Sum Of Algebras Research Articles

Fast algorithms for a multidimensional hypercomplex discrete Fourier transform with radix-3 decomposition

Combined algorithms for the multidimensional hypercomplex discrete Fourier transform (HDFT) of a real signal with data representation in the Hamilton-Eisenstein generalized codes are synthesized. The complexity of arithmetic operations in a commutative-associative hypercomplex algebra and its representation in generalized codes are obtained. It is shown that there exist only two essentially different commutative-associative hypercomplex algebras: the direct sums of real or complex algebras. The computational complexity of the algorithm synthesized is estimated.

Unitary orbits of normal operators in von Neumann algebras

The starting points for this paper are simple descriptions of the norm and strong* closures of the unitary orbit of a normal operator in a von Neumann algebra. The statements are in terms of spectral data and do not depend on the or cardinality of the algebra. We relate this to several known results and derive some consequences, of which we list a few here. Exactly when the ambient von Neumann algebra is a direct sum of sigma-finite algebras, any two normal operators have the same norm-closed unitary orbit if and only if they have the same strong*-closed unitary orbit if and only if they have the same strong-closed unitary orbit. But these three closures generally differ from each other and from the unclosed unitary orbit, and we characterize when equality holds between any two of these four sets. We also show that in a properly infinite von Neumann algebra, the strong-closed unitary orbit of any operator, not necessarily normal, meets the center in the (non-void) left essential central spectrum of Halpern. One corollary is a type III Weyl-von Neumann-Berg theorem involving containment of essential central spectra.

KK ‐theory of amalgamated free products of C *‐algebras

AbstractJ. Cuntz has conjectured the existence of two cyclic six terms exact sequences relating the KK ‐groups of the amalgamated free product A 1 ∗︁ B A 2 to the KK ‐groups of A 1, A 2 and B. First we establish automatic existence of strict and absorbing homomorphisms. Then we use this result to verify the conjecture when B is a countable direct sum of matrix algebras and the embeddings of B into A 1 and A 2 are quasiunital. Inspired by the proof we achieve the following nice classification result: A separable C *‐algebra B is a countable direct sum of matrix algebras if and only if the unitary group of the multiplier algebra U M (B) is compact in the strict topology. Finally we prove the conjecture when the amalgamated free product has the property that any asymptotically split extension of A 1 ∗︁ B A 2 is split. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Fuzzy multi-objective immune optimization algorithm-based conceptual design

Scheme design is the key process of conceptual design.Because scheme design is a combination optimization problem,it is difficult that how to avoid combination explore to produce some feasible schemes.The model of conceptual de- sign is built from the point of views of fuzzy multi-objective optimization.The theorem of multi-fuzzy-set algebra sum and the theorem of fuzzy set compare are presented,and then ac- cording to the relationships between structures and functions of the production,the fuzzy mathematical model of the multi-objective optimization of conceptual design is provided. In order to solve the multi-objective optimization question,an effective multi-objective optimal algorithm based on the evolu- tions of the immune cells,whose convergence and diversity are better than SPEA.An example of transmission scheme design of the belt conveyor is given to show that this method is able to produce the feasible schemes set and realize the automatic scheme design.

On embeddings of some quotient algebras of free sums of Lie algebras

Let (B i ) i∈I be a set of Lie algebras; let X be a free Lie algebra; let \(F = \left( {\sum\limits_{i \in I} {^ * B_i } } \right)\) * X be their free sum; let R be an ideal of F such that R ⋂ B i = 1 (i ∈ I); let V be a variety of Lie algebras such that V(R) is an ideal of F. Under some restrictions, we construct an embedding of F/V(R) into the verbal wreath product of a free algebra of the variety V with F/R.

Hamiltonian and quasi-Hamiltonian structures associated with semi-direct sums of Lie algebras

The trace variational identity is generalized to zero curvature equations associated with non-semi-simple Lie algebras or, equivalently, Lie algebras possessing degenerate Killing forms. An application of the resulting generalized variational identity to a class of semi-direct sums of Lie algebras in the AKNS case furnishes Hamiltonian and quasi-Hamiltonian structures of the associated integrable couplings. Three examples of integrable couplings for the AKNS hierarchy are presented: one Hamiltonian and two quasi-Hamiltonian.

