Associative Research Articles

Degenerations of 3-dimensional nilpotent associative algebras over an algebraically closed field

Degenerations of 3-dimensional nilpotent associative algebras over an algebraically closed field

Torus actions on free associative algebras, lifting and Białynicki-Birula type theorems

Torus actions on free associative algebras, lifting and Białynicki-Birula type theorems

Associative algebras generated by regular Volterra operators

In this work, we consider genetic algebras, generated by Volterra quadratic stochastic operators, and establish relation between regularity of Volterra operators and associativity of corresponding genetic algebra.

Hochschild (Co)Homology and Applications

In 1945 Gerhard Hochschild published On the cohomology groups of an associative algebra in the Annals of Mathematics and thereby created what is now called Hochschild theory. The subject not only provides interesting homological invariants; it also serves as a link connecting algebra, topology, and geometry. The focus of the meeting was on recent developments, for instance in the study of singularities, deformations, and representations.

Quantizations of transposed Poisson algebras by Novikov deformations

Abstract The notions of the Novikov deformation of a commutative associative algebra and the corresponding classical limit are introduced. We show such a classical limit belongs to a subclass of transposed Poisson algebras, and hence the Novikov deformation is defined to be the quantization of the corresponding transposed Poisson algebra. As a direct consequence, we revisit the relationship between transposed Poisson algebras and Novikov–Poisson algebras due to the fact that there is a natural Novikov deformation of the commutative associative algebra in a Novikov–Poisson algebra. Hence all transposed Poisson algebras of Novikov–Poisson type, including unital transposed Poisson algebras, can be quantized. Finally, we classify the quantizations of 2-dimensional complex transposed Poisson algebras in which the Lie brackets are non-abelian up to equivalence.

Classification of three dimensional anti-dendriform algebras

This article is devoted to the classification of anti-dendriform algebras that are associated with associativity. In particular, the paper is devoted to classifying anti-dendriform algebras associated with null-filiform associative algebras and three-dimensional algebras. It is known that any finite-dimensional associative algebra without a nonzero idempotent element is nilpotent. If an associative algebra has a nonzero idempotent element, then there does not exist a compatible anti-dendriform algebra structure associated with associative algebras.

Crossed modules, non-abelian extensions of associative conformal algebras and wells exact sequences

In this paper, we introduce the notions of crossed modules of associative conformal algebras, two-term strongly homotopy associative conformal algebras, and discuss the relationship between them and the third Hochschild cohomology of associative conformal algebras. We classify the non-abelian extensions by introducing the non-abelian cohomology. We show that non-abelian extensions of an associative conformal algebra can be viewed as Maurer–Cartan elements of a suitable differential graded Lie algebra, and prove that there is a one-one correspondence between the Deligne groupoid of this differential graded Lie algebra and the non-abelian cohomology. For any two associative conformal algebras [Formula: see text] and [Formula: see text], we provide the condition that a pair of automorphisms in Aut(A) × Aut(B) can be extended to an automorphisms of the non-abelian extension algebra of [Formula: see text] by [Formula: see text], and give the fundamental sequence of Wells in the context of associative conformal algebras.

Twisted bimodules and universal enveloping algebras associated to VOAs

For any vertex operator algebra V, finite automorphism g of V of order T and m,n∈(1/T)Z+, we construct a family of associative algebras Ag,n(V) and Ag,n(V)−Ag,m(V)-bimodules Ag,n,m(V) from the point of view of representation theory. We prove that the algebra Ag,n(V) is identical to the algebra Ag,n(V) constructed by Dong, Li and Mason, and that the bimodule Ag,n,m(V) is identical to Ag,n,m(V) which was constructed by Dong and Jiang. We also prove that the Ag,n(V)−Ag,m(V)-bimodule Ag,n,m(V) is isomorphic to U(V[g])n−m/U(V[g])n−m−m−1/T, where U(V[g])k is the subspace of degree k of the (1/T)Z-graded universal enveloping algebra U(V[g]) of V with respect to g and U(V[g])kl is some subspace of U(V[g])k. And we show that all these bimodules Ag,n,m(V) can be defined in a simpler way.

Lie Rota–Baxter operators on the Sweedler algebra H4

If [Formula: see text] is an associative algebra, then we can define the adjoint Lie algebra [Formula: see text] and Jordan algebra [Formula: see text]. It is easy to see that any associative Rota–Baxter (RB) operator on [Formula: see text] induces a Lie and Jordan RB operator on [Formula: see text] and [Formula: see text], respectively. Are there Lie (Jordan) RB operators, which are not associative RB operators? In this paper, we explore these questions for the Sweedler algebra [Formula: see text], which is a 4-dimensional non-commutative Hopf algebra. More precisely, we describe the RB operators on the adjoint Lie algebra [Formula: see text].

