Group Algebras Satisfying a Certain Lie Identity

ABSTRACT

Let K be a field of characteristic p ≠ 2 and let G be any group. A characterization of group algebras KG satisfying the Lie identity [[x,y],[u,v],[z,t]] = 0 for all x,y,u,v,z,t ? KG is obtained.     

Asymptotic Behavior of Poisson Kernels on NA Groups

On a Lie group S = NA, that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, a probability measure μ is considered and treated as a distribution according to which transformations s ∈ S acting on N = S/A are sampled. Under natural conditions, formulated some over thirty years ago, there is a μ-invariant measure m on N. Properties of m have been intensively studied by a number of authors. The present article deals with the situation when μ(A) = ?(s _t  ∈ A), where ?⁺ ? t → s _t  ∈ S is the diffusion on S generated by a second order subelliptic, hypoelliptic, left-invariant operator on S. In this article the most general operators of this kind are considered. Precise asymptotic for m at infinity and for the Green function of the operator are given. To achieve this goal a pseudodifferential calculus for operators with coefficients of finite smoothness is formulated and applied.     

On the Commutative Twisted Group Algebras

Let G be an abelian group and let R be a commutative ring with identity. Denote by R _t G a commutative twisted group algebra (a commutative twisted group ring) of G over R, by ?(R) and ?(R _t G) the nil radicals of R and R _t G, respectively, by G _p the p-component of G and by G ₀ the torsion subgroup of G. We prove that:

1. If R is a ring of prime characteristic p, the multiplicative group R* of R is p-divisible and ?(R) = 0, then there exists a twisted group algebra R _{t 1} (G/G _p ) such that R _t G/?(R _t G) ? R _{t 1} (G/G _p ) as R-algebras;

2. If R is a ring of prime characterisitic p and R* is p-divisible, then ?(R _t G) = 0 if and only if ?(R) = 0 and G _p  = 1; and

3. If B(R) = 0, the orders of the elements of G ₀ are not zero divisors in R, H is any group and the commutative twisted group algebra R _t G is isomorphic as R-algebra to some twisted group algebra R _{t 1} H, then R _t G ₀ ? R _{t 1} H ₀ as R-algebras.

     

Posets of infinite prinjective type and embeddings of Kronecker modules into the category of prinjective peak I-spaces

For a restricted Lie algebra L over a field of characteristic p > 0 we study the Lie nilpotency index t _L (u(L)) of its restricted universal enveloping algebra u(L). In particular, we determine an upper and a lower bound for t _L (u(L)). Finally, under the assumption that L is p-nilpotent and finite-dimensional, we establish when the Lie nilpotency index of u(L) is maximal.

Communicated by I. Shestakov.     

Nilpotent Ideals in Graded Lie Algebras and Almost Constant-Free Derivations

It is proved that if a (?/p ?)-graded Lie algebra L, where p is a prime, has exactly d nontrivial grading components and dim L ₀ = m, then L has a nilpotent ideal of d-bounded nilpotency class and of finite (m,d)-bounded codimension. As a consequence, Jacobson's theorem on constant-free nilpotent Lie algebras of derivations is generalized to the almost constant-free case. Another application is for Lie algebras with almost fixed-point-free automorphisms.     

On v-closed manis valution rings

ABSTRACT

The graded Lie algebra L associated to the Nottingham group with respect to its natural filtration is known to be a loop algebra of the first Witt algebra W ₁ . The fact that the Schur multiplier of W ₁ , in characteristic p > 3, is one-dimensional implies that L is not finitely presented. Consider the universal covering ? ₁ of W ₁ and the corresponding loop algebra M of ? ₁ . In this paper we prove that M itself is finitely presented for p > 3. In characteristic p >  11 the algebra M turns out to be presented by two relations.     

Small Data Scattering for the One-Dimensional Nonlinear Dirac Equation with Power Nonlinearity

We study scattering problems for the one-dimensional nonlinear Dirac equation (?_t + α?_x + iβ)Φ = λ|Φ|^p?1Φ. We prove that if p > 3 (resp. p > 3 + 1/6), then the wave operator (resp. the scattering operator) is well-defined on some 0-neighborhood of a weighted Sobolev space. In order to prove these results, we use linear operators D(t)xD(?t) and t?_x + x?_t ? α/2, where {D(t)}_t∈? is the free Dirac evolution group. For the reader's convenience, in an appendix we list and prove fundamental properties of D(t)xD(?t) and t?_x + x?_t ? α/2.     

Nested Log-Concavity

We give conditions on the coefficients of a polynomial p(x) so that p(x + t) be log-concave or strictly log-concave. Several applications are given: if p(x) is a polynomial with nonnegative and nondecreasing coefficients, then p(x + t) is strictly log-concave for all t ≥ 1; for any polynomial p(x) with positive leading coefficient, there is t ₀ ≥ 0 such that for any t ≥ t ₀ it holds that the coefficients of p(x + t) are positive, strictly decreasing, and strictly log-concave; if p(x) is a log-concave polynomial with nonnegative coefficients and no internal zeros, then p(x + t) is strictly log-concave for all t > 0; Betti numbers of lexsegment monomial ideals are strictly log-concave.     

Compact Exceptional Simple Kantor Triple Systems Defined on Tensor Products of Composition Algebras∗

In this article we give the classification of compact exceptional simple Kantor triple systems defined on tensor products of composition algebras A =  ₁?  ₂ such that their Kantor algebras ?(φ, A) are real forms of exceptional simple Lie algebras.     

Some Extensions of PSL(2, p 2) are Uniquely Determined by Their Complex Group Algebras

In Tong-Viet's, 2012 work, the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras?

