Last update: 30 May 2014
This is a typed copy of the LOMI preprint Quantization of Lie groups and Lie algebras by L.D. Faddeev, N.Yu. Reshetikhin and L.A. Takhtajan.
Recommended for publication by the Scientific Council of Steklov Mathematical Institute, Leningrad Department (LOMI) 24, September, 1987.
Submitted to the volume, dedicated to Professor's M. Sato 60-th birthday.
The Algebraic Bethe Ansatz, which is the essence of the quantum inverse scattering method, emerges as a natural development of the following different directions in mathematical physics: the inverse scattering method for solving nonlinear equations of evolution [GGK1967], quantum theory of magnets [Bet1931], the method of commuting transfer-matrices in classical statistical mechanics [Bax1982]] and factorizable scattering theory [Yan1967,Zam1979]. It was formulated in our papers [STF1979,TFa1979,Fad1984]. Two simple algebraic formulas lie in the foundation or the method: and Their exact meaning will be explained in the next section. In the original context or the Algebraic Bethe Ansats plays the role of the quantum monodromy matrix of the auxiliary linear problem and is a matrix with operator-valued entries whereas is an ordinary matrix. The second formula can be considered as a compatibility condition for the first one.
Realizations of the formulae and for particular models naturally led to new algebraic objects which can be viewed as deformations of Lie-algebraic structures [KRe1981,Skl1982,SklNONE,Skl1985]. V. Drinfeld has shown [Dri1985,Dri1986-3] that these constructions are adequately expressed in the language of Hopf algebras [Abe1980]. On this way he has obtained a deep generalization of his results of [KRe1981,Skl1982,SklNONE,Skl1985]. Part of these results were also obtained by M. Jimbo [Jim1986,Jim1986-2].
However, from our point of view, these authors did not use formula to the full strength. We decided, using the experience gained in the analysis of concrete models, to look again at the basic constructions of deformations. Our aim is to show that one can naturally define the quantization of simple Lie groups and Lie algebras using exclusively the main formulae and Following the spirit of non-commutative geometry [Con1986] we will quantize the algebra of functions on a Lie group instead of the group itself. The quantization of the universal enveloping algebra of the Lie algebra will be based on a generalization of the relation where is a subalgebra in of distributions with support in the unit element of
We begin with some general definitions. After that we treat two important examples and finally we discuss our constructions from the point of view of deformation theory. In this paper we use a formal algebraic language and do not consider problems connected with topology and analysis. A detailed presentation of our results will be given elsewhere.
Let be an complex vector space (the reader can replace the field by any field of characteristic zero). Consider a non-degenerate matrix satisfying the equation where the lower indices describe the imbedding of the matrix into
Definition 1. Let be an associative algebra over with generators satisfying the following relations where and is a unit matrix in The algebra is called the algebra of functions on the quantum formal group corresponding to the matrix
In the case the algebra is generated by the matrix elements of the group and is commutative.
Theorem 1. The algebra is a bialgebra (a Hopf algebra) with comultiplioation
Let be the dual space to the algebra Comultiplication in induces multiplication in where and Thus has the structure of an associative algebra with unit where
Definition 2. Let be the subalgebra in generated by elements and where Here The matrices act nontrivially on factors number and in the tensor product and coincide there with the matrices where here is the permutation matrix in for The left hand side of the formula (3) denotes the values of and of the matrices-functionals on the homogeneous elements of the algebra of degree When the right hand side of the formula (3) is equal to The algebra is called the algebra of regular functionals on
Due to the equation (1) and the definition 2 is consistent with the relations (2) in the algebra
Remark 1. The apparent doubling of the number of generators of the algebra in comparison with the algebra is explained as follows: due to the formula (3) some of the matrix elements of the matrices are identical or equal to zero. In interesting examples (see below) the matrices are of Borel type.
|1)||In the algebra the following relations take place: where and|
|2)||Multiplication in the algebra induces a comultiplication in so that acquires a structure of a bialgebra.|
The algebra can be considered as a quantization of the universal enveloping algebra, which is defined by the matrix
Let us also remark that in the framework of the scheme presented one can easily formulate the notion of quantum homogeneous spaces.
Definition 3. A subalgebra which is a left coideal: is called the algebra of functions on quantum homogeneous space associated with the matrix
Now we shall discuss concrete examples of the general construction presented, above.
Let a matrix of the form [Jim1986-2] where are matrix units and satisfies equation (1). It is natural to call the corresponding algebra the algebra of functions on the of the group and denote it by
Theorem 3. The element where summation goes over all elements of the symmetric group and is the length of the element generates the center of the algebra
Definition 4. The quotient-algebra of defined by an additional relation is called the algebra of functions on the of the group and is denoted by
Theorem 4. The algebra has an antipode which is given on the generators by: where and The antipode has the properties and where
In the case the matrix is given explicitly by and the relations (2) reduce to the following simple formulae: and In this case
Remark 2. When relations (2) admit the following The algebra with this anti-involution is nothing but the algebra In the case this algebra and the matrix of the form (7) appeared for the first time in [FTa1986]. The subalgebra generated by the elements and is the left coideal and may be called the algebra of functions on the of the symmetric homogeneous space of rank for the group In the case we obtain the of the Lobachevski plane.
