Rank varieties for a class of finite-dimensional local algebras

We develop a rank variety for finite-dimensional modules over a certain class of finite-dimensional local k-algebras, . Included in this class are the truncated polynomial algebras , with k an algebraically closed field and arbitrary. We prove that these varieties characterise projectivity of modules (Dade’s lemma) and examine the implications for the tree class of the stable Auslander-Reiten quiver. We also extend our rank varieties to infinitely generated modules and verify Dade’s lemma in this context.     

Minimal varieties of algebras of exponential growth

The exponent of a variety of algebras over a field of characteristic zero has been recently proved to be an integer. Through this scale we can now classify all minimal varieties of given exponent and of finite basic rank. As a consequence, we describe the corresponding T-ideals of the free algebra and we compute the asymptotics of the related codimension sequences, verifying in this setting some known conjectures. We also show that the number of these minimal varieties is finite for any given exponent. We finally point out some relations between the exponent of a variety and the Gelfand-Kirillov dimension of the corresponding relatively free algebras of finite rank.     

Cyclic homology of truncated quiver algebras

We determine the cyclic homology of truncated quiver algebras over a general commutative unital ring. Thus we improve and generalise the results in [8] and [5], where this problem has been studied for algebras over a field. We use the spectral sequence of a double complex that is obtained from minimal projective bimodule resolution and comparison morphisms.     

Weighted locally gentle quivers and Cartan matrices

We introduce and study the class of weighted locally gentle quivers. This naturally extends the class of gentle quivers and gentle algebras, which have been intensively studied in the representation theory of finite-dimensional algebras, to a wider class of potentially infinite-dimensional algebras. Weights on the arrows of these quivers lead to gradings on the corresponding algebras. For natural grading by path lengths, any locally gentle algebra is Koszul. The class of locally gentle algebras consists of the gentle algebras together with their Koszul duals.Our main result is a general combinatorial formula for the determinant of the weighted Cartan matrix of a weighted locally gentle quiver. We show that this weighted Cartan determinant is a rational function which is completely determined by the combinatorics of the quiver-more precisely by the number and the weight of certain oriented cycles.     

On primitive ideals

We extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the Irreducibility theorem for associated varieties and Duflo theorem on primitive ideals, to much wider classes of algebras. Our general version of the Irreducibility Theorem says that if A is a positively filtered associative algebra such that gr A is a commutative Poisson algebra with finitely many symplectic leaves, then the associated variety of any primitive ideal in A is the closure of a single connected symplectic leaf. Our general version of the Duflo theorem says that if A is an algebra with a triangular structure, see § 2, then any primitive ideal in A is the annihilator of a simple highest weight module. Applications to symplectic reflection algebras and Cherednik algebras are discussed.     

Line-bundle-valued ternary quadratic forms over schemes

We study degenerations of rank 3 quadratic forms and of rank 4 Azumaya algebras, and extend what is known for good forms and Azumaya algebras. By considering line-bundle-valued forms, we extend the theorem of Max-Albert Knus that the Witt-invariant—the even Clifford algebra of a form—suffices for classification. An algebra Zariski-locally the even Clifford algebra of a ternary form is so globally up to twisting by square roots of line bundles. The general, usual and special orthogonal groups of a form are determined in terms of automorphism groups of its Witt-invariant. Martin Kneser’s characteristic-free notion of semiregular form is used.     

PRME AND k-PRIME IDEALS IN Ω-GROUPS

Abstract

In this paper we define two concepts of prime ideals for Ω-groups. The first generalizes the definitions of prime ideal in rings, nearrings, Γ-rings, associative algebras and Lie algebras. The second generalizes a concept defined for groups by ??ukin ([21]). We show that both lead to radicals in the sense of Hoehnke ([10]). Furthermore in the case of rings, Γ-rings, abelian zero-symmetric nearrings and cubic rings these two definitions coincide, thus obtaining a new characterization for the prime ideal. Zero-symmetric Ω-groups are defined analogously to the nearring case and a new characterization in term of ideals is given.     

Degree bounds for Gröbner bases in algebras of solvable type

We establish doubly-exponential degree bounds for Gröbner bases in certain algebras of solvable type over a field (as introduced by Kandri-Rody and Weispfenning). The class of algebras considered here includes commutative polynomial rings, Weyl algebras, and universal enveloping algebras of finite-dimensional Lie algebras. For the computation of these bounds, we adapt a method due to Dubé based on a generalization of Stanley decompositions. Our bounds yield doubly-exponential degree bounds for ideal membership and syzygies, generalizing the classical results of Hermann and Seidenberg (in the commutative case) and Grigoriev (in the case of Weyl algebras).     

