Algebras, hyperalgebras, nonassociative bialgebras and loops

Sabinin algebras are a broad generalization of Lie algebras that include Lie, Malcev and Bol algebras as very particular examples. We present a construction of a universal enveloping algebra for Sabinin algebras, and the corresponding Poincaré-Birkhoff-Witt Theorem. A nonassociative counterpart of Hopf algebras is also introduced and a version of the Milnor-Moore Theorem is proved. Loop algebras and universal enveloping algebras of Sabinin algebras are natural examples of these nonassociative Hopf algebras. Identities of loops move to identities of nonassociative Hopf algebras by a linearizing process. In this way, nonassociative algebras and Hopf algebras interlace smoothly.     

Affine cellular algebras

Graham and Lehrer have defined cellular algebras and developed a theory that allows in particular to classify simple representations of finite dimensional cellular algebras. Many classes of finite dimensional algebras, including various Hecke algebras and diagram algebras, have been shown to be cellular, and the theory due to Graham and Lehrer successfully has been applied to these algebras.We will extend the framework of cellular algebras to algebras that need not be finite dimensional over a field. Affine Hecke algebras of type A and infinite dimensional diagram algebras like the affine Temperley–Lieb algebras are shown to be examples of our definition. The isomorphism classes of simple representations of affine cellular algebras are shown to be parameterised by the complement of finitely many subvarieties in a finite disjoint union of affine varieties. In this way, representation theory of non-commutative algebras is linked with commutative algebra. Moreover, conditions on the cell chain are identified that force the algebra to have finite global cohomological dimension and its derived category to admit a stratification; these conditions are shown to be satisfied for the affine Hecke algebra of type A if the quantum parameter is not a root of the Poincaré polynomial.     

Equations des algèbres Lie triple qui sont des algèbres train

In this paper, we consider equations of Lie triple algebras that are train algebras. We obtain two different types of equations depending on assuming the existence of an idempotent or a pseudo-idempotent.In general Lie triple algebras are not power-associative. However we show that their train equation with an idempotent is similar to train equations of power-associative algebras that are train algebras and we prove that Lie triple algebras that are train algebras of rank $\le 4$ with an idempotent are Jordan algebras.Moreover, the set of non-trivial idempotents has the same expression in Peirce decomposition as that of $e$-stable power-associative algebras.We also prove that the algebra obtained by $2$-gametization process of a Lie triple algebra is a Lie triple one.     

The structure of AS-Gorenstein algebras

In this paper, we define a notion of AS-Gorenstein algebra for N-graded algebras, and show that symmetric AS-regular algebras of Gorenstein parameter 1 are exactly preprojective algebras of quasi-Fano algebras. This result can be compared with the fact that symmetric graded Frobenius algebras of Gorenstein parameter −1 are exactly trivial extensions of finite-dimensional algebras. The results of this paper suggest that there is a strong interaction between classification problems in noncommutative algebraic geometry and those in representation theory of finite-dimensional algebras.     

Finitistic dimension and Igusa–Todorov algebras

The notion of Igusa–Todorov algebras is introduced in connection with the (little) finitistic dimension conjecture, and the conjecture is proved for those algebras. Such algebras contain many known classes of algebras over which the finitistic dimension conjecture holds, e.g., algebras with the representation dimension at most 3, algebras with radical cube zero, monomial algebras and left serial algebras, etc. It is an open question whether all artin algebras are Igusa–Todorov. We provide some methods to construct many new classes of (2-)Igusa–Todorov algebras and thus obtain many algebras such that the finitistic dimension conjecture holds. In particular, we show that the class of 2-Igusa–Todorov algebras is closed under taking endomorphism algebras of projective modules. Hence, if all quasi-hereditary algebras are 2-Igusa–Todorov, then all artin algebras are 2-Igusa–Todorov by [V. Dlab, C.M. Ringel, Every semiprimary ring is the endomorphism ring of a projective module over a quasihereditary ring, Proc. Amer. Math. Soc. 107 (1) (1989) 1–5] and have finite finitistic dimension.     

