научная статья по теме CONTRACTIONS AND DEFORMATIONS OF QUASICLASSICAL LIE ALGEBRAS PRESERVING A NONDEGENERATE QUADRATIC CASIMIR OPERATOR Физика

Текст научной статьи на тему «CONTRACTIONS AND DEFORMATIONS OF QUASICLASSICAL LIE ALGEBRAS PRESERVING A NONDEGENERATE QUADRATIC CASIMIR OPERATOR»

ЯДЕРНАЯ ФИЗИКА, 2008, том 71, № 5, с. 857-862

ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ

CONTRACTIONS AND DEFORMATIONS OF QUASICLASSICAL LIE ALGEBRAS PRESERVING A NONDEGENERATE QUADRATIC

CASIMIR OPERATOR

© 2008 R. Campoamor-Stursberg*

Departamento de Geometría y Topología, Universidad Complutense, Madrid, Spain

Received August 20, 2007

By means of contractions of Lie algebras, we obtain new classes of indecomposable quasiclassical Lie algebras that satisfy the Yang—Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise naturally from noncompact real simple algebras with nonsimple complexification, where we impose that a nondegenerate quadratic Casimir operator is preserved by the limiting process. We further consider the converse problem and obtain sufficient conditions on integrable cocycles of quasiclassical Lie algebras in order to preserve nondegenerate quadratic Casimir operators by the associated linear deformations.

PACS: 02.20.Sv

1. INTRODUCTION

Reductive Lie algebras have been shown to be the most convenient class of algebras for physical applications. They arise naturally as the Lie algebras of compact groups and contain the class of semisimple algebras. Moreover, they have an important property, namely an invariant metric1), of crucial importance in problems like defining Wess-Zumino—Witten (WZW) models. Classically made on semisimple and reductive algebras, models based on nonreductive algebras have been shown to be of physical interest [1]. Other important applications of Lie algebras endowed with a symmetric nondegener-ate invariant form, which we call here quasiclassical2), are, for example, conformal field theory, where they correspond to the Lie algebras admitting a Sug-awara construction, or the Yang—Baxter equations, where quasiclassical algebras provide classes of solutions [2-4].

A Lie algebra L is called quasiclassical if it possesses a bilinear symmetric form (.,.) that satisfies the constraints

([X,Y] ,Z) = (X, [Y,Z]), (1)

if(X, Y) = 0, УХ <E L ^ Y = 0. (2)

E-mail: rutwig@mat.ucm.es

!)This metric arises immediately from the Killing tensor for the

semisimple case.

2)Other authors call algebras like these symmetric self-dual.

The first condition shows that the bilinear form satisfies an associativity condition (also called invariance), while the second expresses nondegener-ateness. Given a basis {X1,...,Xn} of L and the

corresponding structure tensor ^Cf^, we obtain the

expression of (.,.) as

(Xi, Xj) = gij. (3)

Since the form is nondegenerate, we find an inverse to the coefficient matrix of (.,.): gij = (gij)_1. Obviously, semisimple Lie algebras satisfy these requirements for the Killing form. Also reductive and Abelian Lie algebras are trivially quasiclassical although in this case the Killing metric is no more nondegenerate. In [4] it was shown that a necessary and sufficient condition for the existence of such a form is that L admits a nondegenerate quadratic Casimir operator C = ga@xa. Using the realization by differential operators Xj = C^Xk^-, this means

that C is a solution of the following system of partial differential equations:

д

C%xk—C = 0. (4)

Using this characterization, we obtain a useful criterion to test whether a Lie algebra is quasiclassical or not, and in certain situations more practical than various pure algebraic structural results (see, e.g., [5] and references therein). In particular, for any given dimension, the classification of quasiclassical Lie

algebras follows from the classification of isomorphism classes once the invariants of the coadjoint representation have been computed. Therefore the problem of finding metrics reduces to an analytical problem, which is solved in low dimension [6, 7].

This paper is structured as follows: In Section 2 we reformulate the Yang—Baxter equations in terms of triple products, which enables us to obtain some sufficiency criteria basing only on the structure tensor of a quasiclassical algebra. This triple-product formulation is used in combination with contractions of Lie algebras to construct large classes of indecomposable quasiclassical algebras that preserve the quadratic nondegenerate Casimir operator of a semisimple classical Lie algebra. In Section 3 we focus on a kind of inverse problem, namely, deformations of quasiclassical Lie algebras that preserve the quadratic Casimir operator, and therefore, the associated metric. This leads to a characterization of such deformations in terms of integrable cocycles in the adjoint cohomology.

2. YANG-BAXTER EQUATIONS AND QUASICLASSICAL ALGEBRAS

2.1. Yang-Baxter Equations and Triple Products

Yang-Baxter equations (YBE) have been known a long time to embody the symmetries of two-dimensional integrable models [8] and also appear in many problems concerning statistical physics and quantum groups. In addition to the classical semisimple case, nonreductive quasiclassical Lie algebras were recognized to provide some solutions of the YBE when these are rewritten in terms of triple products [9]. With this reformulation, some useful sufficient conditions can be found in dependence of the structure tensor of the quasiclassical Lie algebra.

