Mathematics

To any Frobenius superalgebra A we associate an oriented Frobenius Brauer category and an affine oriented Frobenius Brauer categeory. We define natural actions of these categories on categories of supermodules for general linear Lie superalgebras gl m|n (A) with entries in A . These actions generalize those on module categories for general linear Lie superalgebras and queer Lie superalgebras, which correspond to the cases where A is the ground field and the two-dimensional Clifford algebra, respectively.

The affinoid envelope, U(L) ? of a free, finitely generated Z p -Lie algebra L has proven to be useful within the representation theory of compact p -adic Lie groups. Our aim is to further understand the algebraic structure of U(L) ? , and to this end, we will define a Dixmier module over U(L) ? , and prove that this object is generally irreducible in case where L is nilpotent. Ultimately, we will prove that all primitive ideals in the affinoid envelope can be described in terms of the annihilators of Dixmier modules, and using this, we aim towards proving that these algebras satisfy a version of the classical Dixmier-Moeglin equivalence.

We develop algebraic methods for computations with tensor data. We give 3 applications: extracting features that are invariant under the orthogonal symmetries in each of the modes, approximation of the tensor spectral norm, and amplification of low rank tensor structure. We introduce colored Brauer diagrams, which are used for algebraic computations and in analyzing their computational complexity. We present numerical experiments whose results show that the performance of the alternating least square algorithm for the low rank approximation of tensors can be improved using tensor amplification.

The conjugation action of the complex orthogonal group on the polynomial functions on n×n matrices gives rise to a graded algebra of invariant polynomials. A spanning set of this algebra is in bijective correspondence to a set of unlabeled, cyclic graphs with directed edges equivalent under dihedral symmetries. When the degree of the invariants is n+1 , we show that the dimension of the space of relations between the invariants grows linearly in n . Furthermore, we present two methods to obtain a basis of the space of relations. First, we construct a basis using an idempotent of the group algebra referred to as Young symmetrizers, but this quickly becomes computationally expensive as n increases. Thus, we propose a more computationally efficient method for this problem by repeatedly generating random matrices using a Monte Carlo algorithm.

We study distances on zigzag persistence modules from the viewpoint of derived categories and Auslander--Reiten quivers. The derived category of ordinary persistence modules is derived equivalent to that of arbitrary zigzag persistence modules, depending on a classical tilting module. Through this derived equivalence, we define and compute distances on the derived category of arbitrary zigzag persistence modules and prove an algebraic stability theorem. We also compare our distance with the distance for purely zigzag persistence modules introduced by Botnan--Lesnick and the sheaf-theoretic convolution distance due to Kashiwara--Schapira.

We provide an explicit set of algebraically independent generators for the algebra of invariant differential operators on the Riemannian symmetric space associated with $\SL_n(\R)$.

We shall study the existence of almost split sequences in tri-exact categories, that is, extension-closed subcategories of triangulated categories. Our results unify and extend the existence theorems for almost split sequences in abelian categories and exact categories (that is, extension-closed subcategories of abelian categories), and those for almost split triangles in triangulated categories. As applications, we shall obtain some new results on the existence of almost split sequences in the derived categories of all modules over an algebra with a unity or a locally finite dimensional algebra given by a quiver with relations.

A joint algebraic interpretation of the biorthogonal Askey polynomials on the unit circle and of the orthogonal Jacobi polynomials is offered. It ties their bispectral properties to an algebra called the meta-Jacobi algebra mJ .

We construct a numeric algorithm for completing the proof of a conjecture asserting that all deformed preprojective algebras of generalized Dynkin type are periodic. In particular, we obtain an algorithmic procedure showing that non-trivial deformed preprojective algebras of Dynkin types E 7 and E 8 exist only in characteristic 2. As a consequence, we show that deformed preprojective algebras of Dynkin types E 6 , E 7 and E 8 are periodic and we obtain an algorithm for a classification of such algebras, up to algebra isomorphism. We do it by a reduction of the conjecture to a solution of a system of equations associated with the problem of the existence of a suitable algebra isomorphism φ f : P f ( E n )→P( E n ) described in Theorem 2.1. One also shows that our algorithmic approach to the conjecture is also applicable to the classification of the mesh algebras of generalized Dynkin type.

Recently, we obtained in [7] a new characterization for an orthogonal system to be a simple-minded system in the stable module category of any representation-finite self-injective algebra. In this paper, we apply this result to give an explicit construction of simple-minded systems over self-injective Nakayama algebras.