Terms Of Algebra Research Articles

The first Hochschild cohomology as a Lie algebra

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvability of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including most quantum complete intersections.

Bimodules over Relative Rota-Baxter Algebras and Cohomologies

A relative Rota-Baxter algebra is a generalization of a Rota-Baxter algebra. Relative Rota-Baxter algebras are closely related to dendriform algebras. In this paper, we introduce bimodules over a relative Rota-Baxter algebra that fits with the representations of dendriform algebras. We define the cohomology of a relative Rota-Baxter algebra with coefficients in a bimodule and then study abelian extenfsions of relative Rota-Baxter algebras in terms of the second cohomology group. Finally, we consider homotopy relative Rota-Baxter algebras and classify skeletal homotopy relative Rota-Baxter algebras in terms of the above-defined cohomology.

Compatible L∞-algebras

A compatible L∞-algebra is a graded vector space together with two compatible L∞-algebra structures on it. Given a graded vector space, we construct a graded Lie algebra whose Maurer-Cartan elements are precisely compatible L∞-algebra structures on it. We provide examples of compatible L∞-algebras arising from Nijenhuis operators, compatible V-datas and compatible Courant algebroids. We define the cohomology of a compatible L∞-algebra and as an application, we study formal deformations. Next, we classify ‘strict’ and ‘skeletal’ compatible L∞-algebras in terms of crossed modules and cohomology of compatible Lie algebras. Finally, we introduce compatible Lie 2-algebras and find their relationship with compatible L∞-algebras.

On the Quaternion Transformation and Field Equations in Curved Space-Time

In this paper, we use four-dimensional quaternionic algebra to describe space-time geodesics in curvature form. The transformation relations of a quaternionic variables are established with the help of basis-transformations of quaternion algebra. We deduce the quaternionic covariant derivative that explains how the quaternion components vary with scalar and vector fields. The quaternionic metric tensor and the geodesic equation are also discussed to describe the quaternionic line element in curved space-time. We examine a quaternionic metric tensor equation for the Riemannian Christoffel curvature tensor. We present the quaternionic Einstein’s field-like equation, which indicates that quaternionic matter and geometry are equivalent. Relevance of the work:In recent decades, hypercomplex algebra, viz., quaternion and octonions, has been widely used to explain various branches of physics. In this way, we have investigated quaternionic transformations and field equations in curved space-time. The present novel work will help to explain the characteristics of the curved space-time universe in terms of quaternion algebra. It can also be used to describe quaternionic gravitational waves, the black hole formulation, and so on.

Kripke Contexts, Double Boolean Algebras with Operators and Corresponding Modal Systems

The notion of a context in formal concept analysis and that of an approximation space in rough set theory are unified in this study to define a Kripke context. For any context (G,M,I), a relation on the set G of objects and a relation on the set M of properties are included, giving a structure of the form ((G,R), (M,S), I). A Kripke context gives rise to complex algebras based on the collections of protoconcepts and semiconcepts of the underlying context. On abstraction, double Boolean algebras (dBas) with operators and topological dBas are defined. Representation results for these algebras are established in terms of the complex algebras of an appropriate Kripke context. As a natural next step, logics corresponding to classes of these algebras are formulated. A sequent calculus is proposed for contextual dBas, modal extensions of which give logics for contextual dBas with operators and topological contextual dBas. The representation theorems for the algebras result in a protoconcept-based semantics for these logics.

Hyperbolic relaxation models for thin films down an inclined plane

We present a family of relaxation models for thin films flows where both viscosity and surface tension effects are inherent. In a first step, a first-order hyperbolic approximation to the dissipationless part of the system is presented. The method is based on an augmented Lagrangian approach, where a classical penalty method is used and high-order derivatives in the Lagrangian are promoted to new independent variables, for which hyperbolic closure equations are sought. Then, we show that the viscous terms can be treated either by plugging them directly to the obtained system, making it of the hyperbolic-parabolic type or by casting them into an approximate algebraic source term that is asymptotically equivalent to the former formulation. Finally, the extension of the method to a classical nonlinear surface tension model is also presented. Numerical results, for all the proposed models are shown and compared with experimental results and reference solutions.

