A braided interpretation of fractional supersymmetry in higher dimensions

Abstract

A many variable q-calculus is introduced using the formalism of braided covector algebras. Its properties when certain of its deformation parameters are roots of unity are discussed in detail, and related to fractional supersymmetry. The special cases of two dimensional supersymmetry and fractional supersymmetry are developed in detail.

Hopf Algebra Approaches to Nonlinear Algebraic Equations

“…Majid called a pair (R, F ) a mixed Yang-Baxter pair if in addition both R and F are the Yang-Baxter operators; they are used in the construction of braided matrices (Lu 1994;Majid, 1995). But in some cases, the condition that R and F themselves are the Yang-Baxter operators is not necessary (Dunne, 1999;Majid, 1995;Street, 1998). …”

“…. , mg with the FRT-relation for n ¼ 2; that is The equations above occur in Dunne (1999), Lu (1994), Majid (1995) and Street (1998). In particular, when R is a solution of the Yang-Baxter equation, we get a cobraided bialgebra by taking F ¼ R. In fact, the FRT-relations (2.6) together with the conditions of the pre-braided bialgebra, in this case, induce X sðx ð1Þ y ð1Þ Þy ð2Þ x ð2Þ ¼ X sðx ð2Þ y ð2Þ Þx ð1Þ y ð1Þ ;…”

“…2, we restrict the FRT-construction to the homogeneous cases to give a Hopf algebra approach for solving the mixed Yang-Baxter type equations, particularly, for the Yang-Baxter equation and the Long equation. These types of mixed Yang-Baxter pairs are used to construct braided matrices (Lu, 1994;Majid, 1995), braided covector algebras (Dunne, 1999;Majid, 1995) and are related to the construction of tricycloid (Street, 1998). We explain also the Hopf algebra approach to solving D-equation, the Long equation, and the Yang-Baxter equation using their respective dimodule categories.…”

K-Fractional spin through Q-deformed (super)-algebras

“…q the Q-deformed bosonic oscillator decomposes into two independent oscillators, an (undeformed) boson and a k-fermion. An appealing question is to ask whether is possible to nd Q-deformed fermionic operators exhibiting a similar property of splitting to Q-deformed bosons, when the deformation parameter Q reduces to a root of unity q. T o answer this question, we consider the Q-fermionic oscillator algebra dened by ff ; f (18) we obtain by a direct calculation the following anticommutation relation fF + ; F g= 1 : (19) Moreover, we h a v e ( F ) 2 = 0 : (20) Thus we see that the Q-deformed fermion reproduce the conventional (ordinary) fermion. It should be noted that the Q-deformed fermions were used by H a y ashi [25] t o g i v e Q -fermionic representation of quantum groups U Q (X) where X is a nite Lie algebra of type A n , B n or D n .…”

Tannaka Duality and the FRT-Construction

“…In Section 2, we restrict Tannaka duality to an n-homogeneous category to give a Hopf algebra approach for solving the mixed Yang-Baxter type equations; those mixed Yang-Baxter type pairs are used to construct braided matrices [5,6], braided covector algebra [1,6] and are related to the construction of tricoycloid [9]. In Section 2, we restrict Tannaka duality to an n-homogeneous category to give a Hopf algebra approach for solving the mixed Yang-Baxter type equations; those mixed Yang-Baxter type pairs are used to construct braided matrices [5,6], braided covector algebra [1,6] and are related to the construction of tricoycloid [9].…”