Decomposability of free Łukasiewicz implication algebras

?ukasiewicz implication algebras are {→,1}-subreducts of Wajsberg algebras (MV-algebras). They are the algebraic counterpart of Super-?ukasiewicz Implicational logics investigated in Komori, Nogoya Math J 72:127–133, 1978. The aim of this paper is to study the direct decomposability of free ?ukasiewicz implication algebras. We show that freely generated algebras are directly indecomposable. We also study the direct decomposability in free algebras of all its proper subvarieties and show that infinitely freely generated algebras are indecomposable, while finitely free generated algebras can be only decomposed into a direct product of two factors, one of which is the two-element implication algebra.     

Coset relation algebras

A measurable relation algebra is a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). A large class of examples of such algebras, using systems of groups and coordinated systems of isomorphisms between quotients of the groups, has been constructed. This class of group relation algebras is not large enough to exhaust the class of all measurable relation algebras. In the present article, the class of examples of measurable relation algebras is considerably extended by adding one more ingredient to the mix: systems of cosets that are used to “shift” the operation of relative multiplication. It is shown that, under certain additional hypotheses on the system of cosets, each such coset relation algebra with a shifted operation of relative multiplication is an example of a measurable relation algebra. We also show that the class of coset relation algebras does contain examples that are not representable as set relation algebras. In later articles, it is shown that the class of coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic measurable relation algebra is essentially isomorphic to a coset relation algebra, and the class of group relation algebras is similarly adequate to the task of representing all measurable relation algebras in which the associated groups are finite and cyclic.     

Quasi-Hereditary Extension Algebras

The paper is a continuation of the authors' study of quasi-hereditary algebras whose Yoneda extension algebras (homological duals) are quasi-hereditary. The so-called standard Koszul quasi-hereditary algebras, presented in this paper, have the property that their extension algebras are always quasi-hereditary. In the natural setting of graded Koszul algebras, the converse also holds: if the extension algebra of a graded Koszul quasi-hereditary algebra is quasi-hereditary, then the algebra must be standard Koszul. This implies that the class of graded standard Koszul quasi-hereditary algebras is closed with respect to homological duality. Another immediate consequence is the fact that all algebras corresponding to the blocks of the category O are standard Koszul.     

一类特征单的n-李代数

利用结合代数与n-李代数的张量积构造了一类无限维特征单的n-李代数,且证明了除n=3的情形以外,这类特征单n-李代数的内导子代数是特征单李代数.     

ETALE DESCENT OF DERIVATIONS

We study étale descent of derivations of algebras with values in a module. The algebras under consideration are twisted forms of algebras over rings, and apply to all classes of algebras, notably associative and Lie algebras, such as the multiloop algebras that appear in the construction of extended affine Lie algebras. The main result is Theorem 2.7.     

格蕴涵代数与MV-代数

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Calabi–Yau algebras and weighted quiver polyhedra

Dimer models have been used in string theory to construct path algebras with relations that are 3-dimensional Calabi–Yau Algebras. These constructions result in algebras that share some specific properties: they are finitely generated modules over their centers and their representation spaces are toric varieties. In order to describe these algebras we introduce the notion of a toric order and show that all toric orders which are 3-dimensional Calabi–Yau algebras can be constructed from dimer models on a torus. Toric orders are examples of a much broader class of algebras: positively graded cancellation algebras. For these algebras the CY-3 condition implies the existence of a weighted quiver polyhedron, which is an extension of dimer models obtained by replacing the torus with any two-dimensional compact orientable orbifold.     

Stability of the Blok Theorem

W. Blok proved that varieties of boolean algebras with a single unary operator uniquely determined by their class of perfect algebras (i.e., duals of Kripke frames) are exactly those which are intersections of conjugate varieties of splitting algebras. The remaining ones share their class of perfect algebras with uncountably many other varieties. This theorem is known as the Blok dichotomy or the Blok alternative. We show that the Blok dichotomy holds when perfect algebras in the formulation are replaced by ω-complete algebras, atomic algebras with completely additive operators or algebras admitting residuals. We also generalize the Blok dichotomy for lattices of varieties of boolean algebras with finitely many unary operators. In addition, we answer a question posed by W. Dziobiak and show that classes of lattice-complete algebras or duals of Scott-Montague frames in a given variety are not determined by their subdirectly irreducible members. Received February 14, 2006; accepted in final form March 26, 2007.     