On absolute Galois splitting fields of central simple algebras

A splitting field of a central simple algebra is said to be absolute Galois if it is Galois over some fixed subfield of the centre of the algebra. The paper proves an existence theorem for such fields over global fields with enough roots of unity. As an application, all twisted function fields and all twisted Laurent series rings over symbol algebras (or p-algebras) over global fields are crossed products. An analogous statement holds for division algebras over Henselian valued fields with global residue field.The existence of absolute Galois splitting fields in central simple algebras over global fields is equivalent to a suitable generalization of the weak Grunwald-Wang theorem, which is proved to hold if enough roots of unity are present. In general, it does not hold and counter examples have been used in noncrossed product constructions. This paper shows in particular that a certain computational difficulty involved in the construction of explicit examples of noncrossed product twisted Laurent series rings cannot be avoided by starting the construction with a symbol algebra.     

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.     

Koszul duality for extension algebras of standard modules

We define and investigate a class of Koszul quasi-hereditary algebras for which there is a natural equivalence between the bounded derived category of graded modules and the bounded derived category of graded modules over (a proper version of) the extension algebra of standard modules. Examples of such algebras include, in particular, the multiplicity free blocks of the BGG category O, and some quasi-hereditary algebras with Cartan decomposition in the sense of König.     

Periodic resolutions and self-injective algebras of finite type

We say that an algebra A is periodic if it has a periodic projective resolution as an (A,A)-bimodule. We show that any self-injective algebra of finite representation type is periodic. To prove this, we first apply the theory of smash products to show that for a finite Galois covering B→A, B is periodic if and only if A is. In addition, when A has finite representation type, we build upon results of Buchweitz to show that periodicity passes between A and its stable Auslander algebra. Finally, we use Asashiba’s classification of the derived equivalence classes of self-injective algebras of finite type to compute bounds for the periods of these algebras, and give an application to stable Calabi-Yau dimensions.     

Invariance of selfinjective algebras of quasitilted type under stable equivalences

We prove that the class of finite dimensional selfinjective algebras over a field which admit Galois coverings by the repetitive algebras of the quasitilted algebras, with Galois groups generated by compositions of the Nakayama automorphisms with strictly positive automorphisms, is invariant under stable and derived equivalences. Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday     

Artin-Schelter regular algebras and categories

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin-Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin-Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin-Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the -category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.     

Hochschild homology and split pairs

We study the Hochschild homology of algebras related via split pairs, and apply this to fiber products, trivial extensions, monomial algebras, graded-commutative algebras and quantum complete intersections. In particular, we compute lower bounds for the dimensions of both the Hochschild homology and cohomology groups of quantum complete intersections.     

Hopf structures on ambiskew polynomial rings

We derive necessary and sufficient conditions for an ambiskew polynomial ring to have a Hopf algebra structure of a certain type. This construction generalizes many known Hopf algebras, for example U(sl₂), U_q(sl₂) and the enveloping algebra of the three-dimensional Heisenberg Lie algebra. In a torsion-free case we describe the finite-dimensional simple modules, in particular their dimensions, and prove a Clebsch-Gordan decomposition theorem for the tensor product of two simple modules. We construct a Casimir type operator and prove that any finite-dimensional weight module is semisimple.     

Quantum group actions, twisting elements, and deformations of algebras

We construct twisting elements for module algebras of restricted two-parameter quantum groups from factors of their R-matrices. We generalize the theory of Giaquinto and Zhang to universal deformation formulas for catagories of module algebras and give examples arising from R-matrices of two-parameter quantum groups.     

Baer and Baer *-ring characterizations of Leavitt path algebras

We characterize Leavitt path algebras which are Rickart, Baer, and Baer ?-rings in terms of the properties of the underlying graph. In order to treat non-unital Leavitt path algebras as well, we generalize these annihilator-related properties to locally unital rings and provide a more general characterizations of Leavitt path algebras which are locally Rickart, locally Baer, and locally Baer ?-rings. Leavitt path algebras are also graded rings and we formulate the graded versions of these annihilator-related properties and characterize Leavitt path algebras having those properties as well.Our characterizations provide a quick way to generate a wide variety of examples of rings. For example, creating a Baer and not a Baer ?-ring, a Rickart ?-ring which is not Baer, or a Baer and not a Rickart ?-ring, is straightforward using the graph-theoretic properties from our results. In addition, our characterizations showcase more properties which distinguish behavior of Leavitt path algebras from their $C^{?}$-algebra counterparts. For example, while a graph $C^{?}$-algebra is Baer (and a Baer ?-ring) if and only if the underlying graph is finite and acyclic, a Leavitt path algebra is Baer if and only if the graph is finite and no cycle has an exit, and it is a Baer ?-ring if and only if the graph is a finite disjoint union of graphs which are finite and acyclic or loops.     

Algebras of quasi-Plücker coordinates are Koszul

Motivated by the theory of quasi-determinants, we study non-commutative algebras of quasi-Plücker coordinates. We prove that these algebras provide new examples of non-homogeneous quadratic Koszul algebras by showing that their quadratic duals have quadratic Gröbner bases.     

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.