Poisson algebras are usually defined as structures with two operations, a commutative associative one and an anticommutative one satisfying the Jacobi identity. These operations are tied up by a distributive law, the Leibniz law. We present Poisson algebras as algebras with one operation which enables one to study them as part of nonassociative algebras. We study the algebraic and cohomological...

The aim of this paper is to introduce and study Lie algebras and Lie groups over noncommutative rings. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F we introduce and study the algebra (g,A)(F), which is the Lie subalgebra of F ⊗ A generated by F ⊗ g. In many examples A is the universal enveloping algebra of g. Our description of the algebra (g,A)(F)...

Usually we shall just call A an algebra if the field k is clear from the context. The algebra A is associative if multiplication is associative i.e. for all a, b, c ∈ A, (ab)c = a(bc), and unital if there is a multiplicative identity, i.e. an element usually denoted by 1 such that, for all a ∈ A, 1a = a1 = a. Note that, in this case, 1 = 0 ⇐⇒ A = {0}. Otherwise, the map k → A defined by t 7→ t·...

In this paper, we show that the fundamental concepts behind the Ntrū cryptosystem can be extended to a broader algebra than Dedekind domains. Also, we present an abstract and generalized algorithm for constructing a Ntrū-like cryptosystem such that the underlying algebra can be non-commutative or even non-associative. To prove the main claim, we show that it is possible to generalize Ntrū over ...

We introduce the first hom-associative Weyl algebras over a field of prime characteristic as generalization associative algebra in characteristic. First, we study properties constructed from by general ``twisting'' procedure. Then, with help these results, determine commuter, center, nuclei, and set derivations algebras. also classify them up to isomorphism, show, among other things, that all n...

A Hom-algebra structure is a multiplication on a vector space where the structure is twisted by a homomorphism. The structure of Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov in [4] and extended by Larsson and Silvestrov to quasi-hom Lie and quasi-Lie algebras in [5, 6]. In this paper we introduce and study Hom-associative, Hom-Leibniz, and Hom-Lie admissible algebraic struc...

Recall that a finite-dimensional commutative associative algebra equipped with an invariant nondegenerate symmetric bilinear form is called a Frobenius algebra (here, we do not require an existence of a unit in Frobenius algebra). Any commutative quasi-Frobenius algebra is always Frobenius, i.e., if the identity ab = ba (commutativity) is fulfilled in a quasi-Frobenius algebra, then the identities

The aim of this paper is to introduce and study Lie algebras over noncommutative rings. For any Lie algebra g sitting inside an associative algebra A and any associative algebra F we introduce and study the F -loop algebra (g, A)(F), which is the Lie subalgebra of F ⊗ A generated by F ⊗ g. In most examples A is the universal enveloping algebra of g. Our description of the loop algebra has a str...

We show that, if A is a finite-dimensional *-simple associative algebra with involution (over the field K of real or complex numbers) whose hermitian part H( A, * > is of degree > 3 over its center, if B is a unital algebra with involution over 06, and if (I.11 is an algebra norm on H( A @ B, * 1, then there exists an algebra norm on A @ B whose restriction to H(A @ B, *> is equivalent to 11 . ...