Commutative Associative Algebra Research Articles

A New Method for Designing Post-Quantum Signature Schemes

The current standards of the digital signature algorithms are based on computational difficulty of the discrete logarithm and factorization problems. Expected appearance in near future of the quantum computer that is able to solve in polynomial time each of the said computational puts forward the actual task of the development of the post-quantum signature algorithms that resist the attacks using the quantum computers. Recently, the signature schemes based on the hidden discrete logarithm problem set in finite non-commutative associative algebras had been proposed. The paper is devoted to a further development of this approach and introduces a new practical post-quantum signature scheme possessing small size of public key and signature. The main contribution of the paper is the developed new method for defining the hidden discrete logarithm problem that allows applying the finite commutative groups as algebraic support of the post-quantum digital signature schemes. The method uses idea of applying multipliers that mask the periodicity connected with the value of discrete logarithm of periodic functions set on the base of the public parameters of the signature scheme. The finite 4-dimensional commutative associative algebra the multiplicative group of which possesses 4-dimensional cyclicity is used as algebraic support of the developed signature scheme.

Degenerations of nilpotent associative commutative algebras

We give a complete description of degenerations of complex 5-dimensional nilpotent associative commutative algebras. As corollary, we have the description of all rigid algebras in the variety of 5-dimensional commutative Leibniz algebras.

Functional analytic issues in $\mathbb{Z}_2^n$-geometry

We show that the function sheaf of a $\mathbb{Z}_2^n$-manifold is a nuclear Fr\'echet sheaf of $\mathbb{Z}_2^n$-graded $\mathbb{Z}_2^n$-commutative associative unital algebras. Further, we prove that the components of the pullback sheaf morphism of a $\mathbb{Z}_2^n$-morphism are all continuous. These results are essential for the existence of categorical products in the category of $\mathbb{Z}_2^n$-manifolds. All proofs are self-contained and explicit.

The algebraic classification of nilpotent associative commutative algebras

In this paper, we give a complete algebraic classification of [Formula: see text]-dimensional complex nilpotent associative commutative algebras.

Mathieu–Zhao subspaces of vertex algebras

We introduce a notion of Mathieu–Zhao subspaces of vertex algebras. Among other things, we show that for a vertex algebra [Formula: see text] and its subspace [Formula: see text] that contains [Formula: see text], [Formula: see text] is a Mathieu–Zhao subspace of [Formula: see text] if and only if the quotient space [Formula: see text] is a Mathieu–Zhao subspace of a commutative associative algebra [Formula: see text]. As a result, one can study the famous Jacobian conjecture in terms of Mathieu–Zhao subspaces of vertex algebras. In addition, for a [Formula: see text]-type vertex operator algebra [Formula: see text] that satisfies the [Formula: see text]-cofiniteness condition, we classify all Mathieu–Zhao subspaces [Formula: see text] that contain [Formula: see text].

Analysis in Noncommutative Algebras and Modules

In a previous paper, we developed an analysis in associative commutative algebras and in modules over them, which may be useful in problems of contemporary mathematical and theoretical physics. Here we work out similar methods in the noncommutative case.

On monogenic functions defined in different commutative algebras

The correspondence between a monogenic function in an arbitrary finite-dimensional commutative associative algebra and a finite collection of monogenic functions in a special commutative associative algebra is established.

Modules for map Virasoro conformal algebras

Any nontrivial finite irreducible module for the map Lie conformal algebra $$\hbox {Vir}\bigotimes {\mathcal {A}}$$ , where $$\hbox {Vir}$$ is the Virasoro conformal algebra and $${\mathcal {A}}$$ is an arbitrary finitely generated commutative associative algebra with unity over $${\mathbb {C}}$$ , is proved to be an evaluation module.

