The Natural Quiver of an Artinian Algebra

The motivation of this paper is to study the natural quiver of an artinian algebra, a new kind of quivers, as a tool independing upon the associated basic algebra. In Li (J Aust Math Soc 83:385–416, 2007), the notion of the natural quiver of an artinian algebra was introduced and then was used to generalize the Gabriel theorem for non-basic artinian algebras splitting over radicals and non-basic finite dimensional algebras with 2-nilpotent radicals via pseudo path algebras and generalized path algebras respectively. In this paper, firstly we consider the relationship between the natural quiver and the ordinary quiver of a finite dimensional algebra. Secondly, the generalized Gabriel theorem is obtained for radical-graded artinian algebras. Moreover, Gabriel-type algebras are introduced to outline those artinian algebras satisfying the generalized Gabriel theorem here and in Li (J Aust Math Soc 83:385–416, 2007). For such algebras, the uniqueness of the related generalized path algebra and quiver holds up to isomorphism in the case when the ideal is admissible. For an artinian algebra, there are two basic algebras, the first is that associated to the algebra itself; the second is that associated to the correspondent generalized path algebra. In the final part, it is shown that for a Gabriel-type artinian algebra, the first basic algebra is a quotient of the second basic algebra. In the end, we give an example of a skew group algebra in which the relation between the natural quiver and the ordinary quiver is discussed.     

Κ1 Group of Finite Dimensional Path Algebra

In this paper, by calculating the commutator subgroup of the unit group of finite path algebra κΔ and the unit group abelianized, we explicitly characterize the Κ₁ group of finite dimensional path algebra over an arbitrary field. Received November 2, 1998, Accepted May 7, 1999     

Embedding Distributions in Algebras of Generalized Functions with Singularities

 It is a known open question whether in Rosinger’s nowhere dense quotient algebra distributions can be embedded, ensuring consistency with partial derivatives. In this paper, we construct algebras of generalized functions with nowhere dense singularities similar to Rosinger’s algebra in which such an embedding of distributions is possible. Moreover, in some of the algebras, the embedding preserves the products of smooth functions. Further, we indicate that most of the properties of Rosinger’s algebra, in particular concerning the solvability of nonlinear PDEs, remain valid in the new algebras. Summarizing, we give a positive answer to the above-mentioned open question, if one allows a modification in the definition of the algebra that does not affect its basic properties.     

Embedding Distributions in Algebras of Generalized Functions with Singularities

 It is a known open question whether in Rosinger’s nowhere dense quotient algebra distributions can be embedded, ensuring consistency with partial derivatives. In this paper, we construct algebras of generalized functions with nowhere dense singularities similar to Rosinger’s algebra in which such an embedding of distributions is possible. Moreover, in some of the algebras, the embedding preserves the products of smooth functions. Further, we indicate that most of the properties of Rosinger’s algebra, in particular concerning the solvability of nonlinear PDEs, remain valid in the new algebras. Summarizing, we give a positive answer to the above-mentioned open question, if one allows a modification in the definition of the algebra that does not affect its basic properties. Received September 20, 2001; in final form April 11, 2002     

Reductions in Computational Complexity Using Clifford Algebras

A number of combinatorial problems are treated using properties of abelian null-square-generated and idempotent-generated subalgebras of Clifford algebras. For example, the problem of deciding whether or not a graph contains a Hamiltonian cycle is known to be NP-complete. By considering entries of Λ^k, where Λ is an appropriate nilpotent adjacency matrix, the k-cycles in any finite graph are recovered. Within the algebra context (i.e., considering the number of multiplications performed within the algebra), these problems are reduced to matrix multiplication, which is in complexity class P. The Hamiltonian cycle problem is one of many problems moved from classes NP-complete and #P-complete to class P in this context. Other problems considered include the set covering problem, counting the edge-disjoint cycle decompositions of a finite graph, computing the permanent of an arbitrary matrix, computing the girth and circumference of a graph, and finding the longest path in a graph.     

Notes on piecewise-Koszul algebras

The relationships between piecewise-Koszul algebras and other “Koszul-type” algebras are discussed. The Yoneda-Ext algebra and the dual algebra of a piecewise-Koszul algebra are studied, and a sufficient condition for the dual algebra A ^! to be piecewise-Koszul is given. Finally, by studying the trivial extension algebras of the path algebras of Dynkin quivers in bipartite orientation, we give explicit constructions for piecewise-Koszul algebras with arbitrary “period” and piecewise-Koszul algebras with arbitrary “jump-degree”.     

Structures and Representations of Generalized Path Algebras

It is shown that an algebra Λ can be lifted with nilpotent Jacobson radical r = r(Λ) and has a generalized matrix unit {e _ii } _I with each ē _ii in the center of if Λ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary, Λ is a finite algebra with non-zero unity element over a perfect field k (e.g., a field with characteristic zero or a finite field) if Λ is isomorphic to a generalized path algebra k (D, Ω, ρ) of finite directed graph with weak relations and dim < ∞; Λ is a generalized elementary algebra which can be lifted with nilpotent Jacobson radical and has a complete set of pairwise orthogonal idempotents if Λ is isomorphic to a path algebra with relations. Presented by Idun Reiten.     

