Asymptotic symmetries at null-infinity for the Rarita-Schwinger field with magnetic term

In this paper we study the magnetic charges of the free massless Rarita-Schwinger field in four dimensional asymptotically flat space-time. This is the first step towards extending the study of the dual BMS charges to supergravity. The magnetic charges appear due to the addition of a boundary term in the action. This term is similar to the theta term in Yang-Mills theory. At null-infinity an infinite dimensional algebra is discovered, both for the electric and magnetic charge.

On the geometry of lattices and finiteness of Picard groups

Let ( K , 𝒪 , k ) {(K,\mathcal{O},k)} be a p-modular system with k algebraically closed and 𝒪 {\mathcal{O}} unramified, and let Λ be an 𝒪 {\mathcal{O}} -order in a separable K-algebra. We call a Λ-lattice L rigid if Ext Λ 1 ⁡ ( L , L ) = 0 {{\operatorname{Ext}}^{1}_{\Lambda}(L,L)=0} , in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the Λ-lattices of a given dimension into “varieties of lattices”, we show that there are only finitely many rigid Λ-lattices L of any given dimension. As a consequence we show that if the first Hochschild cohomology of Λ vanishes, then the Picard group and the outer automorphism group of Λ are finite. In particular, the Picard groups of blocks of finite groups defined over 𝒪 {\mathcal{O}} are always finite.

Torsion Pairs and Ringel Duality for Schur Algebras

Let A be a finite-dimensional algebra over an algebraically closed field. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective A–modules P into those of the torsion submodules of P. As an application, we show that blocks of both the classical and quantum Schur algebras S(2,r) and Sq(2,r) in characteristic p > 0 are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain 2pk simple modules for some k.

Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

Relativistic Inversion, Invariance and Inter-Action

A general formula for inversion in a relativistic Clifford–Dirac algebra has been derived. Identifying the base elements of the algebra as those of space and time, the first order differential equations over all quantities proves to encompass the Maxwell equations, leads to a natural extension incorporating rest mass and spin, and allows an integration with relativistic quantum mechanics. Although the algebra is not a division algebra, it parallels reality well: where division is undefined turns out to correspond to physical limits, such as that of the light cone. The divisor corresponds to invariants of dynamical significance, such as the invariant interval, the general invariant quantities in electromagnetism, and the basis set of quantities in the Dirac equation. It is speculated that the apparent 3-dimensionality of nature arises from a beautiful symmetry between the three-vector algebra and each of four sets of three derived spaces in the full 4-dimensional algebra. It is conjectured that elements of inversion may play a role in the interaction of fields and matter.

On Quiver Grassmannians and Orbit Closures for Gen-Finite Modules

We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.

On band modules and $$\tau $$-tilting finiteness

In this paper, motivated by a $$\tau $$ τ -tilting version of the Brauer-Thrall Conjectures, we study general properties of band modules and their endomorphisms in the module category of a finite dimensional algebra. As an application we describe properties of torsion classes containing band modules. Furthermore, we show that a special biserial algebra is $$\tau $$ τ -tilting finite if and only if no band module is a brick. We also recover a criterion for the $$\tau $$ τ -tilting finiteness of Brauer graph algebras in terms of the Brauer graph.

Algebraic Supports and New Forms of the Hidden Discrete Logarithm Problem for Post-quantum Public-key Cryptoschemes

This paper introduces two new forms of the hidden discrete logarithm problem defined over a finite non-commutative associative algebras containing a large set of global single-sided units. The proposed forms are promising for development on their base practical post-quantum public key-agreement schemes and are characterized in performing two different masking operations over the output value of the base exponentiation operation that is executed in framework of the public key computation. The masking operations represent homomorphisms and each of them is mutually commutative with the exponentiation operation. Parameters of the masking operations are used as private key elements. A 6-dimensional algebra containing a set of p3 global left-sided units is used as algebraic support of one of the hidden logarithm problem form and a 4-dimensional algebra with p2 global right-sided units is used to implement the other form of the said problem. The result of this paper is the proposed two methods for strengthened masking of the exponentiation operation and two new post-quantum public key-agreement cryptoschemes. Mathematics subject classification: 94A60, 16Z05, 14G50, 11T71, 16S50.