Closedness Properties of Internal Relations III: Pointed Protomodular Categories

A pointed variety of universal algebras is protomodular in the sense of D. Bourn, if and only if it is classically ideal determined in the sense of A. Ursini (this result is due to D. Bourn and G. Janelidze). We prove a characterization theorem for pointed protomodular categories, which is a (pointed) categorical version of Ursini’s characterization theorem for classically ideal determined varieties, involving classically 0-regular algebras. A suitable simplification of the property of a pair of relations, which is used to define a classically 0-regular algebra, yields a new closedness property of a single binary relation – we show that a finitely complete pointed category is protomodular if and only if every binary internal relation R→A ² in it has this closedness property. Partially supported by South African National Research Foundation, and Georgian National Science Foundation (GNSF/ST06/3-004).     

On Morita Equivalence for Simple Generalized Weyl Algebras

We give a necessary condition for Morita equivalence of simple Generalized Weyl algebras of classical type. We propose a reformulation of Hodges’ result, which describes Morita equivalences in case the polynomial defining the Generalized Weyl algebra has degree 2, in terms of isomorphisms of quantum tori, inspired by similar considerations in noncommutative differential geometry. We study how far this link can be generalized for n ≥ 3.     

Representations of Khovanov–Lauda–Rouquier algebras and combinatorics of Lyndon words

We construct irreducible representations of affine Khovanov–Lauda–Rouquier algebras of arbitrary finite type. The irreducible representations arise as simple heads of appropriate induced modules, and thus our construction is similar to that of Bernstein and Zelevinsky for affine Hecke algebras of type A. The highest weights of irreducible modules are given by the so-called good words, and the highest weights of the ‘cuspidal modules’ are given by the good Lyndon words. In a sense, this has been predicted by Leclerc.     

Weyl’s Theorems for Some Classes of Operators

We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator T_μ of a group algebra L¹(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras.     

Algebras with small homological dimensions

We consider the algebras Λ which satisfy the property that for each indecomposable module X, either its projective dimension pd_Λ X is at most one or its injective dimension id_Λ X is at most one. This clearly generalizes the so-called quasitilted algebras introduced by Happel–Reiten–Smal?. We show that some of the niciest features for this latter class of algebras can be generalized to the case we are considering, in particular the existence of a trisection in its module category. Received: 26 August 1998     

The equational complexity of Lyndon’s algebra

The equational complexity of Lyndon’s nonfinitely based 7-element algebra lies between n − 4 and 2n + 1. This result is based on a new algebraic proof that Lyndon’s algebra is not finitely based. We prove that Lyndon’s algebra is inherently nonfinitely based relative to a rather rich class of algebras. We also show that the variety generated by Lyndon’s algebra contains subdirectly irreducible algebras of all cardinalities except 0, 1, and 4.     

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.     

The number of simple modules of the Hecke algebras of type G(r,1,n)

This paper classifies the simple modules of the cyclotomic Hecke algebras of type G(r,1,n) and the affine Hecke algebras of type A in arbitrary characteristic. We do this by first showing that the simple modules of the cyclotomic Hecke algebras are indexed by the set of “Kleshchev multipartitions”. Received July 24, 1998; in final form February 8, 1999     

Quantum Symmetric Pairs and the Reflection Equation

It is shown that central elements in G. Letzter’s quantum group analogs of symmetric pairs lead to solutions of the reflection equation. This clarifies the relation between Letzter’s approach to quantum symmetric pairs and the approach taken by M. Noumi, T. Sugitani, and M. Dijkhuizen. We develop general tools to show that a Noumi-Sugitani-Dijkhuizen type construction of quantum symmetric pairs can be performed preserving spherical representations from the classical situation. These tools apply to the symmetric pair FII and to all symmetric pairs which correspond to an automorphism of the underlying Dynkin diagram. Hence Noumi-Sugitani-Dijkhuizen type constructions with desirable properties are possible for various symmetric pairs for exceptional Lie algebras. Presented by Susan Montgomery.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-rigidity in von Neumann algebras

We introduce the notion of L ²-rigidity for von Neumann algebras, a generalization of property (T) which can be viewed as an analogue for the vanishing of 1-cohomology into the left regular representation of a group. We show that L ²-rigidity passes to normalizers and is satisfied by nonamenable II₁ factors which are non-prime, have property Γ, or are weakly rigid. As a consequence we obtain that if M is a free product of diffuse von Neumann algebras, or if M=LΓ where Γ is a finitely generated group with β₁ ⁽²⁾(Γ)>0, then any nonamenable regular subfactor of M is prime and does not have properties Γ or (T). In particular this gives a new approach for showing solidity for a free group factor thus recovering a well known recent result of N. Ozawa.     

Some applications of higher commutators in Mal’cev algebras

We establish several properties of Bulatov’s higher commutator operations in congruence permutable varieties. We use higher commutators to prove that for a finite nilpotent algebra of finite type that is a product of algebras of prime power order and generates a congruence modular variety, affine completeness is a decidable property. Moreover, we show that in such algebras, we can check in polynomial time whether two given polynomial terms induce the same function.     

