Generalized polar decomposition method (or briefly GPD method) has been introduced by Munthe-Kaas and Zanna to approximate the matrix exponential. In this paper, we investigate the numerical stability of that method with respect to roundoff propagation. The numerical GPD method includes two parts: splitting of a matrix $Z\in \mathfrak{g}$, a Lie algebra of matrices and computing $\exp(Z)\mathbf{v}$ for a vector $\mathbf{v}$. We show that the former is stable provided that $\|Z\|$ is not so large, while the latter is not stable in general except with some restrictions on the entries of the matrix Z and the vector $\mathbf{v}$.

Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification of $\mathrm{sl}(3)$ quantum knot invariants and also in physics. The $2D$-TQFT of surfaces, on the other hand, is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and comultiplication. In this paper, we explore algebraic operations that branch lines derive under TQFT. In particular, we investigate Lie bracket and bialgebra structures. Relations to the original Frobenius algebra structures are discussed both algebraically and diagrammatically.

We introduce the notion of left (and right) quasi-Loday algebroids and a ``universal space'' for them, called a left (right) omni-Loday algebroid, in such a way that Lie algebroids, omni-Lie algebras and omni-Loday algebroids are particular substructures.

The concept of intuitionistic fuzzy Lie algebra over a fuzzy field is introduced. We study the "necessity" and "possibility" operators on intuitionistic fuzzy Lie algebra over a fuzzy field and give some properties of homomorphic images.

In this paper, we prove the fundamental theorem of color Hopf module similar to the fundamental theorem of Hopf module. As an application, we prove that the graded global dimension of a color Hopf algebra coincides with the projective dimension of the trivial module K

Let A be an associative algebra over a field $k$, and let $\mathcal{M}$ be a finite family of right $A$-modules. A study of the noncommutative deformation functor $\mathrm{Def}_{\mathcal{M}}$ of the family $\mathcal{M}$ leads to the construction of the algebra $\mathcal{O}^A(\mathcal{M})$ of observables and the generalized Burnside theorem, due to Laudal (2002). In this paper, we give an overview of aspects of noncommutative deformations closely connected to the generalized Burnside theorem.

The index of a seaweed Lie algebra can be computed from its associated meander graph. We examine this graph in several ways with a goal of determining families of Frobenius (index zero) seaweed algebras. Our analysis gives two new families of Frobenius seaweed algebras as well as elementary proofs of known families of such Lie algebras.

We study deformations and the moduli space of 3-dimensional complex associative algebras. We use extensions to compute the moduli space, and then give a decomposition of this moduli space into strata consisting of complex projective orbifolds, glued together through jump deformations. The main purpose of this paper is to give a logically organized description of the moduli space, and to give an explicit description of how the moduli space is constructed by extensions.

We have previously introduced the notion of non-commutative phase space (algebra) associated to any associative algebra, defined over a field. The purpose of the present paper is to prove that this construction is useful in non-commutative deformation theory for the construction of the versal family of finite families of modules. In particular, we obtain a much better understanding of the obstruction calculus, that is, of the Massey products.

Let X be a scheme over an algebraically closed field k, and let $x\in\operatorname{Spec} R\subseteq X$ be a closed point corresponding to the maximal ideal $\mathfrak{m}\subseteq R$. Then $\hat{\mathcal{O}}_{X,x}$ is isomorphic to the prorepresenting hull, or local formal moduli, of the deformation functor $\mathrm{Def}_{R/\mathfrak{m}}:\underline{\ell}\rightarrow\mathrm{Sets}$. This suffices to reconstruct $X$ up to etalé coverings. For a noncommutative $k$-algebra $A$ the simple modules are not necessarily of dimension one, and there is a geometry between them. We replace the points in the commutative situation with finite families of points in the noncommutative situation, and replace the geometry of points with the geometry of sets of points given by noncommutative deformation theory. We apply the theory to the noncommutative moduli of three-dimensional endomorphisms.

We construct dynamical Yang-Baxter maps, which are set-theoretical solutions to a version of the quantum dynamical Yang-Baxter equation, by means of homogeneous pre-systems, that is, ternary systems encoded in the reductive homogeneous space satisfying suitable conditions. Moreover, a characterization of these dynamical Yang-Baxter maps is presented.