Schematic su(2)⊕ h3 interaction Hamiltonians, where su(2) plays the role of the pseudo-spin algebra of fermion operators and h3 is the Heisenberg algebra for bosons, are shown to be closely related to certain nonlinear models defined on a single quantum algebra suq(2) of quasifermions. In particular, suq(2) analogues of the Da ProvidenciaSchütte and extended Lipkin models are presented. The con...

Following our approach to metric Lie algebras developed in a previous paper we propose a way of understanding pseudo-Riemannian symmetric spaces which are not semi-simple. We introduce cohomology sets (called quadratic cohomology) associated with orthogonal modules of Lie algebras with involution. Then we construct a functorial assignment which sends a pseudo-Riemannian symmetric space M to a t...

It is well-known that the representation of several classes of residuated lattices involves lattice-ordered groups. An often applicable method to determine the representing group (or groups) from a residuated lattice is based on partial algebras: the monoidal operation is restricted to those pairs which fulfil a certain extremality condition, and else left undefined. The subsequent construction...

Let R be a commutative Noetherian Q-algebra (Q is the field of rational numbers). Let δ be a derivation of R and σ be an automorphism of R. Then we prove the following: 1. If R is a Pseudo-valuation ring, then R[x, δ] is also a Pseudo-valuation ring. 2. If R is a divided ring, then R[x, δ] is also a divided ring. 3. If R is a Pseudo-valuation ring, thenR[x, x−1, σ] is also a Pseudo-valuation ri...

Forest algebras are defined for investigating languages of forests [ordered sequences] of unranked trees, where a node may have more than two [ordered] successors [9]. We show that several parameters on forests can be realized as forest algebra homomorphisms from the free forest algebra into algebras which retain the equational axioms of forest algebras. This includes the number of nodes, the n...

In this note, we introduce a class of algebras that are in some sense related to conformal algebras. This class (called TC-algebras) includes Weyl algebras and some of their (associative and Lie) subalgebras. By a conformal algebra we generally mean what is known as H-pseudo-algebra over the polynomial Hopf algebra H = k[T1, . . . , Tn]. Some recent results in structure theory of conformal alge...

This article is a further contribution to our research [1] into a class of infinite-dimensional Lie algebras L∞(N+, N−) generalizing the standardW∞ algebra, viewed as a tensor operator algebra of SU(1, 1) in a group-theoretic framework. Here we interpret L∞(N+, N−) either as infinite continuations of pseudo-unitary symmetries or as “higher-U(N+, N−)spin extensions” of the diffeomorphism algebra...

A simple axiomatic characterization of the noncommutative Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every quotient Itô algebra has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Lévy (Poisson) algebra. In particular,...

This article is a further contribution to our research [1] into a class of infinite-dimensional Lie algebras L∞(N+, N−) generalizing the standard W∞ algebra, viewed as a tensor operator algebra of SU(1, 1) in a group-theoretic framework. Here we interpret L∞(N+, N−) either as a infinite continuation of pseudo-unitary symmetries U(N+, N−), or as a “higher-U(N+, N−)-spin extension” of the diffeom...

We survey a recent extension of the moving frames method for infinite-dimensional Lie pseudo-groups. Applications include a new, direct approach to the construction of Maurer–Cartan forms and their structure equations for pseudogroups, and new algorithms, based on constructive commutative algebra, for uncovering the structure of the algebra of differential invariants for pseudogroup actions.