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Register a new userConvolution algebras for Relational Groupoids and Reduction
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We introduce the notions of relational groupoids and relational convolution algebras. We provide various examples arising from the group algebra of a group $G$ and a given normal subgroup $H$. We also give conditions for the existence of a Haar system of measures on a relational groupoid compatible with the convolution, and we prove a reduction theorem that recovers the usual convolution of a Lie groupoid.
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We introduce the notion of Haantjes algebra: It consists of an assignment of a family of operator fields on a differentiable manifold, each of them with vanishing Haantjes torsion. They are also required to satisfy suitable compatibility conditions. Haantjes algebras naturally generalize several known interesting geometric structures, arising in Riemannian geometry and in the theory of integrable systems. At the same time, as we will show, they play a crucial role in the theory of diagonalization of operators on differentiable manifolds. Assuming that the operators of a Haantjes algebra are semisimple and commute, we shall prove that there exists a set of local coordinates where all operators can be diagonalized simultaneously. Moreover, in the general, non-semisimple case, they acquire simultaneously, in a suitable local chart, a block-diagonal form.
Let $H$ be the quaternion algebra. Let $g$ be a complex Lie algebra and let $U(g)$ be the enveloping algebra of $g$. We define a Lie algebra structure on the tensor product space of $H$ and $U(g)$, and obtain the quaternification $g^H$ of $g$. Let $S^3g^H$ be the set of $g^H$-valued smooth mappings over $S^3$. The Lie algebra structure on $S^3g^H$ is induced naturally from that of $g^H$. On $S^3$ exists the space of Laurent polynomial spinors spanned by a complete orthogonal system of eigen spinors of the tangential Dirac operator on $S^3$. Tensoring $U(g)$ we have the space of $U(g)$-valued Laurent polynomial spinors, which is a Lie subalgebra of $S^3g^H$. We introduce a 2-cocycle on the space of $U(g)$-valued Laurent polynomial spinors by the aid of a tangential vector field on $S^3$. Then we have the corresponding central extension $hat g(a)$ of the Lie algebra of $U(g)$-valued Laurent polynomial spinors. Finally we have the a Lie algebra $hat g=hat g(a)+Cd$ which is obtained by adding to $hat g(a)$ a derivation $d$ which acts on $hat g(a)$ as the radial derivation. When $g$ is a simple Lie algebra with its Cartan subalgebra $h$, We shall investigate the weight space decomposition of $(hat g, ad(hat h))$, where $hat h=h+Ca+Cd$ . The previo
We study admissible and equivalence point transformations between generalized multidimensional nonlinear Schrodinger equations and classify Lie symmetries of such equations. We begin with a wide superclass of Schrodinger-type equations, which includes all the other classes considered in the paper. Showing that this superclass is not normalized, we partition it into two disjoint normalized subclasses, which are not related by point transformations. Further constraining the arbitrary elements of the superclass, we construct a hierarchy of normalized classes of Schrodinger-type equations. This gives us an appropriate normalized superclass for the non-normalized class of multidimensional nonlinear Schrodinger equations with potentials and modular nonlinearities and allows us to partition the latter class into three families of normalized subclasses. After a preliminary study of Lie symmetries of nonlinear Schrodinger equations with potentials and modular nonlinearities for an arbitrary space dimension, we exhaustively solve the group classification problem for such equations in space dimension two.
The purpose of this paper is to define the concept of multi-Dirac structures and to describe their role in the description of classical field theories. We begin by outlining a variational principle for field theories, referred to as the Hamilton-Pontryagin principle, and we show that the resulting field equations are the Euler-Lagrange equations in implicit form. Secondly, we introduce multi-Dirac structures as a graded analog of standard Dirac structures, and we show that the graph of a multisymplectic form determines a multi-Dirac structure. We then discuss the role of multi-Dirac structures in field theory by showing that the implicit field equations obtained from the Hamilton-Pontryagin principle can be described intrinsically using multi-Dirac structures. Furthermore, we show that any multi-Dirac structure naturally gives rise to a multi-Poisson bracket. We treat the case of field theories with nonholonomic constraints, showing that the integrability of the constraints is equivalent to the integrability of the underlying multi-Dirac structure. We finish with a number of illustrative examples, including time-dependent mechanics, nonlinear scalar fields and the electromagnetic field.
Dimensional reduction occurs when the critical behavior of one system can be related to that of another system in a lower dimension. We show that this occurs for directed branched polymers (DBP) by giving an exact relationship between DBP models in D+1 dimensions and repulsive gases at negative activity in D dimensions. This implies relations between exponents of the two models: $gamma(D+1)=alpha(D)$ (the exponent describing the singularity of the pressure), and $ u_{perp}(D+1)= u(D)$ (the correlation length exponent of the repulsive gas). It also leads to the relation $theta(D+1)=1+sigma(D)$, where $sigma(D)$ is the Yang-Lee edge exponent. We derive exact expressions for the number of DBP of size N in two dimensions.
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