Non-embeddable simple relation algebras

This paper presents solutions or partial solutions for several problems in the theory of relation algebras. In a simple relation algebra an element x satisfying the condition (a) must be an atom of . It follows that x must also be an atom in every simple extension of . Andréka, Jónsson and Németi [1, Problem 4] (see [12, Problem P5]) asked whether the converse holds: if x is an atom in every simple extension of a simple relation algebra, must it satisfy (a)? We show that the answer is “no”.? The only known examples of simple relation algebras without simple proper extensions are the algebras of all binary relations on a finite set. Jónsson proposed finding all finite simple relation algebras without simple proper extensions [12, Problem P6]. We show how to construct many new examples of finite simple relation algebras that have no simple proper extensions, thus providing a partial answer for this second problem. These algebras are also integral and nonrepresentable.? Andréka, Jónsson, Németi [1, Problem 2] (see [12, Problem P7]) asked whether there is a countable simple relation algebra that cannot be embedded in a one-generated relation algebra. The answer is “yes”. Givant [3, Problem 9] asked whether there is some k such that every finitely generated simple relation algebra can be embedded in a k-generated simple relation algebra. The answer is “no”. Received November 27, 1996; accepted in final form July 3, 1997.     

Noncommutative classical and quantum mechanics for quadratic Lagrangians (Hamiltonians)

Classical and quantum mechanics based on an extended Heisenberg algebra with additional canonical commutation relations for position and momentum coordinates are considered. In this approach additional noncommutativity is removed from the algebra by a linear transformation of coordinates and transferred to the Hamiltonian (Lagrangian). This linear transformation does not change the quadratic form of the Hamiltonian (Lagrangian), and Feynman’s path integral preserves its exact expression for quadratic models. The compact general formalism presented here can be easily illustrated in any particular quadratic case. As an important result of phenomenological interest, we give the path integral for a charged particle in the noncommutative plane with a perpendicular magnetic field. We also present an effective Planck constant ħ _eff which depends on additional noncommutativity.     

The Loewy Length of the Descent Algebra of Type <Emphasis Type="Italic">D</Emphasis>

In this article the Loewy length of the descent algebra of D _2m + 1 is shown to be m + 2, for m ≥ 2, by providing an upper bound that agrees with the lower bound in Bonnafé and Pfeiffer (2006). The bound is obtained by showing that the length of the longest path in the quiver of the descent algebra of D _2m + 1 is at most m + 1. To achieve this bound, the geometric approach to the descent algebra is used, in which the descent algebra of a finite Coxeter group W is identified with an algebra associated to the reflection arrangement of W.     

From Lie algebras of vector fields to algebraic group actions

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → Der_K(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.     

Cluster-tilted algebras as trivial extensions

Given a finite-dimensional algebra C (over an algebraicallyclosed field) of global dimension at most two, we define itsrelation-extension algebra to be the trivial extension C Ext(DC,C) of C by the C–C-bimoduleExt(DC,C). We give a constructionfor the quiver of the relation-extension algebra in case thequiver of C has no oriented cycles. Our main result says thatan algebra is cluster-tilted if and only if there exists a tilted algebra C such that is isomorphic to the relation-extensionof C.     

Current algebra and the nonlinear σ-model

We present the current algebra of a particular form in the nonlinear σ-model. The algebra has a non-Abelian form with field-dependent structure functions. We comment on the connection of the model with noncommutative space. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 405–412, June, 2007.     

Yang-Baxter algebra and generation of quantum integrable models

We discover an operator-deformed quantum algebra using the quantum Yang-Baxter equation with the trigonometric R-matrix. This novel Hopf algebra together with its q→1 limit seems the most general Yang-Baxter algebra underlying quantum integrable systems. We identify three different directions for applying this algebra in integrable systems depending on different sets of values of the deforming operators. Fixed values on the whole lattice yield subalgebras linked to standard quantum integrable models, and the associated Lax operators generate and classify them in a unified way. Variable values yield a new series of quantum integrable inhomogeneous models. Fixed but different values at different lattice sites can produce a novel class of integrable hybrid models including integrable matter-radiation models and quantum field models with defects, in particular, a new quantum integrable sine-Gordon model with defect. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 470–485, June, 2007.     