On para‐Kähler Lie algebroids and contravariant pseudo‐Hessian structures

AbstractIn this paper, we generalize all the results obtained on para‐Kähler Lie algebras in [3] to para‐Kähler Lie algebroids. In particular, we study exact para‐Kähler Lie algebroids as a generalization of exact para‐Kähler Lie algebras. This study leads to a natural generalization of pseudo‐Hessian manifolds, we call them contravariant pseudo‐Hessian manifolds. Contravariant pseudo‐Hessian manifolds have many similarities with Poisson manifolds. We explore these similarities which, among others, leads to a powerful machinery to build examples of non trivial pseudo‐Hessian structures. Namely, we will show that given a finite dimensional commutative and associative algebra , the orbits of the action Φ of on given by are pseudo‐Hessian manifolds, where . We illustrate this result by considering many examples of associative commutative algebras and show that the resulting pseudo‐Hessian manifolds are very interesting.

On submultiplicative constants of an algebra

If mathcal{A} is a finite-dimensional commutative associative real algebra with norm | cdot| then we say that the rth submultiplicative constant of mathcal{A} is the smallest constant m_{r}(mathcal{A}) for which | x_{1} x_{2} cdots x_{r} | leq m_{r}(mathcal {A}) | x_{1} | | x_{2} | cdots | x_{r} |. For a product algebra, we show that there exist zero divisors where equality is attained in the inequality defining m_{r}( mathcal{A}). We also study rho_{mathcal{A}} = limsup_{r rightarrowinfty} sqrt[r]{m_{r}(mathcal{A})}. We explain how rho_{mathcal{A}} appears in the generalization of the Cauchy–Hadamard criterion for hypercomplex power series. We find the submultiplicative constants and rho_{mathcal{A}} for the real group algebra of the cyclic group of order n as well as the complicated numbers mathcal{C}_{n} = { a_{1}+a_{2}k+ cdots+ a_{n} k^{n-1} | a_{i} inmathbb{R}, k^{n} = -1 } with Euclidean norm. Submultiplicative constants for the n-dual numbers Delta_{n} with Euclidean norm are also calculated or conjectured for n leq6. We show, for n geq2, rho_{Delta_{n}} = 1 for Delta_{n} given the p-norm.

Про моногенні функції на розширеннях комутативної алгебри

Для n-вимірної (2 ⩽ n < 1) комутативної асоціативної алгебри

Hypercomplex method for solving linear PDEs

Algebraic-analytic approach to constructing solutions for given partial differential equations were investigated in many papers. In particular, in papers [1-14]. It involves solving two problems. Problem (P 1) is to describe all the sets of vectors \( e_1, e_2, \ldots, e_d \), which satisfy the characteristic equation (or specify the procedure by which they can be found). And the problem (P 2) is to describe all the components of monogenic (i.e., continuous and differentiable in sense Gateaux) functions. In particular, for the equation (4) we must describe the components of the function \( \Phi(\zeta) = \exp \zeta \). Note that in the papers [15, 16] a constructive description of all analytic functions with values is obtained in an arbitrary finite-dimensional commutative associative algebra over the field \(\mathbb{C}\). The Theorem 5.1 of the paper [17] states that it is enough to limit the study of monogenic functions in algebras with the basis of \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1} \} \), where \( \eta_1, \eta_2, \ldots, \eta_{n-1} \) are nilpotents. In addition, in [18] it is showed that in each algebra with a basis of the form \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1} \} \) the equation (3) has solutions. That is, the problems (P 1) and (P 2) are completely solved on the classes of commutative associative algebras with the basis \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1}\} \). It is worth noting that in a finite-dimensional algebra a decomposition of monogenic functions has a finite number of components, and therefore, it generates a finite number of solutions of a given partial differential equations. In this paper, we propose a procedure for constructing an infinite number of families of solutions of given linear differential equations with partial derivatives with constant coefficients. We use monogenic functions that are defined on some sequences of commutative associative algebras over the field of complex numbers. To achieve this goal, we first study the solutions of the so-called characteristic equation on a given sequence of algebras. Further, we investigate monogenic functions on the sequence of algebras and study their relation with solutions of partial deferential equations. The proposed method is used to construct solutions of some equations of mathematical physics. In particular, for the three-dimensional Laplace equation and the wave equation, for the equation of transverse oscillations of the elastic rod and the conjugate equation, a generalized biharmonic equation and the two-dimensional Helmholtz equation. We note that this method yields all analytic solutions of the two-dimensional Laplace equation and the two-dimensional biharmonic equation (Goursat formula).