Representation theory of q-rook monoid algebras

We show that, over an arbitrary field, q-rook monoid algebras are iterated inflations of Iwahori-Hecke algebras, and, in particular, are cellular. Furthermore we give an algebra decomposition which shows a q-rook monoid algebra is Morita equivalent to a direct sum of Iwahori-Hecke algebras. We state some of the consequences for the representation theory of q-rook monoid algebras.

Relative amenability and the non-amenability of B(l1)

AbstractIn this paper we begin with a short, direct proof that the Banach algebra B(l1) is not amenable. We continue by showing that various direct sums of matrix algebras are not amenable either, for example the direct sum of the finite dimensional algebras is no amenable for 1 ≤ p ≤ ∞, p ≠ 2. Our method of proof naturally involves free group algebras, (by which we mean certain subalgebras of B(X) for some space X with symmetric basis—not necessarily X = l2) and we introduce the notion of ‘relative amenability’ of these algebras.

Semidirect sums of Lie algebras and discrete integrable couplings

A relation between semidirect sums of Lie algebras and integrable couplings of lattice equations is established, and a practicable way to construct integrable couplings is further proposed. An application of the resulting general theory to the generalized Toda spectral problem yields two classes of integrable couplings for the generalized Toda hierarchy of lattice equations. The construction of integrable couplings using semidirect sums of Lie algebras provides a good source of information on complete classification of integrable lattice equations.

Semilattices of Groups and Nonstable K-Theory of Extended Cuntz Limits

We give an elementary characterization of those abelian monoids M that are direct limits of countable sequences of finite direct sums of monoids of the form either (Z/nZ) ⊔ {0} or Z ⊔ {0}. This characterization involves the Riesz refinement property together with lattice-theoretical properties of the collection of all subgroups of M (viewed as a semigroup), and it makes it pos- sible to express M as a certain submonoid of a direct product � × G, where � is a distributive semilattice with zero and G is an abelian group. When applied to the monoids V (A) appearing in the nonstable K-theory of C*-algebras, our results yield a full description of V (A) for C*-inductive limits A of finite sums of full matrix algebras over either Cuntz algebras On, where 2 ≤ n < ∞, or corners of O1 by projections, thus extending to the case including O1 earlier work by the authors together with K.R. Goodearl.

Algebraic structures connected with pairs of compatible associative algebras

We study associative multiplications in semisimple associative algebras over ℂ compatible with the usual one or, in other words, linear deformations of semi-simple associative algebras over ℂ. It turns out that these deformations are in one-to-one correspondence with representations of certain algebraic structures, which we call M-structures in the matrix case and PM-structures in the case of direct sums of several matrix algebras. We also investigate various properties of PM-structures, provide numerous examples and describe an important class of PM-structures. The classification of these PM-structures naturally leads to affine Dynkin diagrams of A,D,E-types.

On Embeddings of Some Factor Algebras of Free Sums of Lie Algebras

On Embeddings of Some Factor Algebras of Free Sums of Lie Algebras

Ermakov's Superintegrable Toy and Nonlocal Symmetries

We investigate the symmetry properties of a pair of Ermakov equations. The system is superintegrable and yet possesses only three Lie point symmetries with the al- gebra sl(2,R). The number of point symmetries is insucient and the algebra unsuitable for the complete specification of the system. We use the method of reduction of order to reduce the nonlinear fourth-order system to a third-order system comprising a linear second-order equation and a conservation law. We obtain the representation of the com- plete symmetry group from this system. Four of the required symmetries are nonlocal and the algebra is the direct sum of a one-dimensional Abelian algebra with the semidirect sum of a two-dimensional solvable algebra with a two-dimensional Abelian algebra. The prob- lem illustrates the diculties which can arise in very elementary systems. Our treatment demonstrates the existence of possible routes to overcome these problems in a systematic fashion.

Semi-direct sums of Lie algebras and continuous integrable couplings

A relation between semi-direct sums of Lie algebras and integrable couplings of continuous soliton equations is presented, and correspondingly, a feasible way to construct integrable couplings is furnished. A direct application to the AKNS spectral problem leads to a novel hierarchy of integrable couplings of the AKNS hierarchy of soliton equations. It is also indicated that the study of integrable couplings using semi-direct sums of Lie algebras is an important step towards complete classification of integrable systems.

