We introduce new Bézier-like formulas with quaternionic weights for rational curves and surfaces in Euclidean 3-space. They are useful for representation of Möbius invariant geometric objects. Any Bézier curves and surfaces on 2-spheres can be converted to the quaternionic Bézier (QB) form of twice lower degree. Our focus is on bilinear QB-surfaces: we derive their implicitization formula and s...

We describe the classical o(3, 2) r-matrices as generating the quantum deformations of either D=3 conformal algebra with mass-like deformation parameters or D=4 AdS algebra with dimensionless deformation parameters. We describe the quantization of classical o(3, 2) r-matrices via Drinfeld twist method which locates the deformation in the coalgebra sector. Further we obtain the quantum o(3, 2) a...

This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine algebras and Yangians. In this paper, we study notions of ~-adic nonlocal vertex algebra and ~-adic (weak) quantum vertex algebra, slightly generalizing Etingof-Kazhdan’s notion of quantum vertex operator algebra. For any topologically free C[[~]]-module W , we study ~-adic...

We describe the classical o(3, 2) r-matrices as generating the quantum deformations of either D=3 conformal algebra with mass-like deformation parameters or D=4 AdS algebra with dimensionless deformation parameters. We describe the quantization of classical o(3, 2) r-matrices via Drinfeld twist method which locates the deformation in the coalgebra sector. Further we obtain the quantum o(3, 2) a...

Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this approach to elementary representations of arbitrary points, point pairs, lines, circles, planes and spheres. Apart from including curved objects, conformal geometri...

The aim of this paper is to introduce the notion of commutative pseudo BE-algebras and investigate their properties.We generalize some results proved by A. Walendziak for the case of commutative BE-algebras.We prove that the class of commutative pseudo BE-algebras is equivalent to the class of commutative pseudo BCK-algebras. Based on this result, all results holding for commutative pseudo BCK-...

This is the first paper in a series to study vertex algebra-like objects arising from infinite-dimensional quantum groups (quantum affine algebras and Yangians). In this paper we lay the foundation for this study. For any vector space W , we study what we call quasi compatible subsets of Hom(W,W ((x))) and we prove that any maximal quasi compatible subspace has a natural nonlocal (namely noncom...

In this paper, we show that every surjective $n$-homomorphism ($n$-anti-homomorphism) from a Banach algebra $A$ into a semisimple Banach algebra $B$ is continuous.

A bstract We study the chiral flat space higher-spin algebra, which is global symmetry algebra of theory in 4d Minkowski space. find that it can be constructed as universal enveloping a certain deformation Poincaré quotiented by set quadratic identities. These identities allow us to identify representation latter analogy with AdS holography, we interpret singleton representation. provide two ex...

Let $mathfrak{A}$ be a Banach algebra. We say that a sequence ${D_n}_{n=0}^infty$ of continuous operators form $mathfrak{A}$ into $mathfrak{A}$ is a textit{local higher derivation} if to each $ainmathfrak{A}$ there corresponds a continuous higher derivation ${d_{a,n}}_{n=0}^infty$ such that $D_n(a)=d_{a,n}(a)$ for each non-negative integer $n$. We show that if $mathfrak{A}$ is a $C^*$-algebra t...