Anti-pre-Lie algebras, Novikov algebras and commutative 2-cocycles on Lie algebras

We introduce the notion of anti-pre-Lie algebras as the underlying algebraic structures of nondegenerate commutative 2-cocycles which are the “symmetric” version of symplectic forms on Lie algebras. They can be characterized as a class of Lie-admissible algebras whose negative left multiplication operators make representations of the commutator Lie algebras. We observe that there is a clear analogy between anti-pre-Lie algebras and pre-Lie algebras by comparing them in terms of several aspects. Furthermore, it is unexpected that a subclass of anti-pre-Lie algebras, namely admissible Novikov algebras, correspond to Novikov algebras in terms of q-algebras. Consequently, there is a construction of admissible Novikov algebras from commutative associative algebras with derivations or more generally, admissible pairs. The correspondence extends to the level of Poisson type structures, leading to the introduction of the notions of anti-pre-Lie Poisson algebras and admissible Novikov-Poisson algebras, whereas the latter correspond to Novikov-Poisson algebras.

Generalizations of Snyder model to curved spaces

We consider generalizations of the Snyder algebra to a curved spacetime background with de Sitter symmetry. As special cases, we obtain the algebras of the Yang model and of triply special relativity. We discuss the realizations of these algebras in terms of canonical phase space coordinates, up to fourth order in the deformation parameters. In the case of triply special relativity we also find exact realization, exploiting its algebraic relation with the Snyder model.

On the nilpotent orbits arising from admissible affine vertex algebras

We give a simple description of the closure of the nilpotent orbits appearing as associated varieties of admissible affine vertex algebras in terms of primitive ideals.

Infinite-Dimensional Algebras as Extensions of Kinematic Algebras

Kinematic algebras can be realised on geometric spaces and constrain the physical models that can live on these spaces. Different types of kinematic algebras exist and we consider the interplay of these algebras for non-relativistic limits of a relativistic system, including both the Galilei and the Carroll limit. We develop a framework that captures systematically the corrections to the strict non-relativistic limit by introducing new infinite-dimensional algebras, with emphasis on the Carroll case. One of our results is to highlight a new type of duality between Galilei and Carroll limits that extends to corrections as well. We realise these algebras in terms of particle models. Other applications include curvature corrections and particles in a background electro-magnetic field.

Ideal extensions and directly infinite algebras

Directly infinite algebras , those algebras, E which have a pair of elements x and y where 1 = x y ≠ y x , are well known to have a sub-algebra isomorphic to M ∞ ( K ) , the set of infinite Z + × Z + -indexed matrices which have only finitely many nonzero entries . When this sub-algebra is actually an ideal, we may analyze the algebra in terms of an extension of some algebra A by M ∞ ( K ) , that is, a short exact sequence of K -algebras 0 → M ∞ ( K ) → E → A → 0 . The present article characterizes all trivial (split) extensions of K [ x , x − 1 ] by M ∞ ( K ) by examining the extensions as sub-algebras of infinite matrix algebras . Furthermore, we construct an infinite family of pairwise non-isomorphic extensions { T i : i ≥ 0 } , all of which can be written as an extension 0 → M ∞ ( K ) → T i → K [ x , x − 1 ] → 0 .

Affine Laumon spaces and a conjecture of Kuznetsov

We prove a conjecture of Kuznetsov stating that the equivariant K-theory of affine Laumon spaces is the universal Verma module for the quantum affine algebra U_q(gl_n^). We do so by reinterpreting the action of the quantum toroidal algebra U_q(gl_n^^) on the K-theory from [14] in terms of the shuffle algebra studied in [12], which constructs an embedding of U_q(gl_n^) into U_q(gl_n^^)

The quantum theory of time: a calculus for q-numbers.

In quantum theory, physical systems are usually assumed to evolve relative to a c-number time. This c-number time is unphysical and has turned out to be unnecessary for explaining dynamics: in the timeless approach to quantum theory developed by Page & Wootters 1983 Phys. Rev. D 27, 2885. (doi:10.1103/PhysRevD.27.2885), subsystems of a stationary universe can instead evolve relative to a ‘clock', which is a quantum system with a q-number time observable. Page & Wootters formulated their construction in the Schrödinger picture, which left open the possibility that the c-number time still plays an explanatory role in the Heisenberg picture. I formulate their construction in the Heisenberg picture and demonstrate how to eliminate c-number time from that picture, too. When the Page–Wootters construction is formulated in the Heisenberg picture, the descriptors of physical systems are functions of the clock's q-number time, and derivatives with respect to this q-number time can be defined in terms of the clock's algebra of observables, which results in a calculus for q-numbers.

