Results 1 
6 of
6
Categorified algebra and quantum mechanics
, 2006
"... Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding a combinatorial model for some mathematical entity is a par ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
Interest in combinatorial interpretations of mathematical entities stems from the convenience of the concrete models they provide. Finding a bijective proof of a seemingly obscure identity can reveal unsuspected significance to it. Finding a combinatorial model for some mathematical entity is a particular instance of the process called “categorification”. Examples include the interpretation of as the Burnside rig of the category of finite sets with product and coproduct, and the interpretation of [x] as the category of combinatorial species. This has interesting applications to quantum mechanics, and in particular the quantum harmonic oscillator, via Joyal’s “species”, a new generalization called “stuff types”, and operators between these, which can be represented as rudimentary Feynman diagrams for the oscillator. In quantum mechanics, we want to represent states in an algebra over the complex numbers, and also want our Feynman diagrams to carry more structure than these “stuff operators ” can do, and these turn out to be closely related. We will show how to construct a combinatorial model for the quantum harmonic
Higher dimensional algebra VII: Groupoidification
, 2010
"... Groupoidification is a form of categorification in which vector spaces are replaced by groupoids and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector space ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
Groupoidification is a form of categorification in which vector spaces are replaced by groupoids and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normalordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang–Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field Fq. The third application is to Hall algebras. We explain how the standard construction of the Hall algebra from the category of Fq representations of a simplylaced quiver can be seen as an example of degroupoidification. This in turn provides a new way to categorify—or more precisely, groupoidify—the positive part of the quantum group associated to the quiver.
Groupoidification Made Easy
, 2008
"... Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector space ..."
Abstract

Cited by 7 (0 self)
 Add to MetaCart
(Show Context)
Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present two applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normalordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang– Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field Fq.
Generalised Brownian Motion and Second Quantisation
, 2000
"... A new approach to the generalised Brownian motion introduced by M. ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
A new approach to the generalised Brownian motion introduced by M.
Functors of White Noise Associated to Characters of the Infinite Symmetric Group
, 2001
"... The characters ; of the in nite symmetric group are extended to multiplicative positive de nite functions t; on pair partitions by using an explicit representation due to Versik and Kerov. The von Neumann algebra ; (K) generated by the elds !; (f) with f in an in nite dimensional real Hilbert space ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
The characters ; of the in nite symmetric group are extended to multiplicative positive de nite functions t; on pair partitions by using an explicit representation due to Versik and Kerov. The von Neumann algebra ; (K) generated by the elds !; (f) with f in an in nite dimensional real Hilbert space K is in nite and the vacuum vector is not separating. For a family tN depending on an integer N < f j): The algebras N (` 2 R (Z)) are type I1 factors. Functors of white noise N are constructed and proved to be nonequivalent for dierent values of N . 1
BessisMoussaVillani conjecture and generalized Gaussian random variables
, 2007
"... In this paper we give the solution of BessisMoussaVillani conjecture (BMV) conjecture for the generalized Gaussian random variables G(f) = a(f) + a∗(f), where f is in the real Hilbert space H. The main examples of generalized Gaussian random variables are qGaussian random variables, (−1 ≤ q ≤ 1) ..."
Abstract
 Add to MetaCart
(Show Context)
In this paper we give the solution of BessisMoussaVillani conjecture (BMV) conjecture for the generalized Gaussian random variables G(f) = a(f) + a∗(f), where f is in the real Hilbert space H. The main examples of generalized Gaussian random variables are qGaussian random variables, (−1 ≤ q ≤ 1), related to q −CCR relation and others commutation relations. We will prove that (BMV) conjecture is true for all operators A = G(f), B = G(g); i.e. we will show that the function F (x) = tr(exp(A+ ixB)) is positive definite function on the real line. The case q = 0,i.e. when G(f) are the free Gaussian (Wigner) random variables and the operators A and B are free with respect to the vacuum trace was proved by M.Fannes and D.Petz [23]. 1 Generalized Gaussian Random Variable. Generalized Gaussian random variables, G(f) were introduced in our paper with R.Speicher [16], where the main example was coming from the qCCR relation for q ∈ [−1, 1]: a(f)a∗(g) − qa∗(g)a(f) =< f, g> I,