Variants of bosonization in parabosonic algebra: the Hopf and super-Hopf structures in parabosonic algebra

Abstract

Parabosonic algebra in finite or infinite degrees of freedom is considered as a Z 2 -graded associative algebra, and is shown to be a Z 2 -graded (or: super) Hopf algebra. The super-Hopf algebraic structure of the parabosonic algebra is established directly without appealing to its relation to the osp(1/2n) Lie superalgebraic structure. The notion of super-Hopf algebra is equivalently described as a Hopf algebra in the braided monoidal category CZ 2 M. The bosonisation technique for switching a Hopf algebra in the braided monoidal category H M (where H is a quasitriangular Hopf algebra) into an ordinary Hopf algebra is reviewed. In this paper we prove that for the parabosonic algebra P B , beyond the application of the bosonisation technique to the original super-Hopf algebra, a bosonisation-like construction is also achieved using two operators, related to the parabosonic total number operator. Both techniques switch the same super-Hopf algebra P B to an ordinary Hopf algebra, producing thus two different variants of P B , with ordinary Hopf structure.

Gradings, Braidings, Representations, Paraparticles: Some Open Problems

“…The results of the previous part of the project (i.e., Section 3.) are expected to lead us in suitable choices for the grading and the braiding of s W and as W (in the same manner that the results of [69][70][71] led us to the use of odd-bosons in [103]). Finally it is worth mentioning, that the computational problem we expect to reveal here is the development of a suitable multinomial theorem mixing commuting and anticommuting variables.…”

“…In [103], we take advantage of the super-Hopf structure of P B which has been extensively studied in [ [69][70][71], and based on it, we develop a ''braided interpretation'' of the Green ansatz for parabosons. We further develop a method, for employing this braided interpretation in order to construct analytic expressions for the matrix elements of the Fock-like representations of P B .…”

“…For each of the above cases, we intend to explicitly compute: (a) The group action (i.e., the grading); (b) The braiding (i.e., the family of isomorphisms), the commutation factor (i.e., the bicharacter or equiv: the color function), (c) The (quasi)triangular structure (i.e., the R -matrix) of the corresponding group Hopf algebra.  Finally, in each of the above cases we intend to apply bosonization [48,72] or bosonization-like techniques (in the sense we have done so in [69][70][71]) to obtain ordinary Hopf structures (with no grading and with trivial braiding) with equivalent representation theories.…”

“…H is simultaneously an algebra, a coalgebra and a G -module, all its structure functions (multiplication, comultiplication, unity, counity and antipode) are G -module morphisms. The investigation of such structures for the case of the paraparticle algebras has been an old issue: The "free" Parafermionic P F and parabosonic P B algebras have been shown to be (see the discussion in the introduction, in Section 2 and also [69,70] for a review) isomorphic to the Universal Enveloping Algebra (UEA) of a Lie algebra and a Lie superalgebra (or: 2 -graded Lie algebra) respectively, while the Relative Parabose set algebra P BF has been shown [14,15] to be isomorphic to the UEA of a ( 2 2 )-graded Lie algebra. At the same time the Relative Parafermi set algebra P FB has been shown [12,13] to be isomorphic to the UEA of a Lie superalgebra.…”

“…At the same time the Relative Parafermi set algebra P FB has been shown [12,13] to be isomorphic to the UEA of a Lie superalgebra. In [69,71] we have studied the case of P B , and we establish its braided group structure (here: 2 -graded Hopf structure) independently of its 2 -graded Lie structure.…”