Associative Research Articles

Linearly sofic Lie algebras

We introduce and study linear soficity for Lie algebras, modeled after linear soficity in associative algebras. We introduce equivalent definitions of linear soficity, one involving metric ultraproducts and the other involving almost representations. We prove that any Lie algebra of subexponential growth is linearly sofic. We also prove that a Lie algebra over a field of characteristic 0 is linearly sofic if and only if its universal enveloping algebra is linearly sofic. As examples, we give explicit families of almost representations for the Witt and Virasoro algebras. Finally, we also show that the restricted universal enveloping algebra of a restricted linearly sofic Lie algebra is also linearly sofic.

Wild Local Structures of Automorphic Lie Algebras

We study automorphic Lie algebras using a family of evaluation maps parametrised by the representations of the associative algebra of functions. This provides a descending chain of ideals for the automorphic Lie algebra which is used to prove that it is of wild representation type. We show that the associated quotients of the automorphic Lie algebra are isomorphic to twisted truncated polynomial current algebras. When a simple Lie algebra is used in the construction, this allows us to describe the local Lie structure of the automorphic Lie algebra in terms of affine Kac-Moody algebras.

La centralità del codice etico nella formazione dell’arbitro di calcio

The article includes historical notes on the Italian Football Federation (FIGC). From its birth, in the year 1898, up to the present reality. The Italian Referees Association (AIA) is a component of the FIGC made up of football referees. It consists of a central structure and a territorial network and is governed by specific rules for all its functions. Among these rules, the Code of Ethics is of great importance, a document that defines and indicates values connected to ethics, understood as rules of respect and behavior which are essential for football referees.

Yoneda algebras of the triplet vertex operator algebra

Given a vertex operator algebra V, one can construct two associative algebras, the Zhu algebra A(V) and the C2-algebra R(V). This gives rise to two abelian categories A(V)-Mod and R(V)-Mod, in addition to the category of admissible modules of V. In case V is rational and C2-cofinite, the category of admissible V-modules and the category of all A(V)-modules are equivalent. However, when V is not rational, the connection between these two categories is unclear. The goal of this paper is to study the triplet vertex operator algebra W(p), as an example to compare these three categories, in terms of abelian categories. For each of these three abelian categories, we will determine the associated Ext quiver, the Morita equivalent basic algebra, i.e., the algebra End(⊕L∈IrrPL)op, and the Yoneda algebra Ext⁎(⊕L∈IrrL,⊕L∈IrrL). As a consequence, the category of admissible log-modules for the triplet VOA W(p) has infinite global dimension, as do the Zhu algebra A(W(p)), and the associated graded algebra grA(W(p)) which is isomorphic to R(W(p)). We also describe the Koszul properties of the module categories of W(p), A(W(p)) and grA(W(p)).

Invariant metrics on current Lie algebras

In this work we state conditions for a current Lie algebra to admit an invariant metric, where is a quadratic Lie algebra and is an associative and commutative algebra with unit. We show that if is an indecomposable quadratic Lie algebra, then admits an invariant metric if and only if also admits an invariant metric. In addition, we prove a theorem similar to the double extension for , where is an indecomposable, nilpotent and quadratic Lie algebra.

Pre-twisted calculus and differential equations

In this paper we introduce the φA-differentiability for functions f:U⊂Rk→Rn, where U is an open set, A is the linear space Rn endowed with a unital associative commutative algebra product, and φ:U⊂Rk→A is a differentiable function in the usual sense. We call it pre-twisted differentiability. With respect to the φA-differentiability we introduce: (a) a type Cauchy–Riemann equations, which serve as φA-differentiability criteria, (b) a Cauchy-integral theorem, and (c) φA-differential equations, which can be used to solve linear and nonlinear ODE systems. It has recently been shown that the φA-differentiable functions define a complete solutions for the PDEs of the form Auxx+Buxy+Cuyy=0, which is used in this paper for solving the corresponding Cauchy problems. Furthermore, solutions of φA-differential equations define solutions for linear and nonlinear PDE systems.

RB-operator and Nijenhuis operator on Hom-associative conformal algebra

Hom-associative conformal algebra [Formula: see text] is an associative conformal algebra with a twist map and satisfies the Hom-associative conformal identity. This study aims to introduce the notion of the Rota–Baxter operator [Formula: see text] on Hom-associative conformal algebra [Formula: see text]. We generalize our study to Hom-dendriform and Hom-tridendriform conformal algebras and give their relation to Hom-preLie conformal algebras. We give the interrelation between dendriform (and tridendriform) algebra with Hom-associative conformal Rota–Baxter algebra. Furthermore, we explore the Nijenhus operator on Hom-associative conformal algebra and describe its relation with Hom-associative Rota–Baxter operator.