Remark 3. When the algebra admits the following The algebra with this anti-involution is nothing but the algebra In the case this algebra was introduced in [VSoNONE,Wor1987].
Remark 4. The algebras where is a simple Lie group, can be defined in the following way. For any simple group there exists a corresponding matrix satisfying equation (1), which generalizes the matrix of the form (6) for the case This matrix depends on the parameter and, as where Here is the vector representation of Lie algebra its Cartan-Weyl basis and the set of positive roots. The explicit form of the matrices can be extracted from [Jim1986-2], [Bas1985]. The corresponding algebra is defined by the relations (2) and an appropriate anti-involution compatible with them. It can be called the algebra of functions on the of the Lie group
Let us discuss now the properties of the algebra It follows from the explicit form (6) of the matrix and the definition 2 that the matrices-functionals and are, respectively, the upper- and lower-triangular matrices. Their diagonal parts are conjugated by the element of the maximal length in the Weyl group of the Lie algebra
Theorem 5. The following equality holds: where is the of the universal enveloping algebra of the Lie algebra introduced in [Dri1985] and [Jim1986]. The center of is generated by the elements
Remark 5. It is instructive to compare the relations for the elements which follow from (5), with those given in [Dri1985] and [Jim1986]. The elements can be considered as a of the Cartan-Weyl basis, whereas the elements are the -deformation of the Chevalley basis. It was this basis that was used in [Dri1985] and [Jim1986]; the complicated relations between the elements of the of the Chevalley basis presented in these papers follow from the simple formulae (3), (5) and (6).
Remark 6. It follows from the definition of the algebra that it can be considered as the algebra where and are, respectively, Borel and Cartan subgroups of the Lie group Moreover, in the general case the of the universal enveloping algebra of simple Lie algebra can be considered as a quantization of the group For the infinite-dimensional case (see the next section) this observation provides a key to the formulation of a quantum Riemann problem.
In the case we have the following explicit formulae: where the generators and of the algebra satisfy the relations which appeared for the first time in [KRe1981].
Replace in the general construction of section 1 the finite-dimensional vector space by an infinite-dimensional vector space where is a formal variable (spectral parameter). Denote by the shift operator (multiplication by and consider as a matrix an element satisfying equation (1) and commuting with the operator Infinite-dimensional analogs of the algebras and - the algebras and are introduced as before by definitions 1 and 2; in addition the elements and commute with Theorems 1 and 2 are valid for this case as well.
Let us discuss a meaningful example of this construction. Choose the matrix to be a matrix-valued function defined by the formula where the matrices are given by (4). The role of the elements is now played by the infinite formal Laurent series with the relations The comultiplication in the algebra is given by the formula The following result is the analog of Theorem 3.
Theorem 6. The element generates the center of the algebra
Let us now briefly discuss the properties of the algebra It is generated by the formal Taylor series which, act on the elements of the algebra by the formulae (3). The relations in have the form Due to lack of space we shall not discuss here an interesting question about the connection of the algebra with the of loop algebras, introduced in [Dri1986-3]. We shall only point out that the algebra has a natural limit when In this case the subalgebra generated by elements coincides with the Yangian introduced in the papers [Dri1985,KRe1986].
Consider the contraction of the algebras and when For definiteness let us have in mind the above finite-dimensional example. The algebra when goes into the commutative algebra with the Poisson structure given by the following formula Here are the coordinate functions on the Lie group Passing from the Lie group to its Lie algebra we obtain from (8) the Poisson structure on the Lie algebra Here where form a basis of If we define where and are respectively the nilpotent and Carton components of we can rewrite the formula (9) in the form (This Poison structure and its infinite-dimensional analogs were studied in [RFa1983] (see also [STS1985,FTa1987]). Thus the Lie algebra has the structure of a Lie bialgebra, where the cobracket is defined by the Poisson structure (10).
Analogously, the contraction of the algebra leads to the Lie bialgebra structure on the dual vector space to the Lie algebra The Lie bracket on is dual to the Poisson structure (10) and the cobracket is defined by the canonical Lie-Poisson structure on
This argument clarifies in what sense the algebras and determine deformations of the corresponding Lie group and Lie algebra. Moreover it shows how an additional structure is defined on these "classical" objects. Returning to the relations and we can now say that constitutes a deformation of the Lie-algebraic defining relations with playing the role of generators, being the array of "quantum" structure constants, and generalizing the Jacoby Identity.
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