Piecewise Hereditary Triangular Matrix Algebras

For any positive integer N,we clearly describe all finite-dimensional algebras A such that the upper triangular matrix algebras TN(A) are piecewise hereditary.Consequently,we describe all finite-dimensional algebras A such that their derived categories of N-complexes are triangulated equivalent to derived categories of hereditary abelian categories,and we describe the tensor algebras A (×) K[X]/(XN) for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.     

Indecomposable representations of generalized weyl algebras

For a class of generalized Weyl algebras which includes the Weyl algebras A_n a criterion is given as to when the category of indecomposable weight and generalized weight modules with supports from a fixed orbit is tame. In tame cases indecomposable modules are described.     

Derived equivalences of selfinjective algebras preserve singularities

We prove that the derived equivalences (more generally the stable equivalences of Morita type) of finite dimensional selfinjective algebras over algebraically closed fields preserve the types of singularities in the orbit closures of module varieties. As an application, we obtain that the orbit closures in the module varieties of the Brauer tree algebras are normal and Cohen-Macaulay. Mathematics Subject Classification (2000):14B05, 14L30, 16D50, 16G20     

Homogeneous polynomials with isomorphic milnor algebras

We recall first Mather’s Lemma providing effective necessary and sufficient conditions for a connected submanifold to be contained in an orbit. We show that two homogeneous polynomials having isomorphic Milnor algebras are right-equivalent.     

Diagram algebras, Hecke algebras and decomposition numbers at roots of unity

We prove that the cell modules of the affine Temperley-Lieb algebra have the same composition factors, when regarded as modules for the affine Hecke algebra of type A, as certain standard modules which are defined homologically. En route, we relate these to the cell modules of the Temperley-Lieb algebra of type B, which provides a connection between Temperley-Lieb algebras on n and n−1 strings. Applications include the explicit determination of some decomposition numbers of standard modules at roots of unity, which in turn has implications for certain Kazhdan-Lusztig polynomials associated with nilpotent orbit closures. The methods involve the study of the relationships among several algebras defined by concatenation of braid-like diagrams and between these and Hecke algebras. Connections are made with earlier work of Bernstein-Zelevinsky on the “generic case” and of Jones on link invariants.     

Flow equivalence and orbit equivalence for shifts of finite type and isomorphism of their groupoids

We give conditions for when continuous orbit equivalence of one-sided shift spaces implies flow equivalence of the associated two-sided shift spaces. Using groupoid techniques, we prove that this is always the case for shifts of finite type. This generalises a result of Matsumoto and Matui from the irreducible to the general case. We also prove that a pair of one-sided shift spaces of finite type are continuously orbit equivalent if and only if their groupoids are isomorphic, and that the corresponding two-sided shifts are flow equivalent if and only if the groupoids are stably isomorphic. As applications we show that two finite directed graphs with no sinks and no sources are move equivalent if and only if the corresponding graph $C^{?}$-algebras are stably isomorphic by a diagonal-preserving isomorphism (if and only if the corresponding Leavitt path algebras are stably isomorphic by a diagonal-preserving isomorphism), and that two topological Markov chains are flow equivalent if and only if there is a diagonal-preserving isomorphism between the stabilisations of the corresponding Cuntz–Krieger algebras (the latter generalises a result of Matsumoto and Matui about irreducible topological Markov chains with no isolated points to a result about general topological Markov chains). We also show that for general shift spaces, strongly continuous orbit equivalence implies two-sided conjugacy.     

Construction process for simple Lie algebras

We give here a construction process for the complex simple Lie algebras and the non-Hermitian type real forms which intersect the minimal nilpotent complex adjoint orbit, using a finite dimensional irreducible representation of the conformal group, or of some two-fold covering of it, with highest weight vector a semi-invariant of degree four. This process leads to a five-graded simple complex Lie algebra and the underlying semi-invariant is intimately related to the structure of the minimal nilpotent orbit. We also describe a similar construction process for the simple real Lie algebras of Hermitian type.     

