###### Abstract

We construct two-parameter deformation of an universal enveloping algebra of a polynomial loop algebra , where is a finite-dimensional complex simple Lie algebra (or superalgebra). This new quantum Hopf algebra called the Drinfeldian can be considered as a quantization of in the direction of a classical r-matrix which is a sum of the simple rational and trigonometric r-matrices. The Drinfeldian contains as a Hopf subalgebra, moreover and are its limit quantum algebras when the deformation parameters goes to and goes to , respectively. These results are easy generalized to a supercase, i.e. when is a finite-dimensional contragredient simple superalgebra.

Drinfeldians

Valeriy N. Tolstoy

Institute of Nuclear Physics

Moscow State University

119899 Moscow&Russia^{1}^{1}1e-mail:

## 1 Introduction

As it is well known, an universal enveloping algebra of a polynomial loop (current) Lie algebra , where is a finite-dimensional complex simple Lie algebra, admits two type deformations: a trigonometric deformation and a rational deformation or Yangian [1, 2]. (In the case there also exists an elliptic quantum deformation of ). The algebras , and are quantizations of in the direction of the simplest trigonometric and rational solutions of the classical Yang-Baxter equation over , respectively. These deformations are one-parameter ones. It turns out that also admits two-parameter deformation which is called the Drinfeldian or the rational-trigonometric quantum algebra. The Drinfeldian is a quantization of in the direction of a classical r-matrix which is a sum of the simplest rational and trigonometric r-matrices. The Drinfeldian contains as a Hopf subalgebra, and and are its limit quantum algebras when the deformation parameters of goes to and goes to , respectively. These results are easy generalized to a supercase, i.e. when is a finite-dimensional contragredient simple superalgebra.

## 2 Quantum algebra

Let be a finite-dimensional complex simple Lie algebra
of a rank with a standard Cartan matrix
, with a system of simple roots
and a maximal positive
root , and with a Chevalley basis
{, , }.
Let be a polynomial loop algebra (or a Lie algebra
of polynomial currents over ), and
be a ”central extension” of :
is a central element.^{2}^{2}2More correctly, the element
is a central element of a central extension of the total loop
algebra .
The Lie algebra
(and its universal enveloping algebra )
is generated by the Chevalley basis of and the affine
element and
with the following defining relations: , where

(2.1) | |||

(2.2) | |||

(2.3) | |||

(2.4) | |||

(2.5) | |||

(2.6) | |||

(2.7) |

Here is the adjoint action of in , i.e. for any . The relations (2.6) relate to the case , and the relation (2.7) belongs to the case (in this case and we set ).

Remark. The defining relations for can be obtained from defining relations of the corresponding non-twisted affine Lie algebra by removing relations with a negative affine root vector .

Let be a standard q-deformation of the universal enveloping algebra with Chevalley generators , and with the defining relations

(2.8) | |||

(2.9) |

where is the q-commutator:

(2.10) |

A Hopf structure of is given the following formulas for a comultiplication , an antipode , and a co-unite :

(2.11) | |||

(2.12) | |||

(2.13) |

###### Definition 2.1

The quantum algebra (or a q-deformation of ) is generated (as an associative algebra) by the algebra and the elements , with the relations:

(2.14) | |||

(2.15) |

where , and

(2.16) | |||

(2.17) |

The Hopf structure of is defined by the formulas , (), and , . The comultiplication, the antipode and the co-unite of the element are given by

(2.18) | |||

(2.19) |

Here we put

(2.20) |

if .

Remark. The defining relations for can be obtained from defining relations of the corresponding quantum non-twisted affine algebra by removing relations with the negative affine root vector .

It is easy to check the following result.

###### Proposition 2.1

There is a one-parameter group of Hopf algebra automorphisms of , , given by

(2.21) |

## 3 Drinfeldian

Here we keep the notations of the previous Section and begin with the following important definition.

###### Definition 3.1

The Drinfeldian is generated as an associative algebra over

(3.1) |

(3.2) |

for , and

(3.3) | |||||

for and ,

(3.4) |

for . The Hopf structure of is defined by the formulas , and , . The comultiplication and the antipode of are given by

(3.5) | |||||

(3.6) |

Here , , and the vector is any element of the weight , such that .

###### Theorem 3.1

(i) The Drinfeldian is a two-parameter
quantization of in the direction of
a classical r-matrix which is a sum of the simplest rational
and trigonometric r-matrices.

(ii) The Hopf algebra
is isomorphic to the Yangian
(with the additional central element ).
Moreover, .

Remark. Since the defining relations for and in terms of the Chevalley basis differ only in the right-hand sides of the relations (3.2)-(3.4), therefore the Dynkin diagram of can be also used for classification of the Drinfeldian and the Yangian .

An analog of Proposition 2.1 is the following result.

###### Proposition 3.1

There is a one-parameter group of Hopf algebra automorphisms of , , given by

(3.7) |

## 4 Drinfeldians and Yangians over Lie algebras of rank 2

Explicit description of the Drinfeldians and the Yangians for the cases were given in [5]-[7]. Here we consider the cases .

1. The Drinfeldian and the Yangian . In the case of the Lie algebra there are two positive simple roots and , and the maximal positive root is .

As we already noted the Drinfeldian of the Yangian can be characterized the Dynkin diagram of the corresponding non-twisted Kac-Moody affine Lie algebra . In the case the Dynkin diagram of the corresponding affine Lie algebra is presented by the picture [3]