Polarization and deformations of generalized dendriform algebras

We generalize three results of M. Aguiar on Loday’s dendriform algebras to dendriform algebras associated with algebras satisfying any given set of relations. We adapt the concept of polarization to such algebras, and use it to generalize Aguiar’s results on deformations and filtrations of dendriform algebras. We introduce weak Rota–Baxter operators and use them to prove a generalization of another result by Aguiar, which provides an interpretation of the natural relation between infinitesimal bialgebras and pre-Lie algebras in terms of dendriform algebras.

The Gelfand–Graev representation of classical groups in terms of Hecke algebras

AbstractLet G be a p-adic classical group. The representations in a given Bernstein component can be viewed as modules for the corresponding Hecke algebra—the endomorphism algebra of a pro-generator of the given component. Using Heiermann’s construction of these algebras, we describe the Bernstein components of the Gelfand–Graev representation for $G=\mathrm {SO}(2n+1)$ , $\mathrm {Sp}(2n)$ , and $\mathrm {O}(2n)$ .

The generator rank of subhomogeneous -algebras

Abstract We compute the generator rank of a subhomogeneous $C^*\!$ -algebra in terms of the covering dimension of the pieces of its primitive ideal space corresponding to irreducible representations of a fixed dimension. We deduce that every $\mathcal {Z}$ -stable approximately subhomogeneous algebra has generator rank one, which means that a generic element in such an algebra is a generator. This leads to a strong solution of the generator problem for classifiable, simple, nuclear $C^*\!$ -algebras: a generic element in each such algebra is a generator. Examples of Villadsen show that this is not the case for all separable, simple, nuclear $C^*\!$ -algebras.

Representations of Shifted Quantum Affine Algebras

AbstractWe develop the representation theory of shifted quantum affine algebras $\mathcal {U}_\mu (\hat {\mathfrak {g}})$ and of their truncations, which appeared in the study of quantized K-theoretic Coulomb branches of 3d $N = 4$ SUSY quiver gauge theories. Our approach is based on novel techniques, which are new in the cases of shifted Yangians or ordinary quantum affine algebras as well: realization in terms of asymptotical subalgebras of the quantum affine algebra $\mathcal {U}_q(\hat {\mathfrak {g}})$, induction and restriction functors to the category $\mathcal {O}$ of representations of the Borel subalgebra $\mathcal {U}_q(\hat {\mathfrak {b}})$ of $\mathcal {U}_q(\hat {\mathfrak {g}})$, relations between truncations and Baxter polynomiality in quantum integrable models, and parametrization of simple modules via Langlands dual interpolation. We first introduce the category $\mathcal {O}_\mu $ of representations of $\mathcal {U}_\mu (\hat {\mathfrak {g}})$ and we classify its simple objects. Then we establish the existence of fusion products and we get a ring structure on the sum of the Grothendieck groups $K_0(\mathcal {O}_\mu )$. We classify simple finite-dimensional representations of $\mathcal {U}_\mu (\hat {\mathfrak {g}})$ and we obtain a cluster algebra structure on the Grothendieck ring of finite-dimensional representations. We prove a truncation has only a finite number of simple representations and we introduce a related partial ordering on simple modules. Eventually, we state a conjecture on the parametrization of simple modules of a non-simply-laced truncation in terms of the Langlands dual Lie algebra. We have several evidences, including a general result for simple finite-dimensional representations.

Topological and Frame Properties of Certain Pathological $$C^*$$-Algebras

We introduce a classification of locally compact Hausdorff topological spaces with respect to the behavior of $\sigma$-compact subsets, and relying on this classification we study properties of corresponding $C^*$-algebras in terms of frame theory and the theory of $\mathcal A$-compact operators in Hilbert $C^*$-modules, some pathological examples are constructed.

Global bases for quantum Borcherds–Bozec algebras

We provide an explicit construction of global bases for quantum Borcherds–Bozec algebras and their irreducible highest weight modules. Moreover, we give a new presentation for quantum Borcherds–Bozec algebras in terms of primitive generators.

Vector multi-pole solutions in the [formula omitted]-coupled Hirota equation

In this paper, we study the vector multi-pole (MP) solutions of the r-coupled Hirota equation (RCHE). First of all, based on the Darboux transformation and the N-soliton solution, we derive the explicit formulas of the arbitrary-order vector MP solutions. Then, through the balance between exponential and algebraic terms, we give all asymptotic solitons in the vector double- and triple-pole solutions via an asymptotic analysis method. Furthermore, we analyze the soliton interactions in vector MP solutions. By comparing with the scalar Hirota equation, we find that the RCHE has richer collision properties: the position shift of each component soliton grows logarithmically with r, the asymptotic solitons have the same shape in each component, and their amplitudes are proportional among the components.