Molecules of Post algebras

Following A. Abian, we define amolecule of a partially ordered set to be any nonzero elementm with the property that any two nonzero elements less than or equal tom have a nonzero lower bound. In a Boolean algebra an element is a molecule iff it is an atom; however, in an arbitrary distributive lattice the notion of a molecule is too general to permit explicit characterization. In this note we examine the molecules in a coproduct of two distributive lattices and use our results to show that molecules play the same role in Post algebras as atoms do in Boolean algebras.Presented by R. S. Pierce.I would like to thank the referee for helpful comments that resulted in simplifying a number of proofs in this paper.     

Nijenhuis algebras, NS algebras, and N-dendriform algebras

In this paper, we study (associative) Nijenhuis algebras, with emphasis on the relationship between the category of Nijenhuis algebras and the categories of NS algebras and related algebras. This is in analogy to the well-known theory of the adjoint functor from the category of Lie algebras to that of associative algebras, and the more recent results on the adjoint functor from the categories of dendriform and tridendriform algebras to that of Rota-Baxter algebras. We first give an explicit construction of free Nijenhuis algebras and then apply it to obtain the universal enveloping Nijenhuis algebra of an NS algebra. We further apply the construction to determine the binary quadratic nonsymmetric algebra, called the N-dendriform algebra, that is compatible with the Nijenhuis algebra. As it turns out, the N-dendriform algebra has more relations than the NS algebra.     

Constructing tilted algebras from cluster-tilted algebras

Any cluster-tilted algebra is the relation extension of a tilted algebra. Given the distribution of a cluster-tilting object in the Auslander–Reiten quiver of the cluster category, we present a method to construct all tilted algebras whose relation extension is the endomorphism ring of this cluster-tilting object.     

Deformed virasoro algebras andw-algebras from elliptic algebras

Deformedw _n -algebras in the classical case (i.e., Poisson structures) and in the quantum case (exchange algebras) are constructed from the elliptic algebras . In memory of M. Saveliev When I think about Misha, and this has been happening rather frequently these recent months, it is his enthusiasm and his communicative energy that first come to mind. Something very important to Misha was friendship. We scientists have the priviledge to possess friends, real friends, among our colleagues, and Misha was a friend. Something else was fundamental for Misha, his family, i.e., his parents, his wife, and his daughter. As Professor Manin said, summarizing Misha’s behavior very simply and elegantly, Misha was “a good son, a good husband, a good father.” Misha’s parents died about three years ago, and I remember how he was sad when I met him in Georgia in September 1996 and he informed me about his father’s and mother’s deaths a few months before. After that, he concentrated all his forces to protect his wife and daughter, the two Svetlanas, as he used to say! He was so proud of his gifted daughter, a good musician, a good painter, and a brilliant student. I wish to present them my most respectful thoughts. Misha was a great expert in integrable systems. He was deeply interested in developing models, which he did so nicely in the framework of Toda theories. I learned much from him about Toda theories and also continuous Lie algebras. Misha was highly convinced of the importance of symmetries and was especially interested in the use of infinite symmetries, the W-algebras among them. The landscape of continuous symmetries, and particularly of infinite-dimensional symmetries, is becoming richer and richer these years. I dare to think that Misha would have been interested in the modest seminar I now present. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 2, pp. 311–322, May, 2000.     

Cogroups in algebras over an operad are free algebras

Let be an operad defined over a field of characteristic zero. Let R be a cogroup in the category of complete -algebras. In this article, we show that R is necessarily the completion of a free -algebra. We also handle the case of cogroups in connected graded algebras over an operad, and the case of groups in connected graded coalgebras over an operad. Received: August 26, 1996 and final version, February 4, 1998     

Octonion algebras obtained from associative algebras with involution

A natural octonion algebra structure on the symmetric elements of trace 0 of central simple associative algebras of degree 3 with involution of the second kind is obtained.