Quantized Weyl algebras at roots of unity

We classify the centers of the quantized Weyl algebras that are polynomial identity algebras and derive explicit formulas for the discriminants of these algebras over a general class of polynomial central subalgebras. Two different approaches to these formulas are given: one based on Poisson geometry and deformation theory, and the other using techniques from quantum cluster algebras. Furthermore, we classify the PI quantized Weyl algebras that are free over their centers and prove that their discriminants are locally dominating and effective. This is applied to solve the automorphism and isomorphism problems for this family of algebras and their tensor products.     

Reflection-closed varieties of multisorted algebras and minor identities

The notion of reflection is considered in the setting of multisorted algebras. The Galois connection induced by the satisfaction relation between multisorted algebras and minor identities provides a characterization of reflection-closed varieties: a variety of multisorted algebras is reflection-closed if and only if it is definable by minor identities. Minor-equational theories of multisorted algebras are described by explicit closure conditions. It is also observed that nontrivial varieties of multisorted algebras of a non-composable type are reflection-closed.     

Enumeration of algebras close to absolutely free algebras and binary trees

We study three classes of algebras: absolutely free algebras, free commutative non-associative, and free anti-commutative non-associative algebras. We study asymptotics of the growth for free algebras of these classes and for their subvarieties as well. Mainly, we study finitely generated algebras, also the codimension growth for varieties in theses classes is studied. For these purposes we use ordinary generating functions as well as exponential generating functions. The following subvarieties are studied in these classes: solvable, completely solvable, right-nilpotent, and completely right-nilpotent subvarieties. The obtained results are equivalent to an enumeration of binary labeled and unlabeled rooted trees that do not contain some forbidden subtrees. We enumerate these trees using generating functions. For solvable and right-nilpotent algebras the generating functions are algebraic. For completely solvable and completely right-nilpotent algebras the respective functions are rational. It is known that these three varieties of algebras satisfy Schreier's property, i.e., subalgebras of free algebras are free. For free groups, there is Schreier's formula for the rank of a subgroup of a free group. We find analogues of this formula for these varieties. They are written in terms of series. As an application, we study invariants of finite groups acting on absolutely free algebras.     

Squarefree Vertex Cover Algebras

In this paper we introduce squarefree vertex cover algebras and exhibit a duality for them. We study the question when these algebras are standard graded and when these algebras coincide with the ordinary vertex cover algebras. It is shown that this is the case for simplicial complexes corresponding to principal Borel sets. Moreover, the generators of these algebras are explicitly described.     

Cylindric algebras and finite polyadic algebras

In this survey paper the short history of cylindric and finitary polyadic algebras (term-definitionally equivalent to quasi-polyadic algebras) is sketched, and the two concepts are compared. Roughly speaking, finitary polyadic algebras constitute a subclass of cylindric algebras that include a transposition operator being strong enough. We discuss the following question: should the definition of cylindric algebras include a transposition operator? Results confirm that the existence of a transposition operator ensures representability (by relativised set algebras). The different variants of cylindric algebras including a transposition operator play an important role in the theory of cylindric-like algebras.     

Weak effect algebras

Weak effect algebras are based on a commutative, associative and cancellative partial addition; they are moreover endowed with a partial order which is compatible with the addition, but in general not determined by it. Every BL-algebra, i.e. the Lindenbaum algebra of a theory of Basic Logic, gives rise to a weak effect algebra; to this end, the monoidal operation is restricted to a partial cancellative operation. We examine in this paper BL-effect algebras, a subclass of the weak effect algebras which properly contains all weak effect algebras arising from BL-algebras. We describe the structure of BL-effect algebras in detail. We thus generalise the well-known structure theory of BL-algebras. Namely, we show that BL-effect algebras are subdirect products of linearly ordered ones and that linearly ordered BL-effect algebras are ordinal sums of generalised effect algebras. The latter are representable by means of linearly ordered groups. This research was partially supported by the German Science Foundation (DFG) as part of the Collaborative Research Center “Computational Intelligence” (SFB 531).