We describe the groups of automorphisms two generated free braided associative algebras with involutive diagonal braidings over a field characteristic ≠2. Depending on form braiding, five different automorphism arise as algebras. This list covers all three that in case quantum planes.

We provide a generalization of Mundici's equivalence between unital Abelian lattice-ordered groups and MV-algebras: the category commutative is equivalent to MV-monoidal algebras. Roughly speaking, structures we call are without unary operation $x \mapsto -x$. The primitive operations $+$, $\lor$, $\land$, $0$, $1$, $-1$. A prime example these $\mathbb{R}$, with obvious interpretation operation...

We establish categorical dualities between varieties of isotropic median algebras and suitable categories of operational and relational topological structures. We follow a general duality theory of B.A. Davey and H. Werner. The duality results are used to describe free isotropic median algebras. If the number of free generators is less than five, the description is detailed.

We describe a set of defining relations for automorphism groups of finitely generated free algebras of Nielsen-Schreier varieties. In particular, this gives a representation of the automorphism groups of free Lie algebras by generators and defining relations. Mathematics Subject Classification (2000): Primary 17A36, 17A50, 17B40; Secondary 17B01, 16W20.

Certain canonical resolutions are described for free associative and free Lie algebras in the category of non-associative algebras. These resolutions derive in both cases from geometric objects, which in turn reflect the combinatorics of suitable collections of leaf-labeled trees.

An embedding question for full (i.e. universal) amalgamated free products of C∗–algebras is raised, and solved in part. This is used to give a necessary and sufficient condition for the full amalgamated free product of finite dimensional C∗–algebras to be residually finite dimensional.

We investigate ways of representing ordered sets as algebras and how the order relation is reflected in the algebraic properties of the variety (equational class) generated by these algebras. In particular we consider two different but related methods for constructing an algebra with one binary operation from an arbitrary ordered set with a top element. The two varieties generated by all these ...

J.-L. Loday introduced the concept of coherent unit actions on a regular operad and showed that such actions give Hopf algebra structures on the free algebras. Hopf algebras obtained this way include the Hopf algebras of shuffles, quasi-shuffles and planar rooted trees. We characterize coherent unit actions on binary quadratic regular operads in terms of linear equations of the generators of th...

Nazarov [Naz96] introduced an infinite dimensional algebra, which he called the affine Wenzl algebra, in his study of the Brauer algebras. In this paper we study certain “cyclotomic quotients” of these algebras. We construct the irreducible representations of these algebras in the generic case and use this to show that these algebras are free of rank r(2n − 1)!! (when Ω is u– admissible). We ne...