _n>=〈f ₁, …f _n〉.     

Herz空间上交换子的有界性

得到了一类线性（或次线性）算子与ＢＭＯ函数构成的交换子在Ｈｅｒｚ型空间上的有界性估计。     

Herz型Hardy空间上的一类交换子

证明一大类次线性算子与ＢＭＯ函数构成的交换子在Ｈａｒｚ型Ｈａｒｄｙ空间上的有界性。     

Commutator Invariant Subgroups of Abelian Groups

We describe the commutator invariant subgroups of a nonreduced abelian group. We find out when all commutator invariant subgroups of a separable group and an algebraically compact torsion-free group are fully invariant and describe the E-centers and E-commutants of these and some other groups.     

A Short Proof of Commutator Estimates

The goal of this note is to give, at least for a restricted range of indices, a short proof of homogeneous commutator estimates for fractional derivatives of a product, using classical tools. Both \(L^{p}\) and weighted \(L^{p}\) estimates can be proved by the same argument. When the space dimension is 1, we obtain some new estimates in the unexplored range \(1/3<r\le 1/2\).

     

Weighted Commutator and Weighted BMO Martingales

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Commutator algebras arising from splicing operations

We prove that the variety of Lie algebras arising from splicing operation coincides with the variety CM of centreby-metabelian Lie algebras. Using these Lie algebras we find the minimal dimension algebras generated the variety CM and the variety of its associative envelope algebras. We study the splicing n-ary operation. We show that all n-ary (n > 2) commutator algebras arising from this operation are nilpotent of index 3. We investigate the generalization of the splicing n-ary operation, and we formulate a series of open problems.     

Commutator estimates in von Neumann algebras

Let M be a von Neumann algebra. For every self-adjoint locally measurable operator a, there exists a central self-adjoint locally measurable operator c ₀ such that, given any ? > 0, |[a, u _ε ]| ? (1 ? ε)|aε ? c ₀| for some unitary operator u _ε ∈ M. In particular, every derivation δ: M → I (where I is an ideal in M) is inner, and δ = δ _a for a ∈ I.     

Commutator Criteria for Magnetic Pseudodifferential Operators

The gauge covariant magnetic Weyl calculus has been introduced and studied in previous works. We prove criteria in terms of commutators for operators to be magnetic pseudodifferential operators of suitable symbol classes; neither the statements nor the proofs depend on a choice of a vector potential. We apply this criteria to inversion problems, functional calculus, affiliation results and to the study of the evolution group generated by a magnetic pseudodifferential operator.