On the q-meromorphic Weyl algebra
We introduce a q-analogue MW_q for the meromorphic Weyl algebra, and study the normalization problem and the symmetric powers sym^n(MW_q) for such algebra from a combinatorial viewpoint.
math.QA↗
arXiv subjects
Rafael Diaz
Publications and source records attributed to Rafael Diaz.
On the Gaussian q-Distribution
We present a study of the Gaussian q-measure introduced by Diaz and Teruel from a probabilistic and from a combinatorial viewpoint. A main motivation for the introduction of the Gaussian q-measure is that its moments are exactly the q-analogues of the double factorial numbers. We show that the Gaussian q-measure interpolates between the uniform measure on the interval [-1,1] and the Gaussian measure on the real line.
math.PR↗
Super, quantum and non-commutative species
We introduce an approach to the categorification of rings, via the notion of distributive categories with negative objects, and use it to lay down categorical foundations for the study of super, quantum and non-commutative combinatorics. Via the usual duality between algebra and geometry, these constructions provide categorifications for various types of affine spaces, thus our works may be regarded as a starting point towards the construction of a categorical geometry.
math.CT↗
Rota-Baxter Categories
We introduce Rota-Baxter categories and construct examples of such structures.
math.CT↗
Homological Quantum Field Theory
We show that the space of chains of smooth maps from spheres into a fixed compact oriented manifold has a natural structure of a transversal $d$-algebra. We construct a structure of transversal 1-category on the space of chains of maps from a suspension space $S(Y)$, with certain boundary restrictions, into a fixed compact oriented manifold. We define homological quantum field theories \HL and construct several examples of such structures. Our definition is based on the notions of string topology of Chas and Sullivan, and homotopy quantum field theories of Turaev.
math.KT↗
Invariants from classical field theory
We introduce a method that generates invariant functions from perturbative classical field theories depending on external parameters. Applying our methods to several field theories such as abelian BF, Chern-Simons and 2-dimensional Yang-Mills theory, we obtain, respectively, the linking number for embedded submanifolds in compact varieties, the Gauss' and the second Milnor's invariant for links in S^3, and invariants under area-preserving diffeomorphisms for configurations of immersed planar curves.
math-ph↗
Compositional Bernoulli numbers
We define and study the combinatorial properties of compositional Bernoulli numbers and polynomials within the framework of rational combinatorics.
math.CO↗
On the (3,N) Maurer-Cartan equation
Deformations of the 3-differential of 3-differential graded algebras are controlled by the (3,N) Maurer-Cartan equation. We find explicit formulae for the coefficients appearing in that equation, introduce new geometric examples of N-differential graded algebras, and use these results to study N Lie algebroids.
math.QA↗
Rational Combinatorics
We propose a categorical setting for the study of the combinatorics of rational numbers. We find combinatorial interpretation for the Bernoulli and Euler numbers and polynomials.
math.CO↗
Membrane Topology
We construct membrane homology groups $\h(M)$ associated with each compact connected oriented smooth manifold, and show that $\h(M)$ is matrix graded algebra.
math.GT↗
An example of Feynman-Jackson integral
We review the construction of a q-analogue of the Gaussian measure. We apply that construction to obtain a q-analogue of Feynman integrals and to compute explicitly an example of such integrals.
math-ph↗
$A^{N}_{\infty}$-algebras
We study higher depth algebras. We introduce several examples of such structures starting from the notion of $N$-differential graded algebras and build up to the concept of $A_{\infty}^N$-algebras.
math.QA↗
Symmetric Boolean Algebras
We define Boolean algebras in the linear context and study its symmetric powers. We give explicit formulae for products in symmetric Boolean algebras of various dimensions. We formulate symmetric forms of the inclusion-exclusion principle.
math.CO↗
Homology and manifolds with corners
We define a model for the homology of manifolds and use it to describe the intersection product on the homology of compact oriented manifolds and to define homological quantum field theories which generalizes the notions of string topology introduced by Chas and Sullivan and homotopy quantum field theories introduced by Turaev.
math.GT↗
Homological matrices
We define homological matrices, construct examples of one-dimension restricted homological quantum field theories, and show a relationship between the two theories.
math.KT↗
Feynman-Jackson integrals
We introduce perturbative Feynman integrals in the context of q-calculus generalizing the Gaussian q-integrals introduced by Diaz and Teruel. We provide analytic as well as combinatorial interpretations for the Feynman-Jackson integrals.
math.QA↗
Graphical introduction to classical Lie algebras
We develop a graphical notation to introduce classical Lie algebras. Although this paper deals with well-known results, our pictorial point of view is slightly different to the traditional one. Our graphical notation is fairly elementary and easy to handle and, thus provides an effective tool for computations with classical Lie algebras. More over, it may be regarded as a first and foundational step in the process of uncovering the categorical meaning of Lie algebras.
math.RT↗
On hypergeometric functions and Pochhammer $k$-symbol
We introduce the $k$-generalized gamma function $Γ_k$, beta function $B_k$, and Pochhammer $k$-symbol $(x)_{n,k}$. We prove several identities generalizing those satisfied by the classical gamma function, beta function and Pochhammer symbol. We provided integral representation for the $Γ_k$ and $B_k$ functions.
math.CA↗