On the hypercomplex numbers and normed division algebras in all dimensions: A unified multiplication.

Mathematics is the foundational discipline for all sciences, engineering, and technology, and the pursuit of normed division algebras in all finite dimensions represents a paramount mathematical objective. In the quest for a real three-dimensional, normed, associative division algebra, Hamilton discovered quaternions, constituting a non-commutative division algebra of quadruples. Subsequent investigations revealed the existence of only four division algebras over reals, each with dimensions 1, 2, 4, and 8. This study transcends such limitations by introducing generalized hypercomplex numbers extending across all dimensions, serving as extensions of traditional complex numbers. The space formed by these numbers constitutes a non-distributive normed division algebra extendable to all finite dimensions. The derivation of these extensions involves the definitions of two new π-periodic functions and a unified multiplication operation, designated as spherical multiplication, that is fully compatible with the existing multiplication structures. Importantly, these new hypercomplex numbers and their associated algebras are compatible with the existing real and complex number systems, ensuring continuity across dimensionalities. Most importantly, like the addition operation, the proposed multiplication in all dimensions forms an Abelian group while simultaneously preserving the norm. In summary, this study presents a comprehensive generalization of complex numbers and the Euler identity in higher dimensions, shedding light on the geometric properties of vectors within these extended spaces. Finally, we elucidate the practical applications of the proposed methodology as a viable alternative for expressing a quantum state through the multiplication of specified quantum states, thereby offering a potential complement to the established superposition paradigm. Additionally, we explore its utility in point cloud image processing.

Twists of graded Poisson algebras and related properties

We introduce a Poisson version of the graded twist of a graded associative algebra and prove that every graded Poisson structure on a connected graded polynomial ring A:=k[x1,…,xn] is a graded twist of a unimodular Poisson structure on A, namely, if π is a graded Poisson structure on A, then π has a decompositionπ=πunim+1∑i=1ndeg⁡xiE∧m where E is the Euler derivation, πunim is the unimodular graded Poisson structure on A corresponding to π, and m is the modular derivation of (A,π). This result is a generalization of the same one in the quadratic setting. The rigidity of graded twisting, PH1-minimality, and H-ozoneness are studied. As an application, we compute the Poisson cohomologies of the quadratic Poisson structures on the polynomial ring of three variables when the potential is irreducible, but not necessarily having an isolated singularity.

Modified double brackets and a conjecture of S. Arthamonov

Around 20 years ago, M. Van den Bergh introduced double Poisson brackets as operations on associative algebras inducing Poisson brackets under the representation functor. Weaker versions of these operations, called modified double Poisson brackets, were later introduced by S. Arthamonov in order to induce a Poisson bracket on moduli spaces of representations of the corresponding associative algebras. Moreover, he defined two operations that he conjectured to be modified double Poisson brackets. The first case of this conjecture was recently proved by M. Goncharov and V. Gubarev motivated by the theory of Rota-Baxter operators of nonzero weight. We settle the conjecture by realising the second case as part of a new family of modified double Poisson brackets. These are obtained from mixed double Poisson algebras, a new class of algebraic structures that are introduced and studied in the present work.

A∞ perspective to Sen's formalism

Sen's formalism is a mechanism for eliminating constraints on the dynamical fields that are imposed independently from equations of motion by employing spurious free fields. In this note a cyclic homotopy associative algebra underlying Sen's formalism is developed. The novelty lies in the construction of a symplectic form and cyclic A∞ maps on an extended algebra that combines the dynamical and spurious fields. This algebraic presentation makes the gauge invariance of theories using Sen's formulation manifest.

Semilinear clannish algebras

AbstractWe define a class of associative algebras generalizing ‘clannish algebras’, as introduced by the second author, but also incorporating semilinear structure, like a skew polynomial ring. Clannish algebras generalize the well‐known ‘string algebras’ introduced by Butler and Ringel. Our main result is the classification of finite‐dimensional indecomposable modules for these new algebras.

Inner isotopes associated with automorphisms of commutative associative algebras

The principal observation of the present paper is that an inner isotopy (i.e. a principal isotopy defined by an algebra endomorphism) is a very helpful instrument in constructing and studying interesting classes of nonassociative algebras. By using methods developed in the paper, we define a new class of commutative nonassociative algebras obtained by inner isotopy from commutative associative polynomial algebras. There is a natural bijection between isomorphism classes of our algebras and integer partitions of the algebra dimensions. Among the interesting features of the nonassociative algebras constructed are that these algebras are generic, some of examples are axial and metrized algebras. We completely describe both the set of algebra idempotents and their spectra.