Complexity of Boolean algebras and their Scott rank

We deal with problems associated with Scott ranks of Boolean algebras. The Scott rank can be treated as some measure of complexity of an algebraic system. Our aim is to propound and justify the procedure which, given any countable Boolean algebra, will allow us to construct a Boolean algebra of a small Scott rank that has the same natural algebraic complexity as has the initial algebra. In particular, we show that the Scott rank does not always serve as a good measure of complexity for the class of Boolean algebras. We also study into the question as to whether or not a Boolean algebra of a big Scott rank can be decomposed into direct summands with intermediate ranks. Examples are furnished in which Boolean algebras have an arbitrarily big Scott rank such that direct summands in them either have a same rank or a fixed small one, and summands of intermediate ranks are altogether missing. This series of examples indicates, in particular, that there may be no nontrivial mutual evaluations for the Scott and Frechet ranks on a class of countable Boolean algebras. Supported by RFFR grant No. 99-01-00485, by a grant for Young Scientists from SO RAN, 1997, and by the Federal Research Program (FRP) “Integration”. Translated fromAlgebra i Logika, Vol. 38, No. 6, pp. 643–666, November–December, 1999.     

Weights on polytopes

Theg-theorem describes the possible face-vectors of a simple polytopeP. Much of the author's proof of the necessity of its conditions, while working within the polytope algebra π, in fact only used the spaces of weights onP. Even though this proof was conceptually easier than the original, which employed techniques from algebraic geometry, nevertheless the properties of π which are needed still require some effort to establish, despite a recent simpler approach to π itself. In the earlier paper, doubt was expressed about whether two basic results could be proved directly for weights; later, it appeared that there might also be a possible problem concerning an alternative definition of the product of certain weights. In this paper these questions are settled, in the context of developing an independent theory of an algebra Ω of weights on polytopes. Since the construction of Ω is more approachable than that of π, a yet easier proof of theg-theorem is now available.     

An example of a commutative basic algebra which is not an MV-algebra

Many algebras arising in logic have a lattice structure with intervals being equipped with antitone involutions. It has been proved in [CHK1] that these lattices are in a one-to-one correspondence with so-called basic algebras. In the recent papers [BOTUR, M.—HALAŠ, R.: Finite commutative basic algebras are MV-algebras, J. Mult.-Valued Logic Soft Comput. (To appear)]. and [BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] we have proved that every finite commutative basic algebra is an MV-algebra, and that every complete commutative basic algebra is a subdirect product of chains. The paper solves in negative the open question posed in [BOTUR, M.—HALAŠ, R.: Complete commutative basic algebras, Order 24 (2007), 89–105] whether every commutative basic algebra on the interval [0, 1] of the reals has to be an MV-algebra.     

Brauer algebras of simply laced type

The diagram algebra introduced by Brauer that describes the centralizer algebra of the n-fold tensor product of the natural representation of an orthogonal Lie group has a presentation by generators and relations that only depends on the path graph A _{n − 1} on n − 1 nodes. Here we describe an algebra depending on an arbitrary graph Q, called the Brauer algebra of type Q, and study its structure in the cases where Q is a Coxeter graph of simply laced spherical type (so its connected components are of type A _{n − 1}, D _n , E₆, E₇, E₈). We find its irreducible representations and its dimension, and show that the algebra is cellular. The algebra is generically semisimple and contains the group algebra of the Coxeter group of type Q as a subalgebra. It is a ring homomorphic image of the Birman-Murakami-Wenzl algebra of type Q; this fact will be used in later work determining the structure of the Birman-Murakami-Wenzl algebras of simply laced spherical type.     

Prime étale groupoid algebras with applications to inverse semigroup and Leavitt path algebras

In this paper we give some sufficient and some necessary conditions for an étale groupoid algebra to be a prime ring. As an application we recover the known primeness results for inverse semigroup algebras and Leavitt path algebras. It turns out that primeness of the algebra is connected with the dynamical property of topological transitivity of the groupoid. We obtain analogous results for semiprimeness.     

The homomorphisms between the Dickson–Mùi algebras as modules over the Steenrod algebra

The Dickson–Mùi algebra consists of all invariants in the mod p cohomology of an elementary abelian p-group under the general linear group. It is a module over the Steenrod algebra, A{\mathcal {A}} . We determine explicitly all the A{\mathcal {A}} -module homomorphisms between the (reduced) Dickson–Mùi algebras and all the A{\mathcal {A}} -module automorphisms of the (reduced) Dickson–Mùi algebras. The algebra of all A{\mathcal {A}} -module endomorphisms of the (reduced) Dickson–Mùi algebra is claimed to be isomorphic to a quotient of the polynomial algebra on one indeterminate. We prove that the reduced Dickson–Mùi algebra is atomic in the meaning that if an A{\mathcal {A}} -module endomorphism of the algebra is non-zero on the least positive degree generator, then it is an automorphism. This particularly shows that the reduced Dickson–Mùi algebra is an indecomposable A{\mathcal {A}} -module. The similar results also hold for the odd characteristic Dickson algebras. In particular, the odd characteristic reduced Dickson algebra is atomic and therefore indecomposable as a module over the Steenrod algebra.