Nichols algebras of group type with many quadratic relations

We classify Nichols algebras of irreducible Yetter–Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known finite-dimensional Nichols algebras of nonabelian group type appear in the result of our classification. We find a new finite-dimensional Nichols algebra over fields of characteristic two.     

Invariant Lie Algebras and Lie Algebras with a Small Centroid

A subalgebra of a Lie algebra is said to be invariant if it is invariant under the action of some Cartan subalgebra of that algebra. A known theorem of Melville says that a nilpotent invariant subalgebra of a finite-dimensional semisimple complex Lie algebra has a small centroid. The notion of a Lie algebra with small centroid extends to a class of all finite-dimensional algebras. For finite-dimensional algebras of zero characteristic with semisimple derivations in a sufficiently broad class, their centroid is proved small. As a consequence, it turns out that every invariant subalgebra of a finite-dimensional reductive Lie algebra over an arbitrary definition field of zero characteristic has a small centroid.     

Finite-Dimensional Discrete Systems Integrated in Quadratures

We consider finite-dimensional reductions (truncations) of discrete systems of the type of the Toda chain with discrete time that retain the integrability. We show that for finite-dimensional chains, in addition to integrals of motion, we can construct a rich family of higher symmetries described by the master symmetry. We reduce the problem of integrating a finite-dimensional system to the implicit function theorem.     

On finite discrete operators and equations

For discrete operators generated by Calderon–Zygmund kernels we consider certain finite-dimensional approximations. We compare an initial discrete operator with its finite-dimensional analogue and obtain invertibility conditions and approximation rate. The error estimate for approximate finite-dimensional solution of corresponding equation is given also for a special right-hand side. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite-Dimensional Simple Poisson Modules

We prove a result that can be applied to determine the finite-dimensional simple Poisson modules over a Poisson algebra and apply it to numerous examples. In the discussion of the examples, the emphasis is on the correspondence with the finite-dimensional simple modules over deformations and on the behaviour of finite-dimensional simple Poisson modules on the passage from a Poisson algebra to the Poisson subalgebra of invariants for the action of a finite group of Poisson automorphisms.     

素特征的“李定理”

设L为代数闭域F上有限维李代数,著名的李定理说:若char F=0,则L为可解当且仅当L的任一有限维不可约模为1维的.在这里特征为0及模为有限维两个条件都是本质的.(1)若charF=P>0,则L为交换当且仅当L的任一(有限维)不可约模为1维的;(2)若char F=0,则L为交换当且仅当L的任一(有限维或无限维)不可约模为1维的; (3)若char F=P>7,L为李代数(限制李代数),则L为可解当且仅当L的任一不可约模(限制模)的维数为p的幂.     

Bounded linear mappings of finite rank

This paper comprises a constructive investigation of the relationship between compactness, finite rank and located kernel for a bounded linear mapping into a finite-dimensional normed space. The main result is that a bounded linear mapping of a normed space into a finite-dimensional normed space is constructively compact if and only if its kernel is located. Several examples are given which highlight the constructive distinction between a mapping into a finite-dimensional space and a mapping with finite-dimensional range.     

Simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0

We classify the central simple finite-dimensional noncommutative Jordan superalgebras of characteristic 0. As a corollary, we describe the Poisson brackets on the simple finite-dimensional Jordan superalgebras of characteristic 0.     

Numerical Differentiation by Finite-Dimensional Regularition

Tikhonov regularization of order one is applied to the differentiationoperation. By minimizing the regularization functional overa finite-dimensional space there results a procedure for numericaldifferentiation. This finite-dimensional regularization resultsin a sparse, symmetric, positive definite matrix problem whencubic splines are chosen as the finite-dimensional space. Theeffects of error in the data on the values of the regularizedderivative can be estimated in terms of the norm of the regularizationoperator. Numerical experiments are presented which illustratethe stability and obtainable accuracy of the method.     

Radford's S4 formula for co-Frobenius Hopf algebras

This note extends Radford's formula for the fourth power of the antipode of a finite-dimensional Hopf algebra to co-Frobenius Hopf algebras and studies equivalent conditions to a Hopf algebra being involutory for finite-dimensional and co-Frobenius Hopf algebras.     