Extended commutator algebra for the $q$-oscillator and a related Askey-Wilson algebra

Let $q$ be a nonzero complex number that is not a root of unity. In the $q$-oscillator with commutation relation $ a a^+-qa^+ a =1$, it is known that the smallest commutator algebra of operators containing the creation and annihilation operators $a^+$ and $ a $ is the linear span of $a^+$ and $ a $, together with all operators of the form ${a^+}^l{\left[a,a^+\right]}^k$, and ${\left[a,a^+\right]}^k a ^l$, where $l$ is a nonnegative integer and $k$ is a positive integer. That is, linear combinations of operators of the form $ a ^h$ or $(a^+)^h$ with $h\geq 2$ or $h=0$ are outside the commutator algebra generated by $ a $ and $a^+$. This is a solution to the Lie polynomial characterization problem for the associative algebra generated by $a^+$ and $ a $. In this work, we extend the Lie polynomial characterization into the associative algebra $\mathcal{P}=\mathcal{P}(q)$ generated by $ a $, $a^+$, and the operator $e^{\omega N}$ for some nonzero real parameter $\omega$, where $N$ is the number operator, and we relate this to a $q$-oscillator representation of the Askey-Wilson algebra $AW(3)$.

Differential codimensions and exponential growth

Let A be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra L by derivations, and let cnL(A), n≥1, be its differential codimension sequence. Such sequence is exponentially bounded and expL⁡(A)=limn→∞⁡cnL(A)n is an integer that can be computed, called differential PI-exponent of A.In this paper we prove that for any Lie algebra L, expL⁡(A) coincides with exp⁡(A), the ordinary PI-exponent of A. Furthermore, in case L is a solvable Lie algebra, we apply such result to classify varieties of L-algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth.

Associative algebras of [formula omitted]-graded vertex operator superalgebras and their applications

Let V be a Z-graded vertex operator superalgebra and G a finite automorphism group of V of order T. For g∈G and n∈1TZ+, a series of associative algebras Ag,n(V) are constructed to study the twisted representations of V. In addition, we introduce a G-twisted associative algebra AG,n(V) such that Ag,n(V) is a quotient algebra of AG,n(V) for any g∈G. As an application of the theory of AG,n(V), we obtain a duality theorem of Schur-Weyl type. In particular, for any g∈G, every irreducible g-twisted V-module is a completely reducible VG-module.

Using a q-shuffle algebra to describe the basic module V(Λ_0) for the quantized enveloping algebra Uq(sl^2)

We consider the quantized enveloping algebra Uq(sl^2). and its basic module V(Λ0). This module is infinite-dimensional, irreducible, integrable, and highest-weight. We describe V(Λ0) using a q-shuffle algebra in the following way. Start with the free associative algebra V on two generators x, y. The standard basis for V consists of the words in x, y. In 1995 M. Rosso introduced an associative algebra structure on V, called a q-shuffle algebra. For u, v ∈ {x, y} their q-shuffle product is u ⋆ v = uv + q(u,v)vu, where (u,v) = 2 (resp. (u,v) = − 2) if u = v (resp. u ≠ v). Let U denote the subalgebra of the q-shuffle algebra V that is generated by x, y. Rosso showed that the algebra U is isomorphic to the positive part of Uq(sl^2). In our first main result, we turn U into a Uq(sl^2).-module. Let U denote the Uq(sl^2).-submodule of U generated by the empty word. In our second main result, we show that the Uq(sl^2).-modules U and V(Λ0) are isomorphic. Let V denote the subspace of V spanned by the words that do not begin with y or xx. In our third main result, we show that U = U ∩ V.

Graded Lie-Rinehart algebras

The present work introduces the class of graded Lie-Rinehart algebras as a natural generalization of graded Lie algebras. It is demonstrated that a tight G-graded Lie-Rinehart algebra L over a commutative and associative G-graded algebra A, where G is an abelian group, can be decomposed into the orthogonal direct sums L=⨁i∈IIi and A=⨁j∈JAj, where each Ii and Aj is a non-zero ideal of L and A, respectively. Additionally, both decompositions satisfy that for any i∈I, there exists a unique j∈J such that AjIi≠0 and that any Ii is a graded Lie-Rinehart algebra over Aj. In the case of maximal length, the aforementioned decompositions of L and A are through indecomposable (graded) ideals, and the (graded) simplicity of any Ii and any Aj are also characterized.

