Related papers: Simple and sub-directly irreducible double Boolean…
Recently, Brzezi\'nski, Koenig and K\"ulshammer have introduced the notion of normal exact Borel subalgebra of a quasihereditary algebra. They have shown that there exists a one-to-one correspondence between normal directed bocses and…
We introduce the notion of $\lambda$-double Lie algebra, which coincides with usual double Lie algebra when $\lambda = 0$. We state that every $\lambda$-double Lie algebra for $\lambda\neq0$ provides the structure of modified double Poisson…
A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but subclasses have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties.…
A formal context consists of objects, properties, and the incidence relation between them. Various notions of concepts defined with respect to formal contexts and their associated algebraic structures have been studied extensively,…
Methods of solving big Boolean equations can be broadly classified as algebraic, tabular, numerical and map methods. The most prominent among these classes are the algebraic and map methods. This paper surveys and compares these two types…
We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different…
A self-dual algebras is one isomorphic as a module to the opposite of its dual; a quasi self-dual algebra is one whose cohomology with coefficients in itself is isomorphic to that with coefficients in the opposite of its dual. For these…
We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense) under the action of the dual pair and…
We construct the unique two-parameter vertex algebra which is freely generated of type ${\mathcal W}(2,4,6,\dots)$, and generated by the weights $2$ and $4$ fields. Subject to some mild constraints, all vertex algebras of type ${\mathcal…
We study representations of the double affine Lie algebra associated to a simple Lie algebra. We construct a family of indecomposable integrable representations and identify their irreducible quotients. We also give a condition for the…
Green and Schroll give an easy criterion for a monomial algebra $A$ to be quasi-hereditary with respect to some partial order $\leq_A$. A natural follow-up question is under which conditions a monomial quasi-hereditary algebra $(A, \leq_A)$…
In this paper, we construct a bar-cobar adjunction and a Koszul duality theory for protoperads, which are an operadic type notion encoding faithfully some categories of bialgebras with diagonal symmetries, like double Lie algebras (DLie).…
For a quantum group, we study those right coideal subalgebras, for which all irreducible representations are one-dimensional. If a right coideal subalgebra is maximal with this property, then we call it a Borel subalgebra. Besides the…
In the present paper, we endow the logics of topological quasi Boolean algebras, topological quasi Boolean algebras 5, intermediate algebras of types 1-3, and pre-rough algebras with proper multi-type display calculi which are sound,…
New families of eight-dimensional real division algebras with large derivation algebra are presented: We generalize the classical Cayley-Dickson doubling process starting with a unital algebra with involution over a field F by allowing the…
A class of algebras called down-up algebras was introduced by G. Benkart and T. Roby. We classify the finite dimensional simple modules over Noetherian down-up algebras and show that in some cases every finite dimensional module is…
Our main result is that any topological algebra based on a Boolean space is the extended Stone dual space of a certain associated Boolean algebra with additional operations. A particular case of this result is that the profinite completion…
Let $\pi: Y\rightarrow X$ be a continuous surjection between compact Hausdorff spaces $Y$ and $X$ which is irreducible in the sense that if $F\subsetneq Y$ is closed, then $\pi(F)\neq X$. We exhibit isomorphisms between various Boolean…
Scaled Boolean algebras are a category of mathematical objects that arose from attempts to understand why the conventional rules of probability should hold when probabilities are construed, not as frequencies or proportions or the like, but…
We will study evolution algebras $A$ which are free modules of dimension $2$ over domains. Furthermore, we will assume that these algebras are perfect, that is $A^2=A$. We start by making some general considerations about algebras over…