Quasi-deformations of z[2(F) with base ]I~[~, ~ -1] *)

DANIEL LARSSON **), SERGE1 D. SILVESTROV t)

Centre for Mathematical Sciences, Lund University Box 118, SE-221 O0 Lund, Sweden

Received 15 August 2006

In this paper we construct quasi-deformations of s[2(•) with the Laurent polynomials over the reals as base algebra and general vector fields for s[2(~) in the base of the quasi- deformation.

PACS: 02.10.HA Key words: quasi-deformations, quasi-hom-Lie algebras, almost quadratic algebras

1 I n t r o d u c t i o n

In [1], the authors constructed quasi-deformations of ~[2(]F) with the base for the quasi-deformations being the well-known representation of s[2 (F) by derivations with monomial coefficients of low degrees, not exceeding two, acting on the base algebra A = F[t] of polynomials in a single independent variable t over a field F of characteristic zero. The result of such quasi-deformations has been shown to be multi-parameter families of almost quadratic algebras with the parameters naturally associated with the twisting endomorphisms of the base algebra defining a quasi-deformation of the generating derivation. For special values of some of the parameters, i.e., for special choices of the quasi-deformation replacements of the generating derivation by twisted derivations, these mult i -parameter families have been shown to be closely related to conformal 2[2 (F) enveloping algebras, [2] - - these include Witten's deformation of s[2(F) as a special case, examples of Auslander- regular algebras, Koszul algebras, Ore extensions, color Lie algebras (see [1] for some connections to these structures), and other interesting algebras arising in non-commutative algebraic geometry, [3], quantum field theory, quantum integrable systems and other contexts (a more comprehensive list of references can be found in [1]). The distinctive common feature of the almost quadratic algebras constructed in [1] is the appearance of a twisted generalized 3-term or 6-term Jacobi identities and thus associated structures of quasi-horn-Lie algebras and horn-Lie algebras.

In this article we get further insight into the paradigm of quasi-deformations by constructing quasi-deformations of 8[2 (R) using more general representations of s[2(ll~) in terms of derivations (vector fields) on a suitable function algebra A in an indeterminant t, than in [1], and then specializing to the case A = N[t, t - l ] .

Let ~o be a a-derivation on a unital, associative, commutat ive algebra A, tha t is, a linear mapping on A which satisfies the a-deformed Leibniz rule ~o (ab) =

*) Presented at the 15 TM International Colloquium on "Integrable Systems and Quantum Sym- metries", Prague, 15-17 June 2006.

* * ) E-mail: dlarsson�9 lth. se f) E-mail: ssilvest�9 ith. se

Czechoslovak Journal of Physics, Vol. 56 (2000), No. 10/11 1227

D. Larsson and S.D. Silvestrov

~ ( a ) b + a ( a ) ~ ( b ) . Let Der~(A) denote the vector space of or-derivations on A. Fixing a homomorphism ~ : A ~ A, an element ~o E Der~(A) and an element 5 E A, assume that these objects satisfy the following two conditions:

a ( A n n ( ~ ) ) C_ A n n ( ~ o ) , (1)

~o(a(a)) = 5 ~ ( ~ ( a ) ) for a �9 A, (2)

where A n n ( ~ ) := {a �9 A la . ~ = 0}. Let A . ~ := {a. ~o I a �9 A} denote the cyclic A-submodule of Der~(A) generated by ~ and extend a to A �9 ~ by a(a �9 ~ ) = a (a ) .~ , . The following theorem, from [4], introducing an F-algebra s tructure on A . ~ making it a quasi-horn-Lie algebra, is pivotal for our constructions of quasi-deformations of s[:(lR) in this paper and also of other finite-dimensional or infinite-dimensionai Lie algebras.

T h e o r e m 1. If (1) holds then the map (., .} defined by

(a. ~a,b" ~ , } = (a(a). ~ ) o (b. ~ , ) - (a(b). 2 ~ o (a. ~ ) (3)

for a, b �9 A, where o denotes composition of maps, is a well-defined F-algebra pro- duct on the F-linear space A . ~ . It satisfies the following identities for a, b, c �9 A:

(a. ~ , b. 2~) = ( a ( a ) ~ ( b ) - a ( b ) ~ ( a ) ) . ~ , (4)

(a . ~ , b . ~ } = - ( b . ~ , a . 2 ~ ) , (5)

and if, in addition, (2) holds, we have the deformed six-term Jacobi identity

(~a,b,c (((T(a). 2c,, (b. ~a , c . ~r -~- (~. (a. ~ , (b. ~ , c. ~ ) } ) -- 0, (6)

where �9 denotes cyclic summation with respect to a, b, c.

A quasi-deformation of a Lie algebra 9, represented in terms of derivations, is the procedure of replacing 0 = d/d t in this representation by a-derivations and then using the above theorem to obtain an algebra structure on A . ~ which is then "pulled-back" to the quasi-deformed version of g.

2 O n e - d i m e n s i o n a l v e c t o r f ie lds for ~[2(F)

Let A be a suitable function algebra over ~ in a single indeterminate t (non- algebraic over ~). Pu t E := x(t)O, H := y(t)O and F := z(t)O with 0 denoting derivation with respect to t. Using the Leibniz rule we can express the s tandard s[2(F)-relations [H, E] = 2E, [H, F] = - 2 F and [E, F] = H as

y ( t ) x ' ( t ) - x ( t ) y ' ( t ) = 2 x ( t ) , ( 7 a )

y ( t ) z ' ( t ) - z ( t ) y ' ( t ) = - 2 z ( t ) , (Tb) x ( t ) z ' ( t ) - z ( t ) x ' ( t ) = y(t) , (7c)

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Quasi-deformations o f zl2 ( F ) . . .

where, for instance, x'( t ) := Ox(t). We take the y(t) as given and use the above system of equations to solve for x( t ) and z( t) . As is probably well known, the solution of the system of equations (7a 7c) is given by

{ y( t ) = y ( t ) ,

z( t) = D e x p ( F ( t ) ) ,

x ( t ) - Y(t) - Y(t) 4z(t) 4D

e x p ( - F ( t ) ) ,

(8)

where F( t ) is a primitive for (y '( t) - 2 ) / y ( t ) , tha t is, F' ( t ) = (y '( t) - 2 ) / y ( t ) . In this article we will concentrate on the monomial choice y(t) = ~t-J . I t is of

a special importance, as it is, for example, a general form of an invertible element in the algebra of Laurent polynomials N[t, t - l ] .

In what follows we assume for simplicity tha t In t is a primitive to t -1 in the algebra A. An example of such algebra A is C ~ ( ( 0 , +oc)) , where Itl = t.

I fy ( t ) = ~it j for ~) ~ ~ a n d j E Z, then F' ( t ) = ( y ' ( t ) - 2 ) / y ( t ) = j t -1 - ( 2 / 9 ) t - j , and hence F( t ) = j l n t - 2 t - J+l / [~ i ( - j + 1)] + CF for j r 1, and F( t ) = (1 - 2/.~) ln t + CF for j = 1, where CF is a constant of integration. Thus, put t ing p = De CF, one gets

E = x ( t ) O = - ~ p t J e x p t ] ( - j + l ) t - J + l 0 ,

F = z(t)O -- #t j exp ~)( - j + 1)

H = y(t)O = 9tJO,

t-J+l)o, (9)

when j r 1, and

~2 E = x ( t ) O = --~pt2-Ccg , H = y ( t ) O = ~ltO, F = z ( t ) O = #tcO, (i0)

when j = 1. Here we put c := 1 - 2 9-1. In the case when A = R[t, t - I ] , an invertible y in R[t, t -1] has to be of the

form y(t) = ~]tJ for ~ E ~ and j E Z, and we can have no exponential since this is in general expressed as an infinite power series. Therefore, from (9) and (10), we should have j = 1 and hence (10).

If A is a rational function field, i.e., any non-zero element can be inverted, then the freedom is much greater. The analysis of this much more complex si tuat ion is beyond the scope and size of the present article and we leave it to future publica- tions.

T h e quas i -de fo rma t ion

In R[t, t -1] a general au tomorphism cr is of the form a( t ) ---- qt for some q E ~. The quasi-deformation procedure is based on the replacement of the main generat ing derivation in the vector fields by a a-der ivat ion and replacement of the commuta to r

Czech. J. Phys. 56 (2006) 1229

D. Larsson and S.D. Silvestrov: Quasi-deformations of sI2(F) ...

b racket for vector fields by a twis ted bracke t be tween quas i -de fo rmed vec tor fields. Namely,

~2 t2-~O ~ ~2 E = -~pp -4--p t2 -c0~ =: e , (11)

F =#tcO ~ #t~O~ =: f , (12)

H =[ItO ~ [ItO~ =: h . (13)

We also know f rom [4] t h a t 0~ on Nit, t -1] (a U F D ) is given by 0~ := D ( t ) ( i d - a ) / g , where g = g c d ( ( i d - a ) ( R [ t , t - l ] ) ) = p - i t ( 1 - qtk-1), for p E R and D(t) E R[t, t - I ] . In this pape r we shall only consider the case when D(t) = p E N and k = 1, t h a t is,

Oa = pt - l i d - a , 1 - q

which is a cons tan t mul t ip le of the Jackson q-der iva t ion opera to r . T h e analysis of the general case is well beyond the page l imi ta t ions for this paper .

The c o m m u t a t o r bracket [-, .] deforms to (., .) as descr ibed in T h e o r e m 1. Ex- plicitly, af ter compu ta t i on , we get ins tead of ~[2 (F)- re la t ions [H, E] = 2E, [H, F] = - 2 F and [E, F] = H the following c o m m u t a t i o n re la t ions for the co r respond ing quas i -deformat ion:

(h, e} = qhe - q2 -~eh -- pq~){1 - C}qe, (14a)

(h, f) = q h f - qCfh = pq~){c - 1}qf , (14b) C ~

(e, f) = q2-~ef - q~fe = - ~ { 2 ( 1 - c )}qh , (14c)

where we have used {n}q := 1 + q + q2 + . . . + qn-1.

R e m a r k 1. Notice t ha t t ak ing q = p = 1 ( tha t is, col lapsing the a -de r iva t ions to der ivat ions) , p = - 1 , c = 2, ~ = - 2 , gives us the s t a n d a r d re la t ions for s[2(R) t h a t we s ta r t ed with, and also t ha t a q-deformed Jacob i ident i ty holds as descr ibed in T h e o r e m 1.

This work was partially supported by the Crafoord Foundation, the Royal Physio- graphic Society in Lund, the Swedish Royal Academy of Sciences, the Swedish Founda- tion for International Cooperation in Research and Higher Education (STINT) and the Liegrits network.

R e f e r e n c e s

[1] D. Larsson and S.D. Silvestrov: math.RA/0506172.

[2] L. Le Bruyn: Commun. Alg. 23 (1995) 1325.

[3] L. Le Bruyn and S.P. Smith: Proc. AMS 118 (1993) 725.

[4] J,T. Hartwig, D. Larsson, and S.D. Silvestrov: J. Algebra 295 (2006) 314.

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