At present an algebra of strongly interacting fields is unknown. In this paper it is assumed that the operators of strongly nonlinear field can form a non-associative algebra. It is shown that such algebra can be described as an algebra of some pairs. The comparison of presented techniques with the Green’s functions method in the superconductivity theory is made. A possible application to the Q...

We introduce a notion of “hopfish algebra” structure on an associative algebra, allowing the structure morphisms (coproduct, counit, antipode) to be bimodules rather than algebra homomorphisms. We prove that quasi-Hopf algebras are examples of hopfish algebras. We find that a hopfish structure on the algebra of functions on a finite set G is closely related to a “hypergroupoid” structure on G. ...

We classify all the pairs of a commutative associative algebra with an identity element and its finite-dimensional commutative locally-finite derivation subalgebra such that the commutative associative algebra is derivation-simple with respect to the derivation subalgebra over an algebraically closed field with characteristic 0. Such pairs are the fundamental ingredients for constructing genera...

We study the residuated basic logic (RBL) of residuated basic algebra in which the basic implication of Visser’s basic propositional logic (BPL) is interpreted as the right residual of a non-associative binary operator · (product). We develop an algebraic system SRBL of residuated basic algebra by which we show that RBL is a conservative extension of BPL. We present the sequent formalization LR...

Recently, relative Rota–Baxter (Lie/associative) algebras are extensively studied in the literature from cohomological points of view. In this paper, we consider Leibniz (rRB algebras) as object our study. We construct an L∞-algebra that characterizes rRB its Maurer–Cartan elements. Then define representations algebra and introduce cohomology with coefficients a representation. As applications ...

Given a partial action α of a group G on an associative algebra A, we consider the crossed product A α G. Using the algebras of multipliers, we generalize a result of Exel (1997) on the associativity of A α G obtained in the context of C∗-algebras. In particular, we prove that A αG is associative, provided that A is semiprime. We also give a criterion for the existence of a global extension of ...

We prove that, if A and B are complete normed non-associative algebras, and if B is strongly semisimple and algebraic, then dense range homomorphisms from A to B are continuous. As a consequence, we obtain that homomorphisms from complete normed algebras into a complete normed power-associative algebraic algebra B are continuous if and only if B has no nonzero elements with zero square. We also...

One of the four well-known series of simple Lie algebras of Cartan type is the series of Lie algebras of Special type, which are divergence-free Lie algebras associated with polynomial algebras and the operators of taking partial derivatives, connected with volume-preserving diffeomorphisms. In this paper, we determine the structure space of the divergence-free Lie algebras associated with pair...

For all classical groups (and for their analogs in infinite dimension or over general base fields or rings) we construct certain contractions, called homotopes. The construction is geometric, using as ingredient involutions of associative geometries. We prove that, under suitable assumptions, the groups and their homotopes have a canonical semigroup completion. Introduction: The classical group...