The Real Forms of the Fractional Supergroup SL(2,C)

: The real forms of complex groups (or algebras) are important in physics and mathematics. The Lie group SL ( 2, C ) is one of these important groups. There are real forms of the classical Lie group SL ( 2, C ) and the quantum group SL ( 2, C ) in the literature. Inspired by this, in our study, we obtain the real forms of the fractional supergroups shown with A N 3 ( SL ( 2, C )) , for the non-trivial N = 1 and N = 2 cases, that is, the real forms of the fractional supergroups A 13 ( SL ( 2, C )) and A 23 ( SL ( 2, C )) .


Introduction
Lie groups and Lie algebras are very important in mathematics and physics. In the field of mathematics, since every Lie group is an analytic manifold and every Lie algebra is a vector space that is tangent to the unit in the manifold, then every innovation in these groups (or algebra) contributes to differential geometry [1][2][3].
Lie symmetry methods are widely used in the solution of various differential equations that constitute deterministic models of mechanics, engineering, physics, finance and many other fields [4][5][6]. For example, using Lie symmetry methods in finance, new solutions for stochastic differential equations and stochastic processes have been developed [7,8]. In particle physics or quantum field theories, finite dimensional Lie algebras have been used to explain space-time symmetries and interactions [9].
In addition, super and fractional supersymmetries are important in relativistic quantum field theory [10,11]. The real forms of Lie groups (algebras) such as the Lie group SL(2, C) (or the Lie algebra sl(2, C)) also have important and wide application [1]. Every g real Lie algebra corresponds to a complex Lie algebra g c with the same base and the same structure constant. At the same time, a complex Lie algebra can transition to a real Lie algebra. The process of going from a complex Lie algebra g c to a real Lie algebra g is called realization. For example, the Lie groups SU(2), SU(1, 1) and SL(2, R) (or Lie algebras su(2), su(1, 1) and sl(2, R)), which are the real forms of the classical Lie group SL(2, C) (or classical Lie algebra sl(2, C)), have an important place in the literature [1].
Similarly, there are real forms of the quantum Lie group SL q (2, C) (or quantum Lie algebra sl q (2, C)) [2] such as the Lie group SU(2) and the quantum group SU q (2), which are very important compact groups in mathematical physics. Hence, many applications exist. For example, in paper [12], a theoretical group approach to generalized oscillator algebra A k is proposed, which is defined by the compact group SU(2) (for the Morse system) for the case k < 0, and SU(1, 1) (for the Harmonic oscillator) for the case k > 0. In addition, phase operators and phase states are introduced in the framework of the SU(2) and SU(1, 1) groups in [13].
One of the most important advantages of this approach is that it is an algebraic approach. Thus, it will always be possible and easy to move from algebra to group or from group to algebra. In our study, we obtain the fractional supergroups SU(2), SU(1, 1) and SL(2, R), which are the real forms of the fractional supergroup SL(2, C), in Hopf algebra formalism based on the permutation group S 3 .
In this context, in the second part, after defining the fractional supergroup by giving some useful definitions, we show with an example that the star operation is consistent with the Hopf algebra structure. In Section 3, we define the real forms of the classical Lie group SL(2, C) and fractional supergroup SL(2, C), respectively.

Preliminaries on *-Algebras
In this section, we summarize important definitions and the relations used to calculate the real forms of the fractional supergroup SL(2, C) [1,2,23]. Definition 1. Let A(G) be the Hopf algebra of functions on the Lie group G and Λ N n be the Hopf algebra generated by λ and θ β , where β = 1, 2, . . . , N, satisfying the following relations: The co-algebra operations and antipodes in A N n (G) rely on the structure constants of the fractional superalgebra U N n (g). Here, the dual of the Hopf algebra A N n (G) is the Hopf algebra U N n (g). In our study, we consider the case of n = 3 and N = 1, 2. The definition and duality relations of the fractional superalgbras U N 3 (g) belonging to this situation are given in the Appendix A.
Definition 2. Let A be an associative algebra with unit I. The algebra A is called a *-Algebra if it has the following properties: (a * ) * = a, 3.
(ab) * = b * a * , I * = I. Definition 3. Let A and B be two unital *-algebras. If a homomorphism H satisfies H(a * ) = H(a) * for a A, then H is called a *-homomorphism from A to B. Definition 4. If Hopf algebra A satisfies Definition 2 and the properties below, then A is called a *-Hopf algebra.
1. The real forms of the group SL(2, C) are the groups SU(2), SU(1, 1) and SL(2, R) which are defined respectively by the *-operations below:

Theorem 1. The algebra A(SU
Proof. It is sufficient to show that the Hopf algebra is consistent with (1) in Definition 5.
Proof. The proof can be shown similarly to the proof of Theorem 1. Proof. The proof can be shown similarly to the proof of Theorem 1.

The Real Forms of the Fractional Supergroup SL(2, C)
We consider the real forms of the fractional supergroup SL(2, C) for N = 1, 2 and n = 3 (that is A N 3 (SL(2, C))).

The Real Forms of
is the direct product of the Hopf algebras A(SL(2, C)) and Λ 1 3 .

Definition 7.
The fractional supergroup A 1 3 (SL(2, C)) is the Hopf algebra generated by a mn , λ and θ β where m, n = 1, 2 and β = 1, satisfying the co-algebra operations and antipode: ε(a mn ) = δ mn (4) Theorem 4. The fractional supergroup A 1 3 (SU (2)) is the * −Hopf algebra generated by a mn , λ and θ β , where m, n = 1, 2 and β = 1 satisfying Definition 7 with the *-operations: Proof. Let us show that the co-algebra structure is consistent with the *-operations. The relations (2)(3) in Definition 7 are shown in Theorem 1. Now, let us show that the relations (5-6) are realized. For this, we will use the definitions of the *-Hopf algebra, considering that ∆ and ε are homomorphisms and S is an anti-homomorphism.