A program for constructing finitely presented Lie algebras and superalgebras

Nuclear Instruments and Methods in Physics Research A 389 (1997) 370-373

ELSEVIER

NUCLEAR INSTRUMENTS 8 METHODS IN PHYSICS RESEARCH SecttonA

A program for constructing finitely presented Lie algebras and superalgebras V.P. Gerdt, V.V. Kornyak* Laboratoq of Computing Technique and Automation. Joint Institute for Nuclear Research, 141980. Dubna. A4oscow region, Russia

Abstract The purpose of this paper is to describe a C program FPLSA for investigating finitely presented Lie algebras and superalgebras. The underlying algorithm is based on constructing the complete set of relations called also standard basis or Griibner basis of ideal of free Lie (super) algebra generated by the input set of relations. The program may be used, in particular, to compute the Lie (super)algebra basis elements and its structure constants, to classify the finitely presented algebras depending on the values of parameters in the relations, and to construct the Hilbert series. These problems are illustrated by examples. PACS: 02.1O.Sp; 02.70.R~ Ke_ywords: Lie superalgebras;

Relations;

Grabner

basis

1. Introduction A Lie super-algebra is a Zz-graded algebra L = Lo 0 Li over a commutative ring K with a unit. We shall consider here only zero characteristic rings. The elements of Lo and Li are called euell and odd, respectively. A Lie product [ ,] in the Lie superalgebra is compatible with the Z,-grading, i.e. if u E L,, o E L,, a, ,!I E Z2 = (0, 11, then [u, u] E La+p. The Lie product satisfies the following axioms: [U, v] = - (- l)“D[v, u]. CU,CU,WI1=

skew-symmetry,

(1)

ecu,L,l,WI+ (- l)“TL~, cu,wll,

Jacobi identity,

UE L,, VE L,.

(2)

If Li = 0 one deals with an ordinary Lie algebra. For brevity, we shall often omit the term super (and Lie) in the text. If every element of a Lie algebra can be formed by a linear combination of Lie products of a finite number of elements called generators, then such algebra is called $&e/y generated. If generators satisfy a finite number of relations having the form of polynomials in the algebra,

*Corresponding author. Tel.: + 7 9621 62 552; fax: + 7 9621 65 145; e-mail: [email protected].

then the algebra is called jnitely dejned. A finitely generated and finitely defined algebra is called jinitely presented. Any finite-dimensional algebra is, obviously, a finitely presented one. Nevertheless, infinite-dimensional finitely presented algebras such as Kac and Kac-Moody algebras and different algebras of vector fields, e.g., the “stringy” (Virasoro, Neveu-Schwarz and Ramond) algebras are of special interest for the modern physics. Having a given set of generators and relations one can try to solve the following problems: Obtaining the explicit structure of the algebra, i.e., the set of basis elements and their commutator table. Giving the list of all different algebras depending on the different values of arbitrary parameters if they are present in the relations. Investigating some characteristics of an infinite-dimensional algebra such as its Hilbert series. Here we describe briefly a C program [l] for solving the above problems.

2. General outline of the program For construction of finitely presented Lie algebras several algorithms were elaborated and implemented in Reduce [Z-4]. In [S] the algorithm of paper [2] was

0168-9002/97/$17.00 Copyright c: 1997 Elsevier Science B.V. All rights reserved PII SO168-9002(97)00133-2

V.P. Gerdt, V.V. Komyak/Nucl.

Instr. and Meth. in Ph_vs. Rex A 389 (1997) 370-373

extended to Lie superalgebras. Unfortunately, these algorithms and programs are too slow to be applied to real problems. For our program we used another algorithmic approach whose details are described in Ref. [l]. The input data for the algorithm are: - The ordered set of generators X = {x1, x2, . } with prescribed parities QE ZZ and positive integer weights wi (= 1 by default). _ The set of scalar parameters P = { pl, pz, . } (optionally). _ The set of defining relations R = {rI, rz, . . . }, where ri are Lie polynomials with coefficients from the commutative ring Z[p,, p2, . ] of scalar polynomials. _ The limiting number of relations to be generated. The output data for the algorithm are: _ The interreduced set of consequences of the input relations E = {fl, ). - The list E = {el, e2. ., ) of algebra elements linearly independent modulo R. - The commutator table [ei, ej] = cfje,. _ The table of scalar polynomials in pi which have been treated as nonzero during computation. Particular values of pi providing vanishing of these polynomials may cause branching of computation and, hence, changes of the algebra structure. - The dimensions of homogeneous components, i.e. (initial part of) Hilbert series. There are three steps in the algorithm: (i) Generation of the relation set R”= {fl, F2;, . } of the consequences of the initial set R. This step executes the subsequent multiplying of relations by generators adding nonzero results to the set of relations and substituting these new relations into the other ones. The process terminates if either all newly arising relations are reduced to zero or the number of relations goes up to the limit fixed at input. The second case means that either the algebra is infinite-dimensional or the input limiting number of the relations is too small. (ii) Completion of the set E = {el,e2, }. Some elements ei are obtained at Step 1 as Lie (sub)monomials of fi. However, generally, the set E produced must be completed by those regular commutators of already existing elements which do not occur in E. In doing so one must verify whether new elements are indeed independent. It may happen that there exists a Jacobi identity containing the new element as a term, such that this identity is reduced to a new relation missing in the output of Step 1. If so one should add the new relation to I? and go back to Step 1. Besides, in the case of a Lie superalgebra the Lie squares of the odd elements are also to be added. This step guarantees the termination of the algorithm in the case of finite-dimensional algebra because all possible regular monomials and all relations involving them are constructed. (iii) Construction of the commutator table. At this step the commutators of the elements obtained at Step 2 are

