New Class Of Algebras Research Articles

Confluent operator algebras and the closability property

Certain operator algebras A on a Hilbert space have the property that every densely defined linear transformation commuting with A is closable. Such algebras are said to have the closability property. They are important in the study of the transitive algebra problem. More precisely, if A is a two-transitive algebra with the closability property, then A is dense in the algebra of all bounded operators, in the weak operator topology. In this paper we focus on algebras generated by a completely nonunitary contraction, and produce several new classes of algebras with the closability property. We show that this property follows from a certain strict cyclicity property, and we give very detailed information on the class of completely nonunitary contractions satisfying this property, as well as a stronger property which we call confluence.

Cyclotomic Temperley–Lieb Algebra of Type D and Its Representation Theory

We define a new class of algebras, cyclotomic Temperley–Lieb algebras of type D, in a diagrammatic way, which is a generalization of Temperley–Lieb algebras of type D. We prove that the cyclotomic Temperley–Lieb algebras of type D are cellular. In fact, an explicit cellular basis is given by means of combinatorial methods. After determining all the irreducible representations of these algebras, we give a necessary and sufficient condition for a cyclotomic Temperley–Lieb algebra of type D to be quasi-hereditary.

EQ-algebras

We introduce a new class of algebras called EQ-algebras. An EQ-algebra has three basic binary operations (meet, multiplication and a fuzzy equality) and a top element. These algebras are intended to become algebras of truth values for a higher-order fuzzy logic (a fuzzy type theory, FTT). The motivation stems from the fact that until now, the truth values in FTT were assumed to form either an IMTL-, BL-, or MV-algebra, all of them being special kinds of residuated lattices in which the basic operations are the monoidal operation (multiplication) and its residuum. The latter is a natural interpretation of implication in fuzzy logic; the equivalence is then interpreted by the biresiduum, a derived operation. The basic connective in FTT, however, is a fuzzy equality and, therefore, it is not natural to interpret it by a derived operation. This defect is expected to be removed by the class of EQ-algebras introduced and studied in this paper. From the algebraic point of view, the class of EQ-algebras generalizes, in a certain sense, the class of residuated lattices and so, they may become an interesting class of algebraic structures as such.

Representation Theory of Finite Dimensional Algebras

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

The Jacobian algebras

A new class of algebras (the Jacobian algebras) is introduced and studied in detail. The Jacobian algebras are obtained from the Weyl algebras by inverting (not in the sense of Ore) certain elements. Surprisingly, the Jacobian algebras and the Weyl algebras have little in common. Moreover, they have almost opposite properties.

On n × m-valued Łukasiewicz-Moisil algebras

n×m-valued Łukasiewicz algebras with negation were introduced and investigated in [20, 22, 23]. These algebras constitute a non trivial generalization of n-valued Łukasiewicz-Moisil algebras and in what follows, we shall call them n×m-valued Łukasiewicz-Moisil algebras (or LM n×m -algebras). In this paper, the study of this new class of algebras is continued. More precisely, a topological duality for these algebras is described and a characterization of LM n×m -congruences in terms of special subsets of the associated space is shown. Besides, it is determined which of these subsets correspond to principal congruences. In addition, it is proved that the variety of LM n×m -algebras is a discriminator variety and as a consequence, certain properties of the congruences are obtained. Finally, the number of congruences of a finite LM n×m -algebra is computed.

Weak MV-algebras

Abstract In a recent paper [CHAJDA, I.—KÜHR, J.: A non-associative generalization of MV-algebras, Math. Slovaca 57, (2007), 301–312], authors introduced and studied a non-associative generalization of MV-algebras called NMV-algebras. In contrast to MV-algebras, sections (i.e. principal filters) in NMV-algebras which are proper (i.e. are not MV-algebras), do not admit a structure of an NMV-algebra with respect to the operations defined in a natural way. The aim of the paper is to present a new class of algebras generalizing MV-algebras but sharing the above property.

Nakayama oriented pullbacks and stably hereditary algebras

We introduce a new class of algebras, the Nakayama oriented pullbacks, obtained from pullbacks of surjective morphisms of algebras A ↠ C and B ↠ C . We prove that such a pullback is tilted when A and B are hereditary. We also show that stably hereditary algebras respecting the clock condition are Nakayama oriented pullbacks, and we use results about these pullbacks to show when a stably hereditary algebra is tilted or iterated tilted.

Almost laura algebras

In this paper, we propose a generalization for the class of laura algebras, called almost laura. We show that this new class of algebras retains most of the essential features of laura algebras, especially concerning the important role played by the non-semiregular components in their Auslander–Reiten quivers. Also, we study more intensively the left supported almost laura algebras, showing that these are characterized by the presence of a generalized standard, convex and faithful component. Finally, we prove that almost laura algebras behave well with respect to full subcategories, split-by-nilpotent extensions and skew group algebras.

Cluster-tilted algebras

We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.

Quantum superalgebras at roots of unity and topological invariants of three-manifolds

The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra. The quantum superalgebra U_q(osp(1|2n)) over C is considered with q a primitive N^th root of unity for all integers N >= 3. For such a q, a certain left ideal I of U_q(osp(1|2n)) is also a two-sided Hopf ideal, and the quotient algebra U_q^(N)(osp(1|2n)) = U_q(osp(1|2n)) / I is a Z_2-graded ribbon Hopf algebra. For all n and all N >= 3, a finite collection of finite dimensional representations of U_q^(N)(osp(1|2n)) is defined. Each such representation of U_q^(N)(osp(1|2n)) is labelled by an integral dominant weight belonging to the truncated dominant Weyl chamber. Properties of these representations are considered: the quantum superdimension of each representation is calculated, each representation is shown to be self-dual, and more importantly, the decomposition of the tensor product of an arbitrary number of such representations is obtained for even N. It is proved that the quotient algebra U_q^(N)(osp(1|2n)), together with the set of finite dimensional representations discussed above, form a pseudo-modular Hopf algebra when N >= 6 is twice an odd number. Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-manifolds. This invariant is shown to be different to the topological invariants of 3-manifolds arising from quantum so(2n+1) at roots of unity.