Hereditary coalgebras and representations of species

Let C be a basic indecomposable hereditary K-coalgebra, where K is an arbitrary field. We investigate a technique for studying C and left C-comodules by means of the left valued Gabriel quiver of C, an associated Tits quadratic form and locally nilpotent representations of the Ext-species of C. One of the main aims of the paper is to prove the following result. Let { S ( j ) } j ∈ I C be a complete set of pairwise non-isomorphic simple left C-comodules and, given i , j ∈ I C , we set s i j 1 = dim K Ext C 1 ( S ( i ) , S ( j ) ) . Then every left C-comodule is a direct sum of finite dimensional C-comodules if and only if the following four conditions are satisfied: (a) for any j ∈ I C , the sum ∑ i ∈ I C s i j 1 + ∑ i ∈ I C s j i 1 is finite, (b) the set { i ∈ I C ; ∑ j ∈ I C s j i 1 = 0 } is finite, (c) there is no infinite sequence j 1 , … , j m , … of elements of I C such that Ext C 1 ( S ( j 2 ) , S ( j 1 ) ) ≠ 0 , Ext C 1 ( S ( j 3 ) , S ( j 2 ) ) ≠ 0 , … , Ext C 1 ( S ( j m + 1 ) , S ( j m ) ) ≠ 0 , … , (d) the integral Tits quadratic form q C : Z ( I C ) → Z of C defined by the formula q C ( v ) = ∑ j ∈ I C s j 0 v j 2 − ∑ i , j ∈ I C s i j 1 v i v j is positive definite, where s j 0 = dim K End C S ( j ) , v ∈ Z ( I C ) and Z ( I C ) is the direct sum of I C copies of Z . In this case, we show that C is isomorphic to the (co)tensor coalgebra T F □ ( M ) associated to the Ext-species of C, where F is the direct sum of the division algebras F j = End C S ( j ) , M = ⊕ i , j ∈ I C M j i , and M j i = Hom F i ( Ext C 1 ( S ( i ) , S ( j ) ) , F i ) . We also describe the Auslander–Reiten quiver Γ ( C - comod ) and the structure of the category C - comod of left C-comodules of finite dimension.

Generating the variety of BL-algebras

This paper collects some results from [AFM] and from [AM]. Our purpose is to illustrate some interesting classes of algebras which generate the whole variety of BL-algebras. In particular, we prove that such variety is generated by its finite members and by the class of finite ordinal sums of Lukasiewicz t-norm algebras. Finally, we characterize the BL-chains which generate the whole variety of BL-algebras.

Continuous fields of Kirchberg [formula omitted]-algebras

In this paper we study the C * -algebras associated to continuous fields over locally compact metrizable zero-dimensional spaces whose fibers are Kirchberg C * -algebras satisfying the UCT. We show that these algebras are inductive limits of finite direct sums of Kirchberg algebras and they are classified up to isomorphism by topological invariants.

Comtrans algebras, Thomas sums, and bilinear forms

A comtrans algebra is said to decompose as the Thomas sum of two subalgebras if it is a direct sum at the module level, and if its algebra structure is obtained from the subalgebras and their mutual interactions as a sum of the corresponding split extensions. In this paper, we investigate Thomas sums of comtrans algebras of bilinear forms. General necessary and sufficient conditions are given for the decomposition of the comtrans algebra of a bilinear form as a Thomas sum. Over rings in which 2 is not a zero divisor, comtrans algebras of symmetric bilinear forms are identified as Thomas summands of algebras of infinitesimal isometries of extended spaces, the complementary Thomas summand being the algebra of infinitesimal isometries of the original space. The corresponding Thomas duals are also identified. These results represent generalizations of earlier results concerning the comtrans algebras of finite-dimensional Euclidean spaces, which were obtained using known properties of symmetric spaces. By contrast, the methods of the current paper involve only the theory of comtrans algebras.

Lie Elements and Knuth Relations

AbstractA coplactic class in the symmetric group consists of all permutations in with a given Schensted Q-symbol, and may be described in terms of local relations introduced by Knuth. Any Lie element in the group algebra of which is constant on coplactic classes is already constant on descent classes. As a consequence, the intersection of the Lie convolution algebra introduced by Patras and Reutenauer and the coplactic algebra introduced by Poirier and Reutenauer is the direct sum of all Solomon descent algebras.

Local and semilocal vertex operator algebras

We initiate a general structure theory for vertex operator algebras V. We discuss the center and the blocks of V, the Jacobson radical and solvable radical, and local vertex operator algebras. The main consequence of our structure theory is that if V satisfies some mild conditions, then it is necessarily semilocal, i.e., a direct sum of local vertex operator algebras.