FISHING FOR COMPLEMENTS

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

We construct a matrix algebra $\Lambda(A,B)$ from two given finite dimensional elementary algebras $A$ and $B$ and give some sufficient conditions on $A$ and $B$ under which the derived Jordan--H\"older property (DJHP) fails for $\Lambda(A,B)$. This provides finite dimensional algebras of finite global dimension which do not satisfy DJHP.

In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It was proven to be an invaluable tool in the study of non-linear filtering problems. In the early eighties, Brockett proposed to classify finite dimensional estimation algebras and Mitter conjectured that all functions in finite dimensional estimation algebras are necessarily polynomials of total degree at most one. Despite the massive effort in understanding the finite dimensional estimation algebras, the 20 year old problem of Brockett and Mitter conjecture remains open. In this paper, we give a classification of finite dimensional estimation algebras of maximal rank and solve the Mitter conjecture affirmatively for finite dimensional estimation algebras of maximal rank. In particular, for an estimation algebra E of maximal rank, we give a necessary and sufficient conditions for E to be finite dimensional in terms of the drift fi (x) and observation hj (x). As an important corollary, we show that the number of statistics needed to compute the conditional density of the state given the observation {y(s):0 ≤ s ≤ t} by the algebraic method is n where n is the dimension of the state.

Finite dimensional cosemisimple Hopf algebras in characteristic 0 are semisimple

We study aisles, equivalently t-structures, in the derived category of a hereditary abelian category. Given an aisle, we associate a sequence of subcategories of the abelian category by considering the different homologies of the aisle. We then obtain a sequence, called a narrow sequence. We then prove that a narrow sequence in a hereditary abelian category consists of a nondecreasing sequence of wide subcategories, together with a tilting torsion class in each of these wide subcategories. Studying the extra conditions that the narrow sequences coming from aisles must satisfy we get a bijection between coreflective narrow seqeunces and t-structures in the derived category. In some cases, including the case of finite-dimensional modules over a finite-dimensional hereditary algebra, we refine our results and show that the t-structures are determined by an increasing sequence of coreflective wide subcategories together with a tilting torsion class in the orthogonal of one wide subcategory in the next, effectively decoupling the different tilting torsion theories one chooses in the wide subcategories. These results are sufficient to recover known classifications of t-structures for smooth projective curves, and for finitely generated modules over a Dedekind ring.

The estimation algebra plays an important role in classification of finite dimensional filters. When finite dimensional estimation algebra has maximal rank, Yau et al. [Yau (2003). Complete classification of finite-dimensional estimation algebras of maximal rank. International Journal of Control, 76(7), 657–677; Yau & Hu (2005). Classification of finite-dimensional estimation algebras of maximal rank with arbitrary state-space dimension and Mitter conjecture. International Journal of Control, 78(10), 689–705.] have proved that η must be a degree 2 polynomial. In this paper, we study the structure of finite dimensional exact estimation algebra with state dimension 3 and rank 2. We establish a sufficient and necessary condition for estimation algebra with nonmaximal rank to be finite dimensional. Importantly, in the new filtering system, η needs not to be a degree 2 polynomial and can be of any degree . It is the first time to systematically analyse nonmaximal rank exact estimation algebra in which η is a polynomial of any degree . For Riccati-type equation, estimates have been done from the viewpoints of both classical solution and weak solution respectively. Finally, finite dimensional filters of Benés type are constructed successfully.

For self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this article, we first show that each derived equivalenceFbetween the derived categories of Artin algebrasAandBarises naturally as a functorbetween their stable module categories, which can be used to compare certain homological dimensions ofAwith that ofB. We then give a sufficient condition for the functorto be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classical result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras that are both derived-equivalent and stably equivalent of Morita type; thus, they share many common invariants.

For self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this article, we first show that each derived equivalenceFbetween the derived categories of Artin algebrasAandBarises naturally as a functorbetween their stable module categories, which can be used to compare certain homological dimensions ofAwith that ofB. We then give a sufficient condition for the functorto be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classical result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras that are both derived-equivalent and stably equivalent of Morita type; thus, they share many common invariants.

Difficulties associated with using X-ray crystallography for structural studies of large macromolecular complexes have made single particle cryo-electron microscopy (cryoEM) a key technique in structural biology. The efficient application of the single particle cryoEM approach requires the sample to be vitrified within the holes of carbon films, with particles well dispersed throughout the ice and adopting multiple orientations. To achieve this, the carbon support film is first hydrophilised by glow discharge, which allows the sample to spread over the film. Unfortunately, for transmembrane complexes especially, this procedure can result in severe sample adsorption to the carbon support film, reducing the number of particles dispersed in the ice. This problem is rate-limiting in the single particle cryoEM approach and has hindered its widespread application to hydrophobic complexes. We describe a novel grid preparation technique that allows for good particle dispersion in the ice and minimal hydrophobic particle adhesion to the support film. This is achieved by hydrophilisation of the carbon support film by the use of selected detergents that interact with the support so as to achieve a hydrophilic and neutral or selectively charged surface.

We give two proofs to the following theorem and its generalization: if a finite dimensional algebra $A$ is derived equivalent to a smooth projective scheme, then any derived equivalence between $A$ and another algebra $B$ is standard, that is, isomorphic to the derived tensor functor by a two-sided tilting complex. The main ingredients of the proofs are as follows: (1) between the derived categories of two module categories, liftable functors coincide with standard functors; (2) any derived equivalence between a module category and an abelian category is uniquely factorized as the composition of a pseudo-identity and a liftable derived equivalence; (3) the derived category of coherent sheaves on a certain projective scheme is triangle-objective, that is, any triangle autoequivalence on it, which preserves the the isomorphism classes of complexes, is necessarily isomorphic to the identity functor.

We survey old and new results on the representation theory of selfinjective algebras of quasitilted type, that is, the finite dimensional selfinjective algebras over a field with Galois coverings by the repetitive categories of quasitilted algebras (endomorphism algebras of tilting objects in hereditary abelian categories).

On maximal green sequences in abelian length categories

Finiteness conditions on the Ext-algebra of a cycle algebra

Let Rbe a finite dimensional central simple algebra over a field F A be any n× n matrix over R. By using the method of matrix representation, this paper obtains the structure formula of the minimal polynomial qA (λ) of A over F. By using qA (λ), this paper discusses the structure of right (left) eigenvalues set of A, and obtains the necessary and sufficient condition that a matrix over a finite dimensional central division algebra is similar to a diagonal matrix.

We give necessary and sufficient conditions on an Ore extension $$A[x;\sigma ,\delta ]$$ , where A is a finite dimensional algebra over a field $${\mathbb {F}}$$ , for being a Frobenius extension of the ring of commutative polynomials $${\mathbb {F}}[x]$$ . As a consequence, as the title of this paper highlights, we provide a negative answer to a problem stated by Caenepeel and Kadison.