Some results on L-dendriform algebras

We introduce a notion of L-dendriform algebra due to several different motivations. L-dendriform algebras are regarded as the underlying algebraic structures of pseudo-Hessian structures on Lie groups and the algebraic structures behind the $\mathcal O$-operators of pre-Lie algebras and the related $S$-equation. As a direct consequence, they provide some explicit solutions of $S$-equation in certain pre-Lie algebras constructed from L-dendriform algebras. They also fit into a bigger framework as Lie algebraic analogues of dendriform algebras. Moreover, we introduce a notion of $\mathcal O$-operator of an L-dendriform algebra which gives an algebraic equation regarded as an analogue of the classical Yang-Baxter equation in a Lie algebra.Comment: 15 page

Nonabelian KP hierarchy with Moyal algebraic coefficients

A higher dimensional analogue of the KP hierarchy is presented. Fundamental constituents of the theory are pseudo-differential operators with Moyal algebraic coefficients. The new hierarchy can be interpreted as large-$N$ limit of multi-component (\gl(N) symmetric) KP hierarchies. Actually, two different hierarchies are constructed. The first hierarchy consists of commuting flows and may be thought of as a straightforward extension of the ordinary and multi-component KP hierarchies. The second one is a hierarchy of noncommuting flows, and related to Moyal algebraic deformations of selfdual gravity. Both hierarchies turn out to possess quasi-classical limit, replacing Moyal algebraic structures by Poisson algebraic structures. The language of W-infinity algebras provides a unified point of view to these results.Comment: 37 pages, Kyoto University KUCP-0062/9

Invariant local twistor calculus for quaternionic structures and related geometries

New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalisations as well as 4-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all invariants and invariant operators arise from these universal operators and that they may be used to reduce all invariants problems to corresponding algebraic problems involving homomorphisms between modules of certain parabolic subgroups of Lie groups. Explicit application of the operators is illustrated by the construction of all non-standard operators between exterior forms on a large class of the geometries which includes the quaternionic structures.Comment: 44 page

Reciprocal transformations of Hamiltonian operators of hydrodynamic type: nonlocal Hamiltonian formalism for linearly degenerate systems

Reciprocal transformations of Hamiltonian operators of hydrodynamic type are investigated. The transformed operators are generally nonlocal, possessing a number of remarkable algebraic and differential-geometric properties. We apply our results to linearly degenerate semi-Hamiltonian systems in Riemann invariants. Since all such systems are linearizable by appropriate (generalized) reciprocal transformations, our formulae provide an infinity of mutually compatible nonlocal Hamiltonian structures, explicitly parametrized by arbitrary functions of one variable.Comment: 26 page

An algebraic generalization of Kripke structures

The Kripke semantics of classical propositional normal modal logic is made algebraic via an embedding of Kripke structures into the larger class of pointed stably supported quantales. This algebraic semantics subsumes the traditional algebraic semantics based on lattices with unary operators, and it suggests natural interpretations of modal logic, of possible interest in the applications, in structures that arise in geometry and analysis, such as foliated manifolds and operator algebras, via topological groupoids and inverse semigroups. We study completeness properties of the quantale based semantics for the systems K, T, K4, S4, and S5, in particular obtaining an axiomatization for S5 which does not use negation or the modal necessity operator. As additional examples we describe intuitionistic propositional modal logic, the logic of programs PDL, and the ramified temporal logic CTL.Comment: 39 page

Reduced pre-Lie algebraic structures, the weak and weakly deformed Balinsky-Novikov type symmetry algebras and related Hamiltonian operators

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky–Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler–Kostant–Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky–Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky–Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky–Novikov algebras, including their fermionic version and related multiplicative and Lie structures

Hierarchy of Dirac, Pauli and Klein-Gordon conserved operators in Taub-NUT background

The algebra of conserved observables of the SO(4,1) gauge-invariant theory of the Dirac fermions in the external field of the Kaluza-Klein monopole is investigated. It is shown that the Dirac conserved operators have physical parts associated with Pauli operators that are also conserved in the sense of the Klein-Gordon theory. In this way one gets simpler methods of analyzing the properties of the conserved Dirac operators and their main algebraic structures including the representations of dynamical algebras governing the Dirac quantum modes.Comment: 16 pages, latex, no figure

Algebraic closures and their variations

We study possibilities for algebraic closures, differences between definable and algebraic closures in first-order structures, and variations of these closures with respect to the bounds of cardinalities of definable sets and given sets of formulae. Characteristics for these possibilities and differences are introduced and described. These characteristics are studied for some natural classes of theories. Besides algebraic closure operators with respect to sets of formulae are introduced and studied. Semilattices and lattices for families of these operators are introduced and characteristics of these structures are described