A contribution to Segal algebras

Let A and B be Banach algebras. If B is an abstract Segal algebra in A, we have a bijective correspondence between the strictly irreducible representations of A and those of B. This gives a bijective correspondence for maximal modular left ideals. If A and B have approximate right units, we obtain a bijective correspondence for right resp. Two-sided ideals. For two-sided ideals this correspondence preserves the property of an ideal having approximate right units. This generalizes a Theorem by H. Reiter.It would be sufficient to require B to be a Banachspace.     

UNITARY GRASSMANNIANS OF DIVISION ALGEBRAS

We consider a central division algebra over a separable quadratic extension of a base field endowed with a unitary involution and prove 2-incompressibility of certain varieties of isotropic right ideals of the algebra. The remaining related projective homogeneous varieties are shown to be 2-compressible in general. Together with [17], where a similar issue for orthogonal and symplectic involutions has been treated, the present paper completes the study of Grassmannians of isotropic right ideals of division algebras.     

Block-matrix generalizations of infinite-dimensional schur products and schur multipliers

We consider an extention of the familiar Schur product to a bilinear product on the space of matrices whose entries are either bounded operators on a fixed Hilbert space or bounded "square" operator matrices. We show that this is a "natural" non-commutative extention of the Schur product, which retains many of its properties. The work is done mainly in infinite dimensions, where we concentrate on the maps induced on the space of bounded operator matrices via left or right "Schur block-multiplication" by a fixed "Schur block-multiplier". Our main goal is to study the distinctions between left and right multipliers, as well as the behaviour of ideals of operators under action of maps induced bu such.     

On the Unitary Similarity of Matrix Families

The classical Specht criterion for the unitary similarity between two complex n × n matrices is extended to the unitary similarity between two normal matrix sets of cardinality m. This property means that the algebra generated by a set is closed with respect to the conjugate transpose operation. Similar to the well-known result of Pearcy that supplements Specht's theorem, the proposed extension can be made a finite criterion. The complexity of this criterion depends on n as well as the length l of the algebras under analysis. For a pair of matrices, this complexity can be significantly lower than that of the Specht--Pearcy criterion.     

On Some Ideals in the Free Nonassociative Algebra

The free Lie, right, and left Leibniz algebras are obtained as quotients of the free nonassociative algebra by suitable ideals. In this article, we prove some remarkable properties of these ideals.     

Nakayama Twisted Centers and Dual Bases of Frobenius Cellular Algebras

For a Frobenius cellular algebra, we prove that, if the left (right) dual basis of a cellular basis is again cellular, then the algebra is symmetric. Moreover, some ideals of the center are constructed by using the so-called Nakayama twisted center.     

Representations of centrally-extended Lie algebras over differential operators and vertex algebras

We construct irreducible modules of centrally-extended classical Lie algebras over left ideals of the algebra of differential operators on the circle, through certain irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries. The structures of vertex algebras associated with the vacuum representations of these algebras are determined. Moreover, we prove that under certain conditions, the highest-weight irreducible modules of centrally-extended classical Lie algebras of infinite matrices with finite number of nonzero entries naturally give rise to the irreducible modules of the simple quotients of these vertex algebras. From vertex algebra and its representation point of view, our results with positive integral central charge are high-order differential operator analogues of the well-known WZW models in conformal field theory associated with affine Kac-Moody algebras. Indeed, when the left ideals are the algebra of differential operators, our Lie algebras do contain affine Kac-Moody algebras as subalgebras and our results restricted on them are exactly the representation contents in WZW models. Similar results with negative central charge are also obtained.     

Koszul incidence algebras, affine semigroups, and Stanley-Reisner ideals

We prove a theorem unifying three results from combinatorial homological and commutative algebra, characterizing the Koszul property for incidence algebras of posets and affine semigroup rings, and characterizing linear resolutions of squarefree monomial ideals. The characterization in the graded setting is via the Cohen-Macaulay property of certain posets or simplicial complexes, and in the more general nongraded setting, via the sequential Cohen-Macaulay property.     

Group codes over fields are asymptotically good

Group codes are right or left ideals in a group algebra of a finite group over a finite field. Following the ideas of a paper on binary group codes by Bazzi and Mitter in 2006, we prove that group codes over finite fields of any characteristic are asymptotically good.     

Representations of bornological algebras

Bounded representations of bornological algebras are considered. The left and right bornological radicals in bornological algebras are introduced. It is shown that the left (right) bornological radical of a bornological algebra A is equal to the intersection of all bornologically closed maximal regular left (respectively, right) ideals of A, and both these radicals of A and the Jacobson radical of A coincide if A is an advertive and simplicial bornological algebra (in particular, a bornological Q-algebra). Bibliography: 16 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 9–22.     

Noncommutative Generalizations of Theorems of Cohen and Kaplansky

This paper investigates situations where a property of a ring can be tested on a set of “prime right ideals.” Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every “prime right ideal” is finitely generated (resp. principal), where the phrase “prime right ideal” can be interpreted in one of many different ways. We also use our methods to show that other properties can be tested on special sets of right ideals, such as the right artinian property and various homological properties. Applying these methods, we prove the following noncommutative generalization of a result of Kaplansky: a (left and right) noetherian ring is a principal right ideal ring iff all of its maximal right ideals are principal. A counterexample shows that the left noetherian hypothesis cannot be dropped. Finally, we compare our results to earlier generalizations of Cohen’s and Kaplansky’s theorems in the literature.     