     

Modified Ringel-Hall algebras,naive lattice algebras and lattice algebras

For a given hereditary abelian category satisfying some finiteness conditions, in certain twisted cases it is shown that the modified Ringel-Hall algebra is isomorphic to the naive lattice algebra and there exists an epimorphism from the modified Ringel-Hall algebra to the lattice algebra. Furthermore, the kernel of this epimorphism is described explicitly. Finally, we show that the naive lattice algebra is invariant under the derived equivalences of hereditary abelian categories.     

Skew Boolean algebras derived from generalized Boolean algebras

Every skew Boolean algebra S has a maximal generalized Boolean algebra image given by S/ where is the Green’s relation defined initially on semigroups. In this paper we study skew Boolean algebras constructed from generalized Boolean algebras B by a twisted product construction for which . In particular we study the congruence lattice of with an eye to viewing as a minimal skew Boolean cover of B. This construction is the object part of a functor from the category GB of generalized Boolean algebras to the category LSB of left-handed skew Boolean algebras. Thus we also look at its left adjoint functor . This paper was written while the second author was a Visiting Professor in the Department of Education at the University of Cagliari. The facilities and assistance provided by the University and by the Department are gratefully acknowledged.     

Realizations of AF-algebras as graph algebras, Exel-Laca algebras, and ultragraph algebras

We give various necessary and sufficient conditions for an AF-algebra to be isomorphic to a graph C^∗-algebra, an Exel-Laca algebra, and an ultragraph C^∗-algebra. We also explore consequences of these results. In particular, we show that all stable AF-algebras are both graph C^∗-algebras and Exel-Laca algebras, and that all simple AF-algebras are either graph C^∗-algebras or Exel-Laca algebras. In addition, we obtain a characterization of AF-algebras that are isomorphic to the C^∗-algebra of a row-finite graph with no sinks.     

Schur algebras of Brauer algebras,II

A classical problem of invariant theory and of Lie theory is to determine endomorphism rings of representations of classical groups, for instance of tensor powers of the natural module (Schur–Weyl duality) or of full direct sums of tensor products of exterior powers (Ringel duality). In this article, the endomorphism rings of full direct sums of tensor products of symmetric powers over symplectic and orthogonal groups are determined. These are shown to be isomorphic to Schur algebras of Brauer algebras as defined in Henke and Koenig (Math Z 272(3–4):729–759, 2012). This implies structural properties of the endomorphism rings, such as double centraliser properties, quasi-hereditary, and a universal property, as well as a classification of simple modules.     

Cellular algebras arising from Hecke algebras of type H_n

We study a finite-dimensional quotient of the Hecke algebra of type for general n, using a calculus of diagrams. This provides a basis of monomials in a certain set of generators. Using this, we prove a conjecture of C.K. Fan about the semisimplicity of the quotient algebra. We also discuss the cellular structure of the algebra, with certain restrictions on the ground ring. Received February 24, 1997; in final form May 9, 1997     

Construction of vertex operator algebras from commutative associative algebras

Given a commutative associative algebra A with an associative form (’), we construct a vertex operator algebra V with the weight two space V₂;? A If in addition the form (’) is nondegenerate, we show that there is a simple vertex operator algebra with V₂;? A We also show that if A is semisimple, then the vertex operator algebra constructed is the tensor products of a certain number of Virasoro vertex operator algebras.     

Conservative algebras of 2-dimensional algebras,II

In 1990 Kantor defined the conservative algebra W(n) of all algebras (i.e. bilinear maps) on the n-dimensional vector space. If n>1, then the algebra W(n) does not belong to any well-known class of algebras (such as associative, Lie, Jordan, or Leibniz algebras). We describe automorphisms, one-sided ideals, and idempotents of W(2). Also similar problems are solved for the algebra W₂ of all commutative algebras on the 2-dimensional vector space and for the algebra S₂ of all commutative algebras with trace zero multiplication on the 2-dimensional vector space.     

Hall algebras,hereditary algebras and quantum groups

4-VIII-1994 & 4-X-1994