Convolution equation and operators on the euclidean motion group

Let $G = \mathbb{R}^2\rtimes SO(2)$ be the Euclidean motion group, let g be the Lie algebra of G and let U(g) be the universal enveloping algebra of g. Then U(g) is an infinite dimensional, linear associative and non-commutative algebra consisting of invariant differential operators on G. The Dirac measure on G is represented by $\delta_G$, while the convolution product of functions or measures on G is represented by $\ast$. Among other notable results, it is demonstrated that for each u in U(g), there is a distribution E on G such that the convolution equation $u\ast E = \delta_G$ is solved by method of convolution. Further more, it is established that the (convolution) operator $A^\prime : C^\infty_c(G)\rightarrow C^\infty(G),$, which is defined as $A^\prime f = f\ast T^n\delta(t)$ extends to a bounded linear operator on $L^2(G)$, for $f\in C^\infty_c(G)$, the space of infinitely differentiable functions on G with compact support. Furthermore, we demonstrate that the left convolution operator LT denoted as $L_Tf = T\ast f$ commutes with left translation, for $T\in D^\prime(G)$.

Affine phase retrieval of quaternion signals

Quaternion algebra H is an extension of the complex number field, which is a noncommutative associative algebra. In recent years, quaternionic Fourier analysis has interested some mathematicians due to its applications in signal analysis and image processing. This paper addresses quaternionic affine phase retrieval (QAPR) in quaternion Euclidean spaces H M , which aims to exactly recover a signal in H M from the magnitudes of its affine measurements. We introduce the concepts of QAPR and phaselift operator in H M . Then, we characterize QAPR in terms of the real Jacobian matrix, prove that 5M is the minimal measurement number for QAPR in H M , study the stability of QAPR-sequences, and use the phaselift techniques to give some sufficient conditions on QAPR for H M which provide us with a method to construct QAPR-sequences.

The classified mathematical papers of A. A. Albert: a glimpse into the application of mathematics to cryptologic problems during the 1950s and 1960s

Abraham Adrian Albert “A cubed” (1905–1972) was an algebraist best known academically for his study of associative and non-associative algebras. By the 1940s he had also developed an interest in cryptology and its relationship to mathematics. Throughout the following two decades, Albert would participate in several mathematics programs to study and advise on cryptanalysis. In particular, Albert’s work with the National Security Agency (NSA) set a framework for the development of cryptology as a mathematical discipline. This research aims to present a timeline for and the breadth of Albert’s promotion of cryptology in the post-WWII American mathematical community and to examine, for the first time, reports Albert authored under contract with the NSA. In doing so, we hope to better understand the relationship between the NSA and academic mathematicians at the time of writing. The reports, released in 2020 by the NSA in response to a 2010 Freedom of Information Act request, have been arranged by the author into eleven collections based on content. Each collection will be introduced with their ideas and results given in a roughly chronological order. These collections span ideas in group theory, field theory, matrix algebra, and geometry, most often with application to cryptological systems.

Physical Motivation On Deformation Quantisation

We study the Poisson structures associated with deformation quantisation and its non-degradable factor on the Casimir function. We also describe a filtered associative algebra in a quotient space as a Poisson algebra and the automorphism of the Poisson bracket is discussed.

Hamiltonians for the quantised Volterra hierarchy

This paper builds upon our recent work, published in Carpentier et al (2022 Lett. Math. Phys. 112 94), where we established that the integrable Volterra lattice on a free associative algebra and the whole hierarchy of its symmetries admit a quantisation dependent on a parameter ω. We also uncovered an intriguing aspect: all odd-degree symmetries of the hierarchy admit an alternative, non-deformation quantisation, resulting in a non-commutative algebra for any choice of the quantisation parameter ω. In this study, we demonstrate that each equation within the quantum Volterra hierarchy can be expressed in the Heisenberg form. We provide explicit expressions for all quantum Hamiltonians and establish their commutativity. In the classical limit, these quantum Hamiltonians yield explicit expressions for the classical ones of the commutative Volterra hierarchy. Furthermore, we present Heisenberg equations and their Hamiltonians in the case of non-deformation quantisation. Finally, we discuss commuting first integrals, central elements of the quantum algebra, and the integrability problem for periodic reductions of the Volterra lattice in the context of both quantisations.