INVARIANTS UNDER STABLE EQUIVALENCES OF MORITA TYPE

The aim of this article is to study some invariants of associative algebras under stable equivalences of Morita type.First of all,we show that,if two finite-dimensional selfinjective k-algebras are sta...     

On the structure of geodesic orbit Riemannian spaces

The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterized by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic orbit Riemannian space, we describe the structure of the nilradical and the radical of the Lie algebra of the isometry group. In the final part, we discuss some new tools to study geodesic orbit Riemannian spaces, related to compact Lie group representations with non-trivial principal isotropy algebras. We discuss also some new examples of geodesic orbit Riemannian spaces, new methods to obtain such examples, and some unsolved questions.     

Identities and Invariant Operators on Homogeneous Spaces

We study identities (functional relations between the generators of the transformation group) and also algebras of invariant operators on homogeneous spaces using the method of orbits of the coadjoint representation (coadjoint orbits). This method permits establishing the relation between these two objects and elaborating an algorithm for their construction. A classification of homogeneous spaces is introduced based on the coadjoint orbit method.     

A 2-Calabi–Yau realization of finite-type cluster algebras with universal coefficients

Mathematische Zeitschrift - We categorify various finite-type cluster algebras with coefficients using completed orbit categories associated to Frobenius categories. Namely, the Frobenius...     

Triangulated categories and Kac-Moody algebras

By using the Ringel-Hall algebra approach, we find a Lie algebra arising in each triangulated category with T ²=1, where T is the translation functor. In particular, the generic form of the Lie algebras determined by the root categories, the 2-period orbit categories of the derived categories of finite dimensional hereditary associative algebras, gives a realization of all symmetrizable Kac-Moody Lie algebras. Oblatum 4-XII-1998 & 11-XI-1999?Published online: 21 February 2000     

Reflection functors and symplectic reflection algebras for wreath products

We construct reflection functors on categories of modules over deformed wreath products of the preprojective algebra of a quiver. These functors give equivalences of categories associated to generic parameters which are in the same orbit under the Weyl group action. We give applications to the representation theory of symplectic reflection algebras of wreath product groups.     

Differential Modules over Quadratic Monomial Algebras

We compare the so-called clock condition to the gradability of certain differential modules over quadratic monomial algebras. These considerations show that a stably hereditary or gentle one-cycle algebra is piecewise hereditary if and only if the orbit category of its bounded derived category with respect to a positive power of the shift functor is triangulated.     

Some Landau–Ginzburg models viewed as rational maps

Gasparim, Grama and San Martin (2016) showed that height functions give adjoint orbits of semisimple Lie algebras the structure of symplectic Lefschetz fibrations (superpotential of the LG model in the language of mirror symmetry). We describe how to extend the superpotential to compactifications. Our results explore the geometry of the adjoint orbit from 2 points of view: algebraic geometry and Lie theory.     

On quiver Grassmannians and orbit closures for representation-finite algebras

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective–injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.     

Orbit equivalence,flow equivalence and ordered cohomology

We study self-homeomorphisms of zero dimensional metrizable compact Hausdorff spaces by means of the ordered first cohomology group, particularly in the light of the recent work of Giordano Putnam, and Skau on minimal homeomorphisms. We show that flow equivalence of systems is analogous to Morita equivalence between algebras, and this is reflected in the ordered cohomology group. We show that the ordered cohomology group is a complete invariant for flow equivalence between irreducible shifts of finite type; it follows that orbit equivalence implies flow equivalence for this class of systems. The cohomology group is the (pre-ordered) Grothendieck group of the C^*-algebra crossed product, and we can decide when the pre-ordering is an ordering, in terms of dynamical properties.     

Affine Yangians and deformed double current algebras in type A

We study the structure of Yangians of affine type and deformed double current algebras, which are deformations of the enveloping algebras of matrix W1+∞-algebras. We prove that they admit a PBW-type basis, establish a connection (limit construction) between these two types of algebras and toroidal quantum algebras, and we give three equivalent definitions of deformed double current algebras. We construct a Schur-Weyl functor between these algebras and rational Cherednik algebras.