It is proved here that some simple groups of Lie type are determined by the structure of their complex group algebras. Let p be an odd prime number and S = PSL(2, p ²). In this paper, we prove that, if M is a finite group such that S < M < Aut(S), M = ?₂ × PSL(2, p ²) or M = SL(2, p ²), then M is uniquely determined by its order and some information about its character degrees. Let X ₁(G) be the set of all irreducible complex character degrees of G counting multiplicities. As a consequence of our results, we prove that, if G is a finite group such that X ₁(G) = X ₁(M), then G ? M. This implies that M is uniquely determined by the structure of its complex group algebra.     

A Generalized Global Cartan Decomposition: A Basic Example

Suppose G ₁ ?  G are complex linear simple Lie groups. Let  ₁ ?  be the corresponding pair of Lie algebras. For the Killing-orthogonal  of  ₁ in  we have a vector space direct sum  =  ₁⊕ , which generalizes the classical Cartan decomposition on the Lie algebras level. In this article we study the corresponding problem of a ‘generalized global Cartan decomposition’ on the Lie groups level for the pair of groups ( G , G ₁) = (SL (4,?),Sp (2,?)); here  =  (4,?),  ₁ =  (2,?), and  = {X ?  | X ^? = X}, where X? X ^? is the symplectic involution. We prove that G  =  G ₁exp  ∪ i G ₁exp . The key point of the proof is to study in detail the set exp ; and for that purpose we introduce the J-twisted Pfaffian of size 2n defined on the set of all 2n × 2n matrices X satisfying X ^? = X, which is here a natural counterpart of the standard Pfaffian.     

Generators of Simple Lie Algebras and the Lower Rank of Some Pro-P Groups

Abstract

Let  be a simple classical Lie algebra over a field F of characteristic p > 7. We show that > d () = 2, where d() is the number of generators of . Let G be a profinite group. We say that G has lower rank ≤ l, if there are {G _α} open subgroups which from a base for the topology at the identity and each G _α is generated (topologically) by no more than l elements. There is a standard way to associate a Lie algebra L(G) to a finitely generated (filtered) pro-p group G. Suppose L(G) ?  ? tF _p [t], where  is a simple Lie algebra over F _p , the field of p elements. We show that the lower rank of G is ≤ d () + 1. We also show that if  is simple classical of rank r and p > 7 or p 2r ² ? r, then the lower rank is actually 2.     

Right Coideal Subalgebras of Quantized Universal Enveloping Algebras of Type G 2*

In this article we describe the right coideal subalgebras containing all group-like elements of the two-parameter quantum group U _q (𝔤), where 𝔤 is a simple Lie algebra of type G ₂, while the main parameter of quantization q is not a root of 1. As a consequence, we determine that there are precisely 60 different right coideal subalgebras containing all group-like elements. If the multiplicative order t of q is finite, t > 4, t ≠ 6, then the same classification remains valid for homogeneous right coideal subalgebras of the two-parameter version of the small Lusztig quantum group u _q (𝔤).     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

Remarks on Taylor Series Expansions and Conditional Expectations for Stratonovich SDEs with Complete V‐Commutativity

Abstract

It may happen that there is not a finite maximum order bound for numerical approximations of stochastic processes X = (X _t : 0 ≤ t ≤ T) satisfying Stratonovich stochastic differential equations (SDEs) with some commutative structure along an appropriate functional V(t, X _t ). This statement can be proven with respect to the concept of mean square convergence under the assumption of “infinite smoothness” of drift a(t, x) and diffusion coefficients b ^j (t, x) and with finite initial second moments. As a result, we obtain an infinite series expansion of the conditional expectation 𝔼[V(t, X _t )|? _t ^N ] on any fixed finite time interval [0, T], provided that the information is collected by discretized σ‐field ? _T ^N  = σ{W _{t 0} , W _{t 1} , …, W _{t N?1} , W _T } at N + 1 given time instants t _i  ∈ [0, T] with t ₀ ≤ t ₁ ≤ ··· ≤ t _N?1 ≤ t _N  = T.     

On a Question of Glasby,Praeger, and Xia

A Jordan partition λ(m, n, p) = (λ₁, λ₂, … , λ _m ) is a partition of mn associated with the expression of a tensor V _m  ? V _n of indecomposable KG-modules into a sum of indecomposables, where K is a field of characteristic p and G a cyclic group of p-power order. It is standard if λ _i  = m + n ? 2i + 1 for all i. We answer a recent question of Glasby, Praeger, and Xia who asked for necessary and sufficient conditions for λ(m, n, p) to be standard.     

Multiplicative Lie Isomorphisms Between Prime Rings

Let ? be a prime ring with 1 containing a nontrivial idempotent E, and let ?′ be another prime ring. If Φ:? → ?′ is a multiplicative Lie isomorphism, then Φ(T + S) = Φ(T) + Φ(S) + Z′ _T,S for all T, S ∈ ?, where Z′ _T,S is an element in the center 𝒵′ of ?′ depending on T and S.     

Non-Diffusive Large Time Behavior for a Degenerate Viscous Hamilton–Jacobi Equation

The convergence to non-diffusive self-similar solutions is investigated for non-negative solutions to the Cauchy problem ? _t u = Δ _p u + |? u| ^q when the initial data converge to zero at infinity. Sufficient conditions on the exponents p > 2 and q > 1 are given that guarantee that the diffusion becomes negligible for large times and the L ^∞-norm of u(t) converges to a positive value as t → ∞.     

A Symmetric Presentation for J 1

Abstract

We give a computer-free proof that the sporadic simple group J ₁ is a isomorphic to the progenitor 2*⁵ : A ₅ factorized over a single relation. Precisely, we prove that J ₁ is defined by the presentation ?x, y, t ∣ x ⁵ = y ³ = (xy)² = 1 = t ² = [y, t] = [y, t ^{x 3} ] = (xt)⁷?.