Representation theory of towers of recollement: Theory, notes, and examples

We give an axiomatic framework for studying the representation theory of towers of algebras. We introduce a new class of algebras, contour algebras, generalising (and interpolating between) blob algebras and cyclotomic Temperley–Lieb algebras. We demonstrate the utility of our formalism by applying it to this class.

On a class of Koszul algebras associated to directed graphs

In [I. Gelfand, V. Retakh, S. Serconek, R.L. Wilson, On a class of algebras associated to directed graphs, Selecta Math. (N.S.) 11 (2005), math.QA/0506507] I. Gelfand and the authors of this paper introduced a new class of algebras associated to directed graphs. In this paper we show that these algebras are Koszul for a large class of layered graphs.

G-Vertex Colored Partition Algebras as Centralizer Algebras of Direct Products#

The partition algebras P k (x) have been defined in Martin [Martin, P. (1990). Representations of graph temperley Lieb algebras. Pupl. Res. Inst. Math. Sci. 26:485–503] and Jones [Jones, V. F. R. (1993). The Potts model and the symmetric group. In: Subfactors: Proceedings of the Taniguchi Symposium on Operator Algebra, Kyuzeso. River edge, NJ: World Scientific, pp. 259–267, 1994]. We introduce a new class of algebras for every group G called “G-Vertex Colored Partition Algebras,” denoted by P k (x, G), which contain partition algebras P k (x), as subalgebras. We generalized Jones result by showing that for a finite group G, the algebra P k (n, G) is the centralizer algebra of an action of the direct product S n × G on tensor powers of its permutation module. Further we show that these algebras P k (x, G) contain as subalgebras the “G-Colored Partition Algebras P k (x;G)” introduced in Bloss [Bloss, M. (2003). G-colored partition algebras as centralizer algebras of wreath products. J. Algebra 265:690–710].

Notes onn×m-valued Łukasiewicz Algebras with Negation

The theory of n × m-valued Łukasiewicz algebras with negation was introduced and studied by the author in [14]. It is worth mentioning that in the particular case m = 2 this new class of algebras coincides with that of n-valued Łukasiewicz algebras. In this work, the research on n × m-valued Łukasiewicz algebras with negation is followed. More precisely, it is proved that this class of algebras is a variety in a different way from that indicated in [14]. Besides, the congruences on these algebras are determined and some of their properties are studied. The main result of this paper is that the principal congurences are equationally definable from which it is inferred that this variety is arithmetical. In addition, it is proved that the congruences on these algebras are regular and that every principal one is a factor congruence.1

Poisson orders, symplectic reflection algebras and representation theory

We introduce a new class of algebras called Poisson orders. This class includes the symplectic reflection algebras of Etingof and Ginzburg, many quantum groups at roots of unity, and enveloping algebras of restricted Lie algebras in positive characteristic. Quite generally, we study this class of algebras from the point of view of Poisson geometry, exhibiting connections between their representation theory and some well-known geometric constructions. As an application, we employ our results in the study of symplectic reflection algebras, completing work of Etingof and Ginzburg on when these algebras are finite over their centres, and providing a framework for the study of their representation theory in the latter case.

DEPENDENCE IN ALGEBRAS

The notions of independence and dimension as introduced by Cohn for universal algebras are quite significantly different from the standard ones. This paper compares some of these notions. In particular, necessary and sufficient conditions for spanning and generating to coincide are derived. Moreover, the paper exhibits a new class of algebras which have bases and well-defined dimensions. This class extends that of v*-algebras and is incomparable with the class of v**-algebras.

Algebras with operator and Campbell-Hausdorff formula

We introduce some new classes of algebras and establish within these algebras Campbell-Hausdorff-like formulae. We describe the application of these constructions to the problem of the connectivity of the Feynman graphs corresponding to Green functions in quantum fields theory.

Simple Algebras of Type (1,1) are Associative

In the classification of almost alternative algebras relative quasiequivalence by Albert two new classes of algebras of type (γ, δ) were introduced, namely those of type (1, 1) and ( —1, 0) (1, equations (34), (35), and Theorem 6). Since rings of type (1, 1) and ( — 1, 0) are anti-isomorphic it suffices to consider those of type (1, 1). They may be defined as rings satisfying1and2for all elements x, y, and z of the ring, where the associator (a, b, c) is given by (a, b, c) = (ab)c — a (bc).Actually the identity2'together with (1) implies (2) whenever the characteristic of the ring is different from 2. This may readily be verified by linearizing (2’). Consequently we may use (1) and (2’) as the defining relations for a ring of type (1, 1).

On commutative linear algebras in which division is always uniquely possible

1. We consider commutative linear algebras in 2n units, with coordinates in a general field F, such that n of the units define a sub-algebra forming a field F( J). The elements of the algebra may be exhibited compactly in the form A + BJ, where A and B range over F(J). As multiplication is not associative in general, A and B do not play the role of coordinates, so that the algebra is not binary in the usual significance of the term.t Nevertheless, by the use of this binary niotation, we may exhibit in a very luminous forml the multiplicatioln-tables of certain algebras in four and six unlits, given in an earlier paper. i Proof of the existence of the algebras and of the uniqtueness of division niow presents no difficulty. The form of the corresponding algebra in 2n units becomes obvious. After thus perfecting and extending known results, we attack the problem of the deternmination of all algebras with the prescribed properties. An extensive new class of algebras is obtained.