An LA-group based design of the non-linear component of block cipher

An S-box is a crucial component of modern block ciphers, providing confusion and making them attack-resistant. For the last three decades, researchers have designed S-boxes using techniques mainly based on associative algebras of Galois fields. In this work, the non-associative structure of an LA-group (left almost group) is employed to design 8 × 8 S-boxes. An LA-group-based design S-box offers appealing properties, such as the inverse of zero element, non-associativity, and fewer constraints. These characteristics increase robustness and expand the key space compared to cyclic groups and Galois fields. Furthermore, an RGB image encryption scheme is also presented in this work to judge the impact of proposed S-boxes. The proposed S-boxes create confusion in a plain image “Red, Green, and Blue” channels, whereas for diffusion, the linear transformation K=(i×32)mod127 is operationalized. In the last, we compared our encryption scheme to some existing benchmarks, revealing that our encryption technique is adequate for image security.

Operations on the Hochschild bicomplex of a diagram of algebras

A diagram of algebras is a functor valued in a category of associative algebras. I construct an operad acting on the Hochschild bicomplex of a diagram of algebras. Using this operad, I give a direct proof that the Hochschild cohomology of a diagram of algebras is a Gerstenhaber algebra. I also show that the total complex is an L∞-algebra. The same results are true for the reduced and asimplicial subcomplexes and asimplicial cohomology. This structure governs deformations of diagrams of algebras through the Maurer-Cartan equation.

On the algebra generated by μ ¯ , ∂ ¯ , ∂ , μ \overline{\mu },\overline{\partial },\partial ,\mu

Abstract In this note, we determine the structure of the associative algebra generated by the differential operators μ ¯ , ∂ ¯ , ∂ \overline{\mu },\overline{\partial },\partial , and μ \mu that act on complex-valued differential forms of almost complex manifolds. This is done by showing that it is the universal enveloping algebra of the graded Lie algebra generated by these operators and determining the structure of the corresponding graded Lie algebra. We then determine the cohomology of this graded Lie algebra with respect to its canonical inner differential [ d , − ] \left[d,-] , as well as its cohomology with respect to all its inner differentials.

Symmetric homology and representation homology

Symmetric homology is a natural generalization of cyclic homology, in which symmetric groups play the role of cyclic groups. In the case of associative algebras, the symmetric homology theory was introduced by Z. Fiedorowicz (1991) and was further developed in the work of S. Ault (2010). In this paper, we show that, for algebras defined over a field of characteristic 0 0 , the symmetric homology is naturally equivalent to the (one-dimensional) representation homology introduced by the authors in joint work with G. Khachatryan (2013). Using known results on representation homology, we compute symmetric homology explicitly for basic algebras, such as polynomial algebras and universal enveloping algebras of (DG) Lie algebras. As an application, we prove two conjectures of Ault and Fiedorowicz, including their main conjecture (2007) on topological interpretation of symmetric homology of polynomial algebras.

Nilpotent algebras, implicit function theorem, and polynomial quasigroups

We prove an implicit function theorem for nilpotent (not necessarily associative or Lie) algebras. This allows us to establish a correspondence between such algebras and quasigroups, in the spirit of the classical correspondence between divisible torsion-free nilpotent groups and rational nilpotent Lie algebras. We study the related questions of the commensurators of nilpotent groups, filiform Lie algebras of maximal solvability length and partially ordered algebras.

Tate–Shafarevich groups and algebras

The Tate–Shafarevich set of a group [Formula: see text] defined by Takashi Ono coincides, in the case where [Formula: see text] is finite, with the group of outer class-preserving automorphisms of [Formula: see text] introduced by Burnside. We consider analogs of this important group-theoretic object for Lie algebras and associative algebras and establish some new structure properties thereof. We also discuss open problems and eventual generalizations to other algebraic structures.

Derimorphisms over Algebras and Applications

The new concept “derimorphism” generalizing both derivation and homomorphism is defined. When a derimorphism is invertible, its inverse is a Rota–Baxter operator. The general theory of derimorphism is established. The classification of all derimorphisms over an associative unital algebra is obtained. Contrary to the nonexistence of nontrivial positive derivations, it is shown that nontrivial positive derimorphisms do exist over any pair of opposite orderings over [Formula: see text], the lattice-ordered full matrix algebra and upper triangular matrix algebra over a totally ordered field.

On [formula omitted]-graded vertex algebras associated with cyclic Leibniz algebras with small dimensions

The main goals for this paper are i) to study of the algebraic structure of N-graded vertex algebras VB associated with vertex A-algebroids B when B are cyclic non-Lie left Leibniz algebras, and ii) to explore relations between the vertex algebras VB and the rank one Heisenberg vertex operator algebra. To achieve these goals, we first classify vertex A-algebroids B associated with given cyclic non-Lie left Leibniz algebras B. Next, we use the constructed vertex A-algebroids B to create a family of indecomposable non-simple vertex algebras VB. Finally, we use the algebraic structure of the unital commutative associative algebras A that we found to study relations between certain types of the vertex algebras VB and the vertex operator algebra associated to the rank one Heisenberg algebra.