Related papers: On the restricted Jordan plane in odd characterist…
We consider the restricted Jordan plane in characteristic $2$, a finite-dimensional Nichols algebra quotient of the Jordan plane that was introduced by Cibils, Lauve and Witherspoon. We extend results from \texttt{arXiv:2002.02514} on the…
We consider the super Jordan plane, a braided Hopf algebra introduced--to the best of our knowledge--in works of N. Andruskiewitsch, I. Angiono, I. Heckenberger, and its restricted version in odd characteristic introduced by the same…
We study finite-dimensional commutative algebras, which satisfy the Jacobi identity. Such algebras are Jordan algebras. We describe some of their properties and give a classification in dimensions $n<7$ over algebraically closed fields of…
In this paper, we classify Jordan superalgebras of dimension up to three over an algebraically closed field of characteristic different of two. Our main motivation to obtain such classification comes out from the intention to give an answer…
We compute the simple finite-dimensional modules and the center of the Drinfeld double of the Jordan plane introduced in $\texttt{arXiv:2002.02514}$ assuming that the characteristic is zero.
We classify finite-dimensional Nichols algebras of Yetter-Drinfeld modules with indecomposable support over finite solvable groups in characteristic 0, using a variety of methods including reduction to positive characteristic. As a…
Jordan, Wigner and von Neumann classified the possible algebras of quantum mechanical observables, and found they fell into 4 "ordinary" families, plus one remarkable outlier: the exceptional Jordan algebra. We point out an intriguing…
Nichols algebras are an important tool for the classification of Hopf algebras. Within those with finite GK dimension, we study homological invariants of the super Jordan plane, that is, the Nichols algebra $A=B(V(-1,2))$. These invariants…
We study algebraic properties of the super Jordan plane~$\mathcal B$ introduced by I. Angiono, I. Heckenberger and the first named author. Concretely we show that $\mathcal B$ is super-prime and has a super-simple super-artinian ring of…
We classify pointed Hopf algebras with finite Gelfand-Kirillov dimension whose infinitesimal braiding has dimension 2 but is not of diagonal type, or equivalently is a block. These Hopf algebras are new and turn out to be liftings of either…
We characterize the space of restrictions of real rational functions to certain algebraic Jordan curves in the plane via the Dirichlet-to-Neumann map associated to the domain in the complex plane bounded by the curve and its Bergman kernel.…
We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants such as real roots. The crucial ingredient is a `reflection' in the class of such Nichols algebras. We conclude the classifications of…
We classify Nichols algebras of irreducible Yetter-Drinfeld modules over nonabelian groups satisfying an inequality for the dimension of the homogeneous subspace of degree two. All such Nichols algebras are finite-dimensional, and all known…
We present new examples of finite-dimensional Nichols algebra over fields of characteristic 2 starting from braided vector spaces that are not of diagonal type, admit realizations as Yetter-Drinfeld modules over finite abelian groups and…
The classification, up to isomorphism, of two-dimensional (not necessarily commutative) Jordan algebras over algebraically closed fields and $\mathbb{R}$ is presented in terms of their matrices of structure constants.
The paper is devoted to classify nilpotent Jordan algebras of dimension up to five over an algebraically closed field of characteristic not 2. We obtained a list of 35 isolated non-isomorphic 5-dimensional nilpotent non-associative Jordan…
The paper is devoted to the description of the varieties of complex 5-dimensional nilpotent Jordan superalgebras. We find all representatives for the isomorphism classes, using the Jordan normal form, results of simultaneous matrix…
The exceptional Jordan algebra is the algebra of $3\times 3$ Hermitian matrices with octonionic entries. It is the only one from Jordan's algebraic formulation of quantum mechanics which is not equivalent to the conventional formulation of…
In this paper, we consider the Drinfeld double $\D$ of a $12$-dimensional Hopf algebra $\C$ over an algebraically closed field of characteristic zero whose coradical is not a subalgebra and describe its simple modules, projective covers of…
We continue the study of the Drinfeld double of the Jordan plane, denoted by $\mathcal D$ and introduced in arXiv:2002.02514. The simple finite-dimensional modules were computed in arXiv:2108.13849; it turns out that they factorize through…