Related papers: Recursive boson system in the Cuntz algebra ${\cal…
Bosons and fermions are described by using canonical generators of Cuntz algebras on any permutative representation. We show a fermionization of bosons which universally holds on any permutative representation of the Cuntz algebra ${\cal…
Bosons and fermions are described by using canonical generators of Cuntz algebras on any permutative representation. According to branching laws associated with these descriptions, a certain representation of the Cuntz algebra $\co{2}$…
Embeddings of the CAR (canonical anticommutation relations) algebra of fermions into the Cuntz algebra ${\cal O}_2$ (or ${\cal O}_{2d}$ more generally) are presented by using recursive constructions. As a typical example, an embedding of…
Previously, we have shown that the CAR algebra for fermions is embedded in the Cuntz algebra ${\cal O}_{2}$ in such a way that the generators are expressed in terms of polynomials in the canonical generators of the latter, and it coincides…
Based on an embedding formula of the CAR algebra into the Cuntz algebra ${\mathcal O}_{2^p}$, properties of the CAR algebra are studied in detail by restricting those of the Cuntz algebra. Various $\ast$-endomorphisms of the Cuntz algebra…
We show a correspondence between simple continued fraction expansions of irrational numbers and irreducible permutative representations of the Cuntz algebra ${\cal O}_{\infty}$. With respect to the correspondence, it is shown that the…
We give explicit expression of recurrency formulae of canonical realization for quantum enveloping algebras $U_{q}(sl(n+1,C))$. In these formulas the generators of the algebra $U_{q}(sl(n+1,C))$ are expressed by means of n-canonical q-boson…
We study the semigroup of Bogoliubov endomorphisms of the canonical commutation relations which give rise to representations of the Cuntz algebra $O_\infty$ on Fock space and describe the corresponding Cuntz algebra generators in detail.
One of the traditional ways of introducing bosons and fermions is through creation-annihilation algebras. Historically, these have been associated with emission and absorption processes at the quantum level and are characteristic of the…
A universality of deformed Heisenberg algebra involving the reflection operator is revealed. It is shown that in addition to the well-known infinite-dimensional representations related to parabosons, the algebra has also finite-dimensional…
The rings of symmetric polynomials form an inverse system whose limit, the ring of symmetric functions, is the model for the bosonic Fock space representation of the affine Lie algebra. We categorify this construction by considering an…
We present a unified theory for formal mathematical systems including recursive systems closely related to formal grammars, including the predicate calculus as well as a formal induction principle. We introduce recursive systems generating…
We bosonise the complex-boson realisations of the $W_\infty$ and $W_{1+\infty}$ algebras. We obtain nonlinear realisations of $W_\infty$ and $W_{1+\infty}$ in terms of a pair of fermions and a real scalar. By further bosonising the…
We prove that, for the quon algebra, which interpolates between the Bose and Fermi statistics and depends on a free parameter q,it is possible to build an su(2) irreducible representation. One of the consequences of this fact is that the…
The inhomogeneous single-, two- and three-boson realizations of the more general polynomial angular momentum algebra SU_n(2) are obtained from the Fock representations of SU_n(2) that corresponds to the indecomposable representation on the…
A one-parameter generalized fermion algebra ${\cal B}_{\kappa}(1)$ is introduced. The Fock representation is studied. The associated coherent states are constructed and the polynomial representation, in the Bargmann sense, is derived. A…
Under some hypotheses (symmetry, confluence), we enumerate all quadratically presented algebras, generated by creation and destruction operators, in which number operators exist. We show that these are algebras of bosons, fermions, their…
Usual quantum statistics is written in Fock space but it is not an algebraic theory. We show that at a deeper level it can be algebraically formalized defining the different statistics as (multi-mode) coherent states of the appropriate (but…
In this paper we describe a new family of algebras which in the case of n = 2 reduces to the Fermion algebra and in the limiting case of n tends to infinity reduces to the Boson algebra. These generalized algebras describe particles which…
Representations of the Cuntz algebra $\mathcal{O}_N$ are constructed from interval dynamical systems associated with slow continued fraction algorithms introduced by Giovanni Panti. Their irreducible decomposition formulas are characterized…