Pérez-Izquierdo and Shestakov recently extended the PBW theorem to Malcev algebras. It follows from their construction that for any Malcev algebra M over a field of characteristic 6= 2, 3 there is a representation of the universal nonassociative enveloping algebra U(M) by linear operators on the polynomial algebra P (M). For the nilpotent non-Lie Malcev algebra M of dimension 5, we use this rep...

Dirichlet branes are objects whose transverse coordinates in space are matrix– valued functions. This leads to considering a matrix algebra or, more generally, a Lie algebra, as the classical phase space of a certain dynamics where the multiplication of coordinates, being given by matrix multiplication, is nonabelian. Further quantising this dynamics by means of a ⋆–product introduces noncommut...

We develop notions of a representation of a topological graph E and of a covariant representation of a topological graph E which do not require the machinery of C∗-correspondences and Cuntz– Pimsner algebras. We show that the C∗-algebra generated by a universal representation of E is isomorphic to the Toeplitz algebra of Katsura’s topological-graph bimodule, and that the C∗-algebra generated by...

following the categorical approach to universal algebra through algebraic theories, proposed by f.~w.~lawvere in his phd thesis, this paper aims at introducing a similar setting for general topology. the cornerstone of the new framework is the notion of emph{categorically-algebraic} (emph{catalg}) emph{topological theory}, whose models induce a category of topological structures. we introduce t...

We propose a notion of a quantum universal enveloping algebra for an arbitrary Lie algebra defined by generators and relations which is based on the quantum Lie operation concept. This enveloping algebra has a PBW basis that admits the Kashiwara crystalization. We describe all skew primitive elements of the quantum universal enveloping algebra for the classical nilpotent algebras of the infinit...

Introduction 2 1. Lie algebras: recollections 3 1.1. The basics 3 1.2. Scaling the structure 3 1.3. Filtrations 4 1.4. The Chevalley complex 4 1.5. The functor of primitives 6 1.6. The enhanced adjunction 6 1.7. The symmetric Hopf algebra 8 2. Looping Lie algebras 9 2.1. Group-Lie algebras 10 2.2. Forgetting to group structure 10 2.3. Chevalley complex of group-Lie algebras 11 2.4. Chevalley co...

Universal algebra has underpinned the modern research in formal logic since Garrett Birkoff’s pioneering work in the 1930’s and 1940’s. Since the early 1970’s, the entanglement of logic and algebra has been successfully exploited in many areas of computer science from the theory of computation to Artificial Intelligence (AI). The scientific outcome of the interplay between logic and universal a...

Let g be a Borcherds-Bozec algebra, U ( ) its universal enveloping algebra and q the corresponding quantum algebra. We show that classical limit of is isomorphic to as Hopf algebras . Thus, can regarded deformation also provide explicit formulas for commutation relations among generators