Given a finite-dimensional vector space V with inner product (.,.), then, for any basis [v\,... ,vn}, we set the coefficients in the usual way:

(vi,vj) ■= gij = gji,

and define the raising of indices

Eg'

i=1

vj = > gij V'.

Given a spectral parameter 6, we consider the map R (6): V 0 V ^ V 0 V defined by

n

R (6) (Vi 0 Vj) Rkl (6) Vk 0 vi. (5)

k,l=1

We obtain the YBE in its usual form [8] from the relations

R12 (6) R13 {6') R23 (6") = (6)

= R23 {6'') R13 (60 R12 (6) ,

where 6" = 6' — 6. The equations can be rewritten using triple products, which provides sometimes a more convenient presentation for solutions governed by certain types of purely solvable quasiclassical Lie algebras. Introducing the triple products [9]

V ,vk ,*]e = E Rj 0

J e " ki

i=1 n

[■v',vk ,vi]*e = Riki vi> j=1

the YBE reduces to the relation

n

Y^ [u, \v,vjMo> > íeJXe» j=1

(7)

(8)

Y^ [v, [u,vj,x\e,, [ej,w,y] j=1

e"

where u, v, w,x,y e V .A particularly interesting case is given when the scattering-matrix elements Rj (6) satisfy the following constraint:

Rki (0) - Rjk = 0

In this case, Eq. (8) becomes

n

Y K [v,vj,w\e>, [ej,x,y e]

eJe"

(9)

(10)

j=1

= V [u,vj'Ie30"]e ' 3=1

subjected to the condition

(u, [v, W' x]e) = (v, [u, X' w]e).

Even in this case, solving of the equations is far from being trivial. However, it was found in [9] that if L satisfies the condition

[L, [[L,L] , [L,L]]]=0, (11)

then we have commutation relation [Rjk (d), Rim =0, = 1,2,3. (12)

This in particular implies that the YBE (8) is satisfied. Classes of solvable Lie algebras in arbitrary dimension that satisfy these conditions have been constructed in [9], as well as examples, where the commutation relation (12) is not necessarily satisfied.

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2.2. Contractions of Quasiclassical Lie Algebras

In this subsection we obtain additional classes of indecomposable nilpotent Lie algebras satisfying condition (11). In comparison with previous constructions, the class of algebras obtained here follows naturally from contractions of simple real Lie algebras that preserve the quadratic Casimir operator. We can therefore construct quasiclassical algebras with prescribed inner product.

We recall that a contraction L — L' of a Lie algebra is given by the commutators

[X,Y]' := hm Ф-

-1

^t(X), Ф4(Y)], (13)

Ф^) = t-niXt, иг G Z.

(14)

Contractions can also be extended to invariants. Let F (X-,... ,Xn ) = ah-ip Xi1 ...Xip beaCasimirop-erator of degree p. Expressing it over the transformed basis we get

F ($t (Xi ),..., ®t(Xn ))= (15)

= tnii+...+nip aii-iP Xi1 ...Xip.

Now let

M = max {m1 + ... + mp\ail ^ip = 0} (16) and consider the limit

F'(Xi,...,Xn )= (17)

= l im t-MF($t(Xi),...,$t(Xn)) =

t—>oo

E

a%1 -гРXi, ...X .

with complexification s © s. Let {Xi ,...,Xn} and {Y-1,..., Yn} be a basis of each copy of s such that

[Xi, X,] = CkjXk, [Yi, Yj] = CkjYk, [Xi, Yj] = 0,

i.e., the structure tensor is the same in both copies. Considering the change of basis given by

Xt = Xt + Yi, Yi = V^l (Yi - X^, i = l,...,n,

the structure tensor over the basis {Xi Fi,..., Yn} is expressed by

\Xi,Xj\ = c^xk, [x

Yi\ = CtYk,

гз

(18)

, X n

(19)

where $t is a parametrized family of nonsingular linear maps in L for all t < œ3). Among the various types of contractions existing, we consider here the so-called generalized Inonu—Wigner contractions given by automorphism of the type4)

,Y3]=-C%Xk.

Since s ® s is quasiclassical for being semisimple, it admits quadratic Casimir operators

Ci = gabXa Xb, C2 = ga0 YaYb.

Suitable linear combinations of them provide a non-degenerate quadratic operator on the direct sum. Rewriting these operators in the new basis, we obtain the operators

яЬл

C1 = gij

LiX j Y iY j)

(20)

C2 = g13 (XiYj + YiXj) ,

n^ +...+nip=M

It is not difficult to see that this expression is a Casimir operator of degree p of the contraction. Imposing that the invariant remains unchanged by the contraction implies certain restriction that must not necessarily occur [11]. For our purpose, preservation of a nondegenerate quadratic Casimir operator implies automatically that the contraction is quasiclassical, and the induced inner product is the same. To this extent, let s be a complex semisimple Lie algebra of classical type Ai,Bi, Ci,Di and let sl (l + 1, C),so (2l + 1, C),sp (2l, C), and so (2l, C) be the noncompact simple real Lie algebras

which are easily seen to be nondegenerate. It is natural to ask whether there exist nontrivial contractions of s ® s such that the contraction preserves at least one of the preceding Casimir operators. In this way, the contraction is also quasiclassical with the same bilinear form as the contracted algebra. The suitable operator to be test

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