Frobenius algebra with relaxed associativity constraint

Frobenius algebra is formulated within the Abelian monoidal category of operad of graphs. A not necessarily associative algebra [Formula: see text] is said to be a Frobenius algebra if there exists a [Formula: see text]-module isomorphism. A new concept of a solvable Frobenius algebra is introduced: an algebra [Formula: see text] is said to be a solvable Frobenius algebra if it possesses a nonzero one-sided [Formula: see text]-module morphism with nontrivial radical. In the category of operad of graphs, we can express the necessary and sufficient conditions for an algebra to be a solvable Frobenius algebra. The notion of a solvable Frobenius algebra makes it possible to find all commutative nonassociative Frobenius algebras (Conjecture 10.1), and to find all Frobenius structures for commutative associative Frobenius algebras. Frobenius algebra allows [Formula: see text]-permuted opposite algebra to be extended to [Formula: see text]-permuted algebras.

Center of the quantum affine vertex algebra in type A

We consider the quantum vertex algebra associated with the double Yangian in type A as defined by Etingof and Kazhdan. We show that its center is a commutative associative algebra and construct algebraically independent families of topological generators of the center at the critical level.

Power associative nilalgebras of dimension 9

In [19] authors described some properties about commutative power associative nilalgebras of nilindex 5. Here we will get new results about the structure of this class of algebras. Those results will allow us to prove that every commutative power associative algebra of dimension 9 and nilindex 5 over a field of characteristic different from 2, 3 and 5 is solvable. Consequently the famous Albert's conjecture ([1], Problem 1.1) is setted for dimension ≤9 and characteristic 0 since the case of dimension 9 and nilindex ≠5 have already been examined in [27], [11], [14] and [20].

Simple Poisson–Farkas Algebras and Ternary Filippov Algebras

We establish connection between the differentiably simple associative commutative algebras with unity and the simple Filippov algebras.

Q-Deformations of 3-Lie Algebras

q-Deformations of 3-Lie algebras and representations of q-3-Lie algebras are discussed. A q-3-Lie algebra [Formula: see text], where [Formula: see text] and [Formula: see text], is a vector space A over a field 𝔽 with 3-ary linear multiplications [ , , ]q and [Formula: see text] from [Formula: see text] to A, and a map [Formula: see text] satisfying the q-Jacobi identity [Formula: see text] for all [Formula: see text]. If the multiplications satisfy that [Formula: see text] and [Formula: see text] is skew-symmetry, then [Formula: see text] is called a type (I)-q-3- Lie algebra. From q-Lie algebras, group algebras and commutative associative algebras, q-3-Lie algebras and type (I)-q-3-Lie algebras are constructed. Also, the non-trivial onedimensional central extension of q-3-Lie algebras is investigated, and new q-3-Lie algebras [Formula: see text], and [Formula: see text] are obtained.

Lie algebras induced by a nonzero field derivation

Given a finite-dimensional associative commutative algebra A over a field F, we define the structure of a Lie algebra using a nonzero derivation D of A. If A is a field and charF > 3; then the corresponding algebra is simple, presenting a nonisomorphic analog of the Zassenhaus algebra W 1(m).

Center of the quantum affine vertex algebra associated with trigonometric R-matrix

We consider the quantum vertex algebra associated with the trigonometric R-matrix in type A as defined by Etingof and Kazhdan. We show that its center is a commutative associative algebra and construct an algebraically independent family of topological generators of the center at the critical level.

Infinite-dimensional 3-Lie algebras and their connections to Harish-Chandra modules

We construct two kinds of infinite-dimensional 3-Lie algebras from a given commutative associative algebra, and show that they are all canonical Nambu 3-Lie algebras. We relate their inner derivation algebras to Witt algebras, and then study the regular representations of these 3-Lie algebras and the natural representations of the inner derivation algebras. In particular, for the second kind of 3-Lie algebras, we find that their regular representations are Harish-Chandra modules, and the inner derivation algebras give rise to intermediate series modules of the Witt algebras and contain the smallest full toroidal Lie algebras without center.