Finitely Deep Matrices

In the algebraic study of deep matrices ? _X () on a finite set of indices over a field, Christopher Kennedy has recently shown that there is a unique proper ideal  whose quotient is a central simple algebra. He showed that this ideal, which doesn't appear for infinite index sets, is itself a central simple algebra. In this article we extend the result to deep matrices with a finite set of 2 or more indices over an arbitrary coordinate algebra A, showing that when the coordinates are simple there is again such a unique proper ideal, and in general that the lattice of ideals of ? _X (A)/ and  are isomorphic to the lattice of ideals of the coordinate algebra A.     

Graded irreducible representations of Leavitt path algebras: A new type and complete classification

We present a new class of graded irreducible representations of a Leavitt path algebra. This class is new in the sense that its representation space is not isomorphic to any of the existing simple Chen modules. The corresponding graded simple modules complete the list of Chen modules which are graded, creating an exhaustive class: the annihilator of any graded simple module is equal to the annihilator of either a graded Chen module or a module of this new type.Our characterization of graded primitive ideals of a Leavitt path algebra in terms of the properties of the underlying graph is the main tool for proving the completeness of such classification. We also point out a problem with the characterization of primitive ideals of a Leavitt path algebra in Rangaswamy (2013) [15].     

Normal Subalgebras, I

We extend the notions of normal subalgebras, clots and ideals of an algebra A in a variety of (universal) algebras, from the familiar case of a single constant to the case of any number of constants. The first idea is that a subalgebra of A is normal when it is the inverse image under some morphism of the subalgebra generated by constants in the target. We argue that a better approach is obtained by considering pullbacks of γ _B and g?:?A?→?B, where g?:?A?→?B is some morphism and γ _B is the morphism from the initial algebra of the variety to B. Examples are shown in Heyting algebras, boolean algebras and unitary rings. Ideals and clots are generalizations of this notion, defined instead by closure under derived operations which have the right behavior on constants. There are several characterizations of these notions; some of them aiming at a categorical generalization. We deal with an (extended) notion of subtractivity, showing that it implies that ideals coincide with normal subalgebras, and it is connected with notions of coherence of congruences, allowing a characterization of protomodular varieties.     

Relationship between genetic algebras and semicommutative matrices

Three kinds of noncommutative Gonshor genetic algebras are defined and characterized in terms of matrices. A necessary condition for an algebra to have one of these properties is the semicommutativity of a set of matrices representing the left (and the right) transformations induced by basis elements. For Gonshor genetic algebras which are interpretable, bounds for the train roots of the algebraare given. In terms of matrices this result yields bounds for the eigenvalues of a set ofcertain stochastic semicommutative matrices.     

Torus-invariant prime ideals in quantum matrices, totally nonnegative cells and symplectic leaves

The algebra of quantum matrices of a given size supports a rational torus action by automorphisms. It follows from work of Letzter and the first named author that to understand the prime and primitive spectra of this algebra, the first step is to understand the prime ideals that are invariant under the torus action. In this paper, we prove that a family of quantum minors is the set of all quantum minors that belong to a given torus-invariant prime ideal of a quantum matrix algebra if and only if the corresponding family of minors defines a non-empty totally nonnegative cell in the space of totally nonnegative real matrices of the appropriate size. As a corollary, we obtain explicit generating sets of quantum minors for the torus-invariant prime ideals of quantum matrices in the case where the quantisation parameter q is transcendental over ${\mathbb{Q}}$ .     

Unitary representations of the generalized Virasoro current algebra

We investigate Verma modules V over the generalized Virasoro current algebrag, which is the semidirect sum of the Virasoro algebra and the central extension of a commutative algebra. It is shown that an arbitrary unitary representation with highest weight of algebrag is isomorphic to the tensor product of a unitary Fock representation ofg (or of a one-dimensional representation ofg) and a unitary representation with highest weight of the Virasoro algebra (considered as a representation of algebrag). This result is used to obtain formulas for the determinants of the matrices defining the Shapovalov form on Verma module V.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 4, pp. 532–538, April, 1990.     

Several observations on Toeplitz and Hankel circulants

An application of the unitary similarity with the discrete Fourier transform to the algebra of diagonal matrices yields the algebra of circulants. It turns out that if, in this construction, the unitary similarity is replaced by the unitary congruence, then the class of the so-called Hankel circulants is obtained. The causes and certain effects of this fact are discussed. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 121–127.     

A note on ideals in synaptic algebras

The notion of a synaptic algebra was introduced by David Foulis. Synaptic algebras unite the notions of an order-unit normed space, a special Jordan algebra, a convex effect algebra and an orthomodular lattice. In this note we study quadratic ideals in synaptic algebras which reflect its Jordan algebra structure. We show that projections contained in a quadratic ideal from a p-ideal in the orthomodular lattice of projections in the synaptic algebra and we find a characterization of those quadratic ideals which are generated by their projections.