###### Abstract

Drinfeld twist is applied to the Lie algebra so that a two-parametric deformation of it is obtained, which is identical to the Jordanian deformation of the obtained by Aneva . The same twist element is applied to deform the Lie superalgebra , since the is embedded into the . By making use of the FRT-formalism, we construct a deformation of the Lie supergroup

OWUAM-029

July 10, 1998

DRINFELD TWIST FOR TWO-PARAMETRIC DEFORMATION OF
AND ^{2}^{2}2Presented at the 7th
International Colloquium ”Quantum Groups and Integrable Systems”,
Prague, 18–20 June 1998.

N. Aizawa

Department of Applied Mathematics, Osaka Women’s University

Sakai, Osaka 590-0035, Japan

## 1 Introduction

It is well known that deformation of Lie groups or Lie algebras is not unique. As for its multiparametric quantum deformation is classified recently [1]. In this note, a two-parametric deformation of by the Drinfeld twist [2] is considered and it is shown that the deformed is identical to the one obtained before [4]. The twist for used here is also applicable to deform the Lie superalgebra , since The universal R-matrix for the deformed is constructed according to the method of twisting. This enables us, using the FRT-formalism, to construct a deformation of the Lie supergroup The same scenario has been carried out for a one-parametric deformation of and (Note that )[3]. The present work follows the line adapted in [3].

The deformation by the Drinfeld twist has been developed in recent years. For example, multiparametric twists for the Drinfeld-Jimbo deformation of simple Lie algebras [8], one-parametric twist for [9], and twist for Poincaré algebra [10], Heisenberg algebra [11], esoteric quantum groups [12], [13], and [3, 14] have been considered.

## 2 Drinfeld twist

This section is devoted to a brief review of the Drinfeld twist. Drinfeld develops it in his study of quasi Hopf algebras, we however restrict ourselves to ordinary Hopf algebras throughout this note.

Let be a Hopf algebra with coproduct , counit and antipode Let be an invertible element in satisfying the conditions

(2.1) |

then we obtain a new Hopf algebra with the same algebraic relations as . However, Hopf algebraic mappings of are different, they are twisted by the twist element . The coproduct, the counit and the antipode for are given by

(2.2) |

where

When the algebra has a universal R-matrix , the universal R-matrix for is given by

(2.3) |

This shows that if is cocommutative, then is a triangular Hopf algebra. This is the case that we would like to consider, since we shall start with a Lie algebra.

## 3 Twist for

The Lie algebra has elements and and defined by the relations

(3.1) |

A twist element for is given by

(3.2) |

with

where and are deformation parameters.

Acoording to (2.2), the coproduct and the antipode are deformed and they read

(3.3) |

The universal R-matrix for the non-cocommutative coproduct given above reads

(3.4) |

A counit is necessary for a Hopf algebra, it is undeformed, that is, Thus we arrive to the definition

Definition 1. The triangular Hopf algebra generated by satisfying the relations (3.1) and (3.3) is said to be the two-parametric deformation of by twisting or

Let us take the particular nonliear combinations of generators,

(3.5) | |||||

Then and satisfy the commutation relations,

(3.6) | |||

The Hopf algebra mappings for these generators are given by

Therefore the algebra generated by is nothing but the one introduced in [4]. The authors of [4] define the algebra so as to be dual to the Jordanian matrix quantum group [5, 6].

## 4 Twist for

The Lie superalgebra has four even and four odd elements denoted by and , respectively. They satisfy the relations

(4.1) |

It is easily seen from the above relations that the even elements form a subalgebra and the universal enveloping algebra is generated by . The observation of implies that the twist element for can be used to twist . Using (3.2), the twisted coproduct for the odd elements are given by

(4.2) | |||||

and the antipode is

(4.3) |

The counit is undeformed and given by The Hopf algebra mappings for even elements have already been given in (3.3).

Definition 2. The triangular Hopf algebra generated by satisfying the relations (4.1), (4.2) and (4.3) is said to be the two-parametric deformation of by twisting or

The universal R-matrix for is same as the one for Noting that the fundamental representation of is same as , we obtain the R-matrix in the fundamental representation of , and see that it is a direct sum of four matrices

(4.4) |

where

(4.5) |

The matrix is the R-matrix (3.4) in the fundamental representation of

## 5 Two-parametric deformation of

Using the R-matrix (4.4) and graded version of FRT-formalism [7], a matrix quantum supergroup dual to can be constructed.

Introducing a supermatrix

where

and are even elements and are odd elements, the FRT-formalism guarantees that the commutation relations for the entries of are given by RMM-relation and their Hopf algebra mappings are given by

(5.1) |

The commutation relations for the entries of read

(5.2) |

The last relation shows that the submatrix satisfy the same algebraic relations as the [6]. As in [6], a determinant for is defined by , then it is not difficult to see that is not a center of deformed and the noncommutativity is independent of

Assuming that the is invertible, the explicit form of the inverse matrix of can be obtain (see [6] for the formula).

We define a superdeterminant for the quantum supermatrix by

(5.3) |

This has the same form as the undeformed case (it is also called Berezinian in undeformed case). Direct computation shows that the commute with all elements of the deformed so that we can safely set A coproduct and a counit for are obvious from (5.1), however, an antipode is not. It is necessary to assume that the combination has a inverse. Then the inverse matrix of is given by [3]

(5.4) |

where is the unit matrix.

Definition 3. An algebra generated by the entries of satisfying (5.2), (5.1), (5.4) and is said to be the two-parametric deformation of or

We finally give some remarks. The twist element (3.2) is also applicable to deform , since . The universal R-matrix for the obtained algebra is used to deform the supergroup . The inclusion of a odd element of into a twist element may be possible. The inclusion of a odd elements is found for recently [14].

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