A QUASI-NEWTON METHOD IN INFINITE-DIMENSIONAL SPACES AND ITS APPLICATION FOR SOLVING A PARABOLIC INVERSE PROBLEM

1.IntroductionQuasi-Newtonmethodsplayanimportantroleinnumericallysolvingnon--linearsystemsofequationsontheEuclideanspaces.Blltitseemsthatthequasi-Newtonmethodshavenotbeenapplieddirectlytosolvinginverseproblemsinpartialdifferentialequations(PDE)uptonowifwe…     

Finite-dimensional homogeneously simple algebras of associative type

In this paper, we describe finite-dimensional homogeneously simple algebras of associative type whose 1-component is a full matrix algebra. In addition, we prove that a finite-dimensional division ring of associative type over an algebraically closed field is isomorphic to a group algebra.     

Sums,couplings, and completions of almost Pontryagin spaces

An almost Pontryagin space can be written as the direct and orthogonal sum of a Hilbert space, a finite-dimensional anti-Hilbert space, and a finite-dimensional neutral space. In this paper orthogonal sums of almost Pontryagin spaces and completions to almost Pontryagin spaces are studied.     

A Bo?ejko-Picardello type inequality for finite-dimensional CAT(0) cube complexes

We prove a Bo?ejko-Picardello type inequality for finite-dimensional CAT(0) cube complexes and as a consequence we obtain that a group acting properly on a finite-dimensional CAT(0) cube complex is weakly amenable with the Cowling-Haagerup constant 1.     

The trace space and Kauffman's knot invariants

The traces in the construction of Kauffman's knot invariants are studied. The trace space is determined for a semisimple finite-dimensional quantum Hopf algebra and the best lower bound of the dimension of the trace space is given for a unimodular finite-dimensional quantum Hopf algebra.

     

Nilpotency of <Emphasis Type="Italic">n</Emphasis>-tuple Lie algebras and associative <Emphasis Type="Italic">n</Emphasis>-tuple algebras

We obtain conditions for the nilpotency of finite-dimensional n-tuple Lie algebras and finite-dimensional associative n-tuple algebras. The established conditions are analogous to theorems of Engel and Wedderburn for Lie algebras and associative algebras.     

On the exponent of finite-dimensional non-cosemisimple Hopf algebras

In 1999, Kashina introduced the exponent of a Hopf algebra. In this article, we prove that the exponent of a finite-dimensional non-cosemisimple Hopf algebra with Chevalley property in characteristic 0 is infinite, and the exponent of a finite-dimensional non-cosemisimple pointed Hopf algebra in positive characteristic is finite.     

Continuous-time approximations for the nonlinear filtering problem

The paper deals with a possible approach to the problem of finite-dimensional filters in the nonlinear case, when the signal is a diffusion process and the observations are corrupted by additive white noise. The approach considers a sequence of finite-dimensional recursive filters that approximate the actual optimal one. The approximating filters are given in terms of functionals of continuous-time Markov chains that converge weakly to the original diffusion. These functionals can be recursively computed via a finite-dimensional Zakai equation, for which the solution is given in terms of a robust input-output relation.Work partially supported by the Consiglio Nazionale delle Ricerche (Italy) under Contract 79.00700.01.     

Hom-Lie algebra structures on semi-simple Lie algebras

Hom-Lie algebras can be considered as a deformation of Lie algebras. In this note, we prove that the hom-Lie algebra structures on finite-dimensional simple Lie algebras are trivial. We find when a finite-dimensional semi-simple Lie algebra admits non-trivial hom-Lie algebra structures and the isomorphic classes of non-trivial hom-Lie algebras are determined.     

An efficient discretization scheme for solving ill-posed problems

In this paper, we consider a finite-dimensional approximation scheme combined with Tikhonov regularization for solving ill-posed problems. Error estimates are obtained by an a priori parameter choice strategy and the results show that the amount of discrete information required for solving the problem is far less than the traditional finite-dimensional approach.