Abstract L. Moret-Bailly constructed families $${\mathfrak {C}}\rightarrow {\mathbb {P}}^1$$ C → P 1 of genus 2 curves with supersingular Jacobian. In this paper we first classify the reducible fibers a family usi...

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...

In this study, we develop a general method to solve the linear elliptic quaternionic matrix equations by means of real representation quaternion matrices. A pseudocode for our approach that provides solution is expressed. Moreover, apply well-known Slyvester and Kalman Yakubovich over algebra.

A Riemannian manifold is said to be Kähler if the holonomy group is contained in U(n). It is quaternion Kähler if the holonomy group is contained in Sp(n)Sp(1). It is known that quaternion Kähler manifolds of dimension ≥ 8 are Einstein, so the scalar curvature s splits these manifolds according to whether s > 0, s = 0 or s < 0. Ricci flat quaternion Kähler manifolds include hyperkähler manifold...

Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split two real nonlinear systems, we derive explicit formulas for computing x2+bx+c=0 quaternion

In this paper, the authors study when the closure (in the Zariski topology) of orbits of representations of quivers of type A are rationally smooth. This is done by considering the corresponding quantized enveloping algebra U and studying the action of the bar involution on PBW bases. Using Ringel’s Hall algebra approach to quantized enveloping algebras and also AuslanderReiten quivers, we can ...

We show that the E 11 representation that contains the space-time translation generators also contains the rank two and five totally anti-symmetric representations of A 10. By studying the behaviour of these latter A 10 representations under SL(32), which we argue is contained in the Cartan involution invariant sub-algebra of E 11 , we find that the rank two and five totally anti-symmetric repr...

We exploit various inclusions of algebraic groups to give a new construction of groups of type E 8 , determine the Killing forms of the resulting E 8 's, and define an invariant of central simple algebras of degree 16 with orthogonal involution " in I 3 " , equivalently, groups of type D 8 with a half-spin representation defined over the base field. The determination of the Killing form is done...

Let $A$ be an associative algebra over a field of characteristic zero. A central polynomial is the free that takes values $A.$ In this survey, we present some recent results about exponential growth codimension sequence and proper in setting algebras with involution graded by finite group.

Let F be an arbitrary field (of arbitrary characteristic). Let A be a central simple F -algebra endowed with a quadratic pair σ (if charF 6= 2 then σ is simply an orthogonal involution on A). We show that the Witt index of σ over the function field of the Severi-Brauer variety of A is divisible by the Schur index of the algebra A.