Von Neumann Regular Rings Satisfying Weak Comparability

The notion of weak comparability was first introduced by K.C. O’Meara, to prove that directly finite simple regular rings satisfying weak comparability must be unit-regular. In this paper, we shall treat (non-necessarily simple) regular rings satisfying weak comparability and give some interesting results. We first show that directly finite regular rings satisfying weak comparability are stably finite. Using the result above, we investigate the strict cancellation property and the strict unperforation property for regular rings satisfying weak comparability, and we show that these rings have the strict unperforation property, which means that nA ≺ nB implies A ≺ B for any finitely generated projective modules A, B and any positive integer n.      

Generalized Poisson brackets and lie algebras of type H in characteristic 0

The Lie algebra of Cartan type H which occurs as a subalgebra of the Lie algebra of derivations of the polynomial algebra was generalized by the first author to a class which included a subalgebra of the derivations of the Laurent polynomials . We show in this paper that these generalizations of Cartan type H algebras are isomorphic to certain generalizations of the classical algebra of Poisson brackets, and that it can be generalized further. In turn, these algebras can be recast in a form that is an adaption of a class of Lie algebras of characteristic p that was defined in 1958 be R. Block. A further generalization of these algebras is the main topic of this paper. We show when these algebras are simple, find their derivations, and determine all possible isomorphisms between two of these algebras. Received December 20, 1996; in final form September 15, 1997     

On Constantive Simple and Order-Primal Algebras

A finite algebra is said to be order-primal if its clone of all term operations is the set of all operations defined on A which preserve a given partial order ≤ on A. In this paper we study algebraic properties of order-primal algebras for connected ordered sets (A; ≤). Such order-primal algebras are constantive, simple and have no non-identical automorphisms. We show that in this case cannot have only unary fundamental operations or only one at least binary fundamental operation. We prove several properties of the varieties and the quasi-varieties generated by constantive and simple algebras and apply these properties to order-primal algebras. Further, we use the properties of order-primal algebras to formulate new primality criteria for finite algebras.* Research supported by the Hungarian research grant No. TO34137 and by the János Bolyai grant.** Research supported by the Thailand Research Fund.     

Projective resolutions and Yoneda algebras for algebras of dihedral type: The family $$D(3\mathcal{Q})$$

This paper provides a method for the computation of Yoneda algebras for algebras of dihedral type. The Yoneda algebras for one infinite family of algebras of dihedral type (the family ) in K. Erdmann’s notation) are computed. The minimal projective resolutions of simple modules were calculated by an original computer program implemented by one of the authors in the C++ language. The algorithm of the program is based on a diagrammatic method presented in this paper and inspired by that of D. Benson and J. Carlson. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 10, No. 4, pp. 65–89, 2004.     

The representation dimension of domestic weakly symmetric algebras

Auslander’s representation dimension measures how far a finite dimensional algebra is away from being of finite representation type. In [1], M. Auslander proved that a finite dimensional algebra A is of finite representation type if and only if the representation dimension of A is at most 2. Recently, R. Rouquier proved that there are finite dimensional algebras of an arbitrarily large finite representation dimension. One of the exciting open problems is to show that all finite dimensional algebras of tame representation type have representation dimension at most 3. We prove that this is true for all domestic weakly symmetric algebras over algebraically closed fields having simply connected Galois coverings.     

Completions of GBL-algebras: negative results

We provide a simple sufficient criterion to show that a given variety of GBL-algebras does not admit (local) completions. As corollaries, we obtain that no variety of GBL-algebras containing Chang’s chain, no nontrivial variety of ℓ-groups, nor the variety of product algebras admit completions. The first result strengthens a result of Gehrke and Priestley. Received August 10, 2006; accepted in final form March 8, 2007.     

Join-semilattices whose sections are residuated PO-monoids

We generalize the concept of an integral residuated lattice to join-semilattices with an upper bound where every principal order-filter (section) is a residuated semilattice; such a structure is called a sectionally residuated semilattice. Natural examples come from propositional logic. For instance, implication algebras (also known as Tarski algebras), which are the algebraic models of the implication fragment of the classical logic, are sectionally residuated semilattices such that every section is even a Boolean algebra. A similar situation rises in case of the Lukasiewicz multiple-valued logic where sections are bounded commutative BCK-algebras, hence MV-algebras. Likewise, every integral residuated (semi)lattice is sectionally residuated in a natural way. We show that sectionally residuated semilattices can be axiomatized as algebras (A, r, →, ⇝, 1) of type 〈3, 2, 2, 0〉 where (A, →, ⇝, 1) is a {→, ⇝, 1}-subreduct of an integral residuated lattice. We prove that every sectionally residuated lattice can be isomorphically embedded into a residuated lattice in which the ternary operation r is given by r(x, y, z) = (x · y) ∨ z. Finally, we describe mutual connections between involutive sectionally residuated semilattices and certain biresiduation algebras. This work was supported by the Czech Government via the project MSM6198959214.