Related papers: A coadjoint orbit-like construction for Jordan sup…
We apply the quaternionic Jordan form to classify the hypercomplex nilpotent almost abelian Lie algebras in all dimensions and to carry out the complete classification of 12-dimensional hypercomplex almost abelian Lie algebras. Moreover, we…
We classify the space-like biharmonic surfaces in 3-dimension pseudo-Riemannian space form, and construct explicit examples of proper biharmonic hypersurfaces in general ADS space.
To an arbitrary Lie superalgebra $L$ we associate its Jordan double ${\mathcal Jor}(L)$, which is a Jordan superalgebra. This notion was introduced by the second author before. Now we study further applications of this construction. First,…
Almost paracontact almost paracomplex Riemannian manifolds of the lowest dimension are studied. Such structures are constructed on hyperspheres in 4-dimensional spaces, Euclidean and pseudo-Euclidean, respectively. The obtained manifolds…
The nontrivial subspaces with primitive coproducts are found in the deformed universal enveloping algebras. They can form carrier spaces for additional Jordanian twists. The latter can be used to construct sequences of twists for algebras…
Let $\mathrm{SL}(n,\mathbb{R})$ be the special linear group and $\mathfrak{sl}(n,\mathbb{R})$ its Lie algebra. We study geometric properties associated to the adjoint orbits in the simplest non-trivial case, namely, those of…
Given a Jordan algebra $A$ and a vector space $V$, we describe and classify all Jordan algebras containing $A$ as a subalgebra of codimension ${\rm dim}_k (V)$ in terms of a non-abelian cohomological type object ${\mathcal J}_{A} \, (V, \,…
A new algebra, hitherto not encountered in the usual Lie algebraic varieties or supervarieties, is introduced. The paper explores the rich and novel structure of the algebra, and it compares it on the one hand with the Jordan-Lie…
We show that the pseudo-Riemannian geometry of submanifolds can be formulated in terms of higher order multi-linear maps. In particular, we obtain a Poisson bracket formulation of almost (para-)K\"ahler geometry.
We determine the isomorphism classes of Jordan algebras in dimension two over the field of real numbers. Using techniques of non-standard analysis we study the properties of the variety of Jordan algebras, and also the contractions among…
We study the hyperkaehler geometry of a regular semisimple adjoint orbit of SL(k,C) via the algebraic geometry of the corresponding reducible spectral curve.
For a given conformal algebra $C$, we write down the correspondence between identities of the coefficient algebra $\Coeff C$ and identities of $C$ itself as of pseudoalgebra. In particular, we write down the defining relations of Jordan,…
The Arens-Michael functor in noncommutative geometry is an analogue of the analytification functor in algebraic geometry: out of the ring of "algebraic functions" on a noncommutative space it constructs the ring of "holomorphic functions"…
We compute minimal sets of generators for the S_n-modules (n <= 4) of multilinear polynomial identities of arity n satisfied by the Jordan product and the Jordan diproduct (resp. pre-Jordan product) in every triassociative (resp.…
The orbit decomposition is given under the automorphism group on the real split Jordan algebra of all hermitian matrices of order three corresponding to any real split composition algebra, or the automorphism group on the complexification,…
We study relations between the two-parameter $\U_q(sl(n))$-invariant deformation quantization on $sl^*(n)$ and the reflection equation algebra. The latter is described by a quantum permutation on $\End(\C^n)$ given explicitly. The…
We study Jordan types of linear forms for graded Artinian Gorenstein algebras having arbitrary codimension. We introduce rank matrices of linear forms for such algebras that represent the ranks of multiplication maps in various degrees. We…
We present explicit formulas for deformation quantization on the co-adjoint orbits of the real diamond Lie group. From this we obtain quantum half-plans, quantum hyperbolic cylinders, quantum hyperbolic paraboloids via Fedosov deformation…
Here we demonstrate the emergence of Grassmann variables in matrix models based on the exceptional Jordan algebra. The Grassmann algebras are built naturally using the octonion algebra. We argue the appearance of Grassmann variables…
The main purpose of this paper is to provide a second cohomology group of a (metric) Hom-Jacobi algebra with coefficients in a given representation. Moreover, we show that second cohomology group classifies abelian extensions of a (metric)…