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computed directly. The commutators produced are reduced modulo the relation set d. If the above algorithm terminates due to the input limiting number relations we cannot distinguish two alternatives: either the algebra is finite-dimensional but the limiting number is too small or the algebra is infinitedimensional. In the last case the truncated output makes sense only if all &E k are homogeneous. In this case we obtain an initial part of the whole Griibner basis, and the set E forms a subbasis of the Lie (super)algebra under construction. Otherwise, the algorithm termination means we have a reduced and, hence, minimal finite Griibner basis. Generally, it does not mean that the algebra is necessarily finite-dimensional. However, if at the last iteration of Step 2 no new elements ei were obtained, then we are done with a finitely dimensional algebra. The algorithm has been implemented in the C language. We used Borland C + + version 4.5 and GCC version 2.6.3 compilers under MS-DOS 6.22. We hope, however, that the C text can be compiled by other compilers because it contains only the standard C statements. The source code has a total length of almost 8000 lines and contains about 150 C functions realizing: top-level algorithms, Lie (super)algebra operations, manipulation with scalar polynomials, multiprecision integer arithmetic, substitutions. list processing, input and output handling, etc. Detailed instructions concerning compiling and running of the program with description of its limitations can be found in Ref. [6]. The version of the program suitable for running on both “low ending” (IBM PC) and “big ending” (SUN) machines can be obtained by anonymous ftp://canc.can.nl/pub/kornyak.

3. Examples of program application We present here some examples of application program to the problems listed in Introduction.

of the

3.1. Computation of the explicit structure of an algebra The following Serve relations hold for Cartan elements hi and Chevalley generators e,, fi corresponding to positive and negative simple roots of any (semi)simple complex Lie algebra L. [hi. h,] = 0,

(3)

Cei, .I;1=

b,h,,

(4)

[hi. ej] = ujiej,

(5)

[hi, jj] = - aji,fj,

(6)

(adei)’ -‘jcej = 0.

(7)

(ndJ)’ -“~yr,= 0,

(8)

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where aij is the Cartan matrix. i, j = 1, , r = rank L. For the exceptional Lie algebra Es the number of Serre relations is 290. The program generates the Grobner basis which contains 23074 relations involving Lie monomials up to degree 58 while the Lie algebra basis elements go up to 29th degree. The commutator table contains 7710 non-zero commutators. The task requires 3 min 14 s of computing time (on 66 MHz AT/486 PC) and 815 516 bytes of memory. This is an example of rather large problem demonstrating abilities of the program. 3.2. Problem

of classi$cation

The following relations arise in investigation symmetries of N = 1 supersymmetric fermionic sion of the Korteweg-de Vries equation:

[7] of exten-

program non-zero P,

4,

prints out the list of expressions in the process of computation: P - 1,

4 - 3,

+ [Cx*, Cr3 x211,

x31 - 3CY, XII = 0, (9) (10) (11) (12)

to be

q - 6.

A possible branching of the algebra structure can be investigated by equating these expressions to zero. For generic values of p and 4 we obtain an eight-dimensional nilpotent Lie superalgebra. For the cases p = 0 or q = 3 or q = 6 the program indicates infinite dimensionality which remains to be proved. In Ref. [7] the algebra for 4 = 6 is proved to be infinite-dimensional and identified with the product of the eight-dimensional nilpotent algebra and the positive subalgebra of the twisted Kac-Moody superalgebra C@’ (2). 3.3. Hilbert series for an injinite-dimensional If L is a graded

3CCCY,x11, x21,x11

assumed

algebra

algebra,

L = @ Li. we define the Hilbert H(t) = c

series H(t) of L as follows:

dim&?.

The following defining [8] of Lie superalgebras [x,x]

relations arose in investigation defