Gelfand–Kirillov Dimension and Local Finiteness of Jordan Superpairs Covered by Grids and Their Associated Lie Superalgebras

Abstract

In this paper we show that a Lie superalgebra L graded by a 3-graded irreducible root system has Gelfand–Kirillov dimension equal to the Gelfand–Kirillov dimension of its coordinate superalgebra A, and that L is locally finite if and only A is so. Since these Lie superalgebras are coverings of Tits–Kantor–Koecher superalgebras of Jordan superpairs covered by a connected grid, we obtain our theorem by combining two other results. Firstly, we study the transfer of the Gelfand–Kirillov dimension and of local finiteness between these Lie superalgebras and their associated Jordan superpairs, and secondly, we prove the analogous result for Jordan superpairs: the Gelfand–Kirillov dimension of a Jordan superpair V covered by a connected grid coincides with the Gelfand– Kirillov dimension of its coordinate superalgebra A, and V is locally finite if and only if A is so.     

Jordan Superderivations

Abstract

We extend Herstein's theorem on Jordan derivations of prime rings to superalgebras. Our main result states that a prime associative superalgebra admits a proper Jordan superderivation only in the case when its even part is commutative.     

Polynomial identities of bicommutative algebras,Lie and Jordan elements

ABSTRACT

An algebra with identities a(bc)?=?b(ac), (ab)c?=?(ac)b is called bicommutative. We construct list of identities satisfied by commutator and anti-commutator products in a free bicommutative algebra. We give criterions for elements of a free bicommutative algebra to be Lie or Jordan.     

INDECOMPOSABLE INJECTIVE MODULES AND A THEOREM OF KAPLANSKY

Abstract

Every tripotent e of a generalized Jordan triple system of second order uniquely defines a decomposition of the space of the triple into a direct sum of eight components. This decomposition is a generalization of the Peirce decomposition for the Jordan triple system. The relations between components are studied in the case when e is a left unit.     

Additive Derivations on Jordan Banach Pairs

Abstract

We show the invariance of “almost all” primitive ideals under additive derivations on a Jordan Banach pair and we extend the well known result of Johnson and Sinclair to the Jordan Banach pairs framework.     

Decidability for modules over a group ring - III

ABSTRACT

Representations of simple Jordan superalgebras of Hermitian 3?×?3 matrices over the exceptional simple alternative superalgebras B (1,2) and B (4,2) of characteristic 3 are studied. Every irreducible bimodule over these superalgebras up to isomorphism is either a regular bimodule or its opposite. As corollaries,some analogues of the Kronecker factorization theorem are proved for Jordan superalgebras that contain H₃(B (1,2)) and H₃(B(4,2)).     

Asymptotic Diagonalization and Normalization of Linear q -Difference Equations

Asymptotic diagonalizations of linear differential equations are studied by several authors. The problems for linear difference equations are investigated recently by Bodine and Sacker. In their work, the full spectrum condition plays essential role. Here we consider a related problem for q-difference equations, |q| < 1, which do not satisfy the full spectrum condition. Our tool is the Arnold normal form for matrix.     

Explicit Classifications of Orbits in Jordan Algebra and Freudenthal Vector Space Over the Exceptional Lie Groups

ABSTRACT

We give the explicit classifications of orbits in the Jordan algebra 𝔍 over the group E _6(?26) and the Freudenthal, R -vector space 𝔓 over the group E _7(?25).

Communicated by E. Zelmanov     

A projection and contraction method for symmetric cone complementarity problem

ABSTRACT

This paper considers the symmetric cone complementarity problem. A new projection and contraction method is presented which only requires some projection calculations and functional computations. It is proved that the iteration sequence produced by the proposed method converges to a solution of the symmetric cone complementarity problem under the condition that the underlying transformation is monotone. Numerical experiments also show the effectiveness of this method.     

The Tits–Kantor–Koecher Construction and Birepresentations of the Jordan Superpair SH(1, n)

Abstract

We introduce the concept of bimodule over a Jordan superpair and the Tits– Kantor–Koecher construction for bimodules. Using the construction we obtain the classification of irreducible bimodules over the Jordan superpair SH(1, n). We also prove semisimplicity for a class of finite dimensional SH(1, n)-bimodules for n ≥ 3.     

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.     

The Dimensions of Certain Symmetry Classes of Tensors

Commutative Rings, by Irving Kaplansky. Revised edition. The University of Chicago Press, Chicago and London, 1974, ix+182pp.($9.75)

Symmetry Groups and their Applications, by Willard Miller, Jr. Academic Press.     

On Composition series relative to a module

ABSTRACT

A commutative algebra with the identity (a * b) * (c * d) ? (a * d) * (c * b) = (a, b, c) * d ? (a, d, c) * b is called Novikov–Jordan. Example: K[x] under multiplication a * b = ?(ab) is Novikov–Jordan. A special identity for Novikov–Jordan algebras of degree 5 is constructed. Free Novikov–Jordan algebras with q generators are exceptional for any q ≥ 1.

     

New Interior-Point Algorithm for Symmetric Optimization Based on a Positive-Asymptotic Barrier Function

Abstract

We define a new interior-point method (IPM), which is suitable for solving symmetric optimization (SO) problems. The proposed algorithm is based on a new search direction. In order to obtain this direction, we apply the method of algebraically equivalent transformation on the centering equation of the central path. We prove that the associated barrier cannot be derived from a usual kernel function. Therefore, we introduce a new notion, namely the concept of the positive-asymptotic kernel function. We conclude that this algorithm solves the problem in polynomial time and has the same complexity as the best known IPMs for SO.     

Additive preservers of idempotence and Jordan homorphisms between rings of square matrices

Suppose R is an idempotence-diagonalizable ring. Let n and m be two arbitrary positive integers with n ≥ 3. We denote by Mn（R） the ring of all n x n matrices over R. Let （Jn（R）） be the additive subgroup of Mn（R） generated additively by all idempotent matrices. Let JJ = （Jn（R）） or Mn（R）. We describe the additive preservers of idempotence from JJ to Mm（R） when 2 is a unit of R. Thereby, we also characterize the Jordan （respectively, ring and ring anti-） homomorphisms from Mn （R） to Mm （R） when 2 is a unit of R.