Completely Positive Definite Maps Over Topological^＊—algebras

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Congruence Permutable Symmetric Extended de Morgan Algebras

An algebra A is said to be congruence permutable if any two congruences on it are permutable. This property has been investigated in several varieties of algebras, for example, de Morgan algebras, p-algebras, Kn,0-algebras. In this paper, we study the class of symmetric extended de Morgan algebras that are congruence permutable. In particular we consider the case where A is finite, and show that A is congruence permutable if and only if it is isomorphic to a direct product of finitely many simple algebras.     

一类特征单的n-李代数

利用结合代数与n-李代数的张量积构造了一类无限维特征单的n-李代数,且证明了除n=3的情形以外,这类特征单n-李代数的内导子代数是特征单李代数.     

L-R smash products for multiplier Hopf algebras

The theory of L-R smash product is extended to multiplier Hopf algebras and a sufficient condition for L-R smash product to be regular multiplier Hopf algebras is given. In particular, the result of the paper implies Delvaux＇s main theorem in the case of smash products.     

Derivations of Tensor Product Algebras

It is known that under certain finite dimensionality condition the derivation algebra of tensor product of two algebras can be obtained in terms of the derivation algebras and the centroids of the involved algebras. We extend this theorem to infinite dimensional case and as an application, we determine the derivation algebra of the fixed point algebra of the tensor product of two algebras, with respect to the tensor product of two finite order automorphisms. These provide the framework for calculating the derivations of some infinite dimensional Lie algebras.     

On the Hochschild cohomology ring of tensor products of algebras

We prove that, as Gerstenhaber algebras, the Hochschild cohomology ring of the tensor product of two algebras is isomorphic to the tensor product of the respective Hochschild cohomology rings of these two algebras, when at least one of them is finite dimensional. In case of finite dimensional symmetric algebras, this isomorphism is an isomorphism of Batalin–Vilkovisky algebras. As an application, we explain by examples how to compute the Batalin–Vilkovisky structure, in particular, the Gerstenhaber Lie bracket, over the Hochschild cohomology ring of the group algebra of a finite abelian group.     

A proof of the Pfister Factor Conjecture

It is shown that any split product of quaternion algebras with orthogonal involution is adjoint to a Pfister form. This settles the Pfister Factor Conjecture formulated by D.B. Shapiro. A more general problem on decomposability for algebras with involution is posed and solved in the case where the algebra is equivalent to a quaternion algebra.     

On homomorphisms between products of median algebras

Homomorphisms of products of median algebras are studied with particular attention to the case when the codomain is a tree. In particular, we show that all mappings from a product \({\mathbf{A_1} \times\ldots\times {\mathbf{A}_{n}}}\) of median algebras to a median algebra \({\mathbf{B}}\) are essentially unary whenever the codomain \({\mathbf{B}}\) is a tree. In view of this result, we also characterize trees as median algebras and semilattices by relaxing the defining conditions of conservative median algebras.     

On the Ternary Approach to Clifford Structures and Ising Lattices

We continue to modify and simplify the Ising-Onsager-Zhang procedure for analyzing simple orthorhombic Ising lattices by considering some fractal structures in connection with Jordan and Clifford algebras and by following Jordan-von Neumann-Wigner (JNW) approach. We concentrate on duality of complete and perfect JNW-systems, in particular ternary systems, analyze algebras of complete JNW-systems, and prove that in the case of a composition algebra we have a self-dual perfect JNW-system related to quaternion or octonion algebras. In this context, we are interested in the product table of the sedenion algebra.     

n-Ary Mal'tsev Algebras

By analogy with n-Lie algebras, which are a natural generalization of Lie algebras to the case of n-ary multiplication, we define the concept of an n-ary Mal'tsev algerba. It is shown that exceptional algebras of a vector cross product are ternary central simple Mal'tsev algebras, which are not 3-Lie algebras if the characteristic of a ground field is distinct from 2 and 3. The basic result is that every n-ary algebra of the vector cross product is an n-ary central simple Mal'tsev algebra.     

The realization of positive definite matrices via planar networks and mixing-type sub-cluster algebras

As an improvement of the combinatorial realization of totally positive matrices via the essential positive weightings of certain planar network by S.Fomin and A.Zelevinsky [7], in this paper,we give a test method of positive definite matrices via the planar networks and the so-called mixing-type sub-cluster algebras respectively,introduced here originally.This work firstly gives a combinatorial realization of all matrices through planar network,and then sets up a test method for positive definite matrices by LDU-decompositions and the horizontal weightings of all lines in their planar networks.On the other hand,mainly the relationship is built between positive definite matrices and mixing-type sub-cluster algebras.     

Graded Lie Algebras and Dynamical Systems

We define the graded Lie algebras generated by ergodic transformation with invariant measure. This algebra is the central extension of Lie algebras which is associated with the usual crossed product. At the same time, it is the special case of algebras with continuum root systems which were defined by the author and M. Saveliev at the beginning of Nineties. Examples of systems with a discrete spectrum are considered.     

Forms and Baer ordered *-fields

It is well known that for a quaternion algegra, the anisotropy of its norm form determines if the quaternion algebra is a division algebra. In case of biquaternio algebra, the anisotropy of the associated Albert form (as defined in [LLT]) determines if the biquaternion algebra is a division ring. In these situations, the norm forms and the Albert forms are quadratic forms over the center of the quaternion algebras; and they are strongly related to the algebraic structure of the algebras. As it turns out, there is a natural way to associate a tensor product of quaternion algebras with a form such that when the involution is orthogonal, the algebra is a Baer ordered *-field iff the associated form is anisotropic.     

Tensor product of partially-additive monoids

Partially-additive monoids (pams) were introduced by Arbib and Manes in order to provide an algebraic semantics for programming languages. In this paper, we prove that the categoryP a m of pams and additive maps is a closed category whose monoids are partially-additive semirings. We follow the tensor product construction of R. Guitart [7] for categories of algebras which generalize the case of modules. Nevertheless, the problem here is more difficult owing to the fact that pams are partial algebras rather than algebras. Thus, we have to make some modifications to Guitart’s approach.     

Reelle Jordanalgebren mit endlicher Spur

We study real Jordan algebras of arbitrary dimension which admit an associative, positive definite trace form and have suitable continuity properties. We construct certain completions until we arrive at a class of Jordan algebras corresponding to associative W^*-algebras of finite type. For these Jordan algebras we derive some basic properties concerning positive elements and idempotents. In contrast to other related investigations exceptional Jordan algebras are not excluded.     

Tensor Product of Massey Products

In this paper, we interpret Massey products in terms of realizations （twitsting cochains） of certain differential graded coalgebras with values in differential graded algebras. In the case where the target algebra is the cobar construction of a differential graded commutative Hopf algebra, we construct the tensor product of realizations and show that the tensor product is strictly associative, and commutative up to homotopy.     

Clifford geometric parameterization of inequivalent vacua

We propose a geometric method to parameterize inequivalent vacua by dynamical data. Introducing quantum Clifford algebras with arbitrary bilinear forms we distinguish isomorphic algebras—as Clifford algebras—by different filtrations (resp. induced gradings). The idea of a vacuum is introduced as the unique algebraic projection on the base field embedded in the Clifford algebra, which is however equivalent to the term vacuum in axiomatic quantum field theory and the GNS construction in C^*‐algebras. This approach is shown to be equivalent to the usual picture which fixes one product but employs a variety of GNS states. The most striking novelty of the geometric approach is the fact that dynamical data fix uniquely the vacuum and that positivity is not required. The usual concept of a statistical quantum state can be generalized to geometric meaningful but non‐statistical, non‐definite, situations. Furthermore, an algebraization of states takes place. An application to physics is provided by an U (2)‐symmetry producing a gap equation which governs a phase transition. The parameterization of all vacua is explicitly calculated from propagator matrix elements. A discussion of the relation to BCS theory and Bogoliubov–Valatin transformations is given. Copyright © 2001 John Wiley & Sons, Ltd.     

算子代数的斜积

引入了算子代数的一种新运算"斜积",证明了在这个新定义的斜积运算下算子代数的自反性保持不变.研究发现,斜积运算对应的子空间格是拓扑意义下的格的直积关系.这个新发现的重要意义在于由此可从已知的自反子空间格生成更多更复杂的新自反格,从而得到新的自反代数.在此基础上,本文对KS-代数保持性等其他非自伴代数类的性质也作了相应研究.     

A class of Lie algebras arising from intersection matrices

We find a class of Lie algebras, which are defined from the symmetrizable generalized intersection matrices. However, such algebras are different from generalized intersection matrix algebras and intersection matrix algebras. Moreover, such Lie algebras generated by semi-positive definite matrices can be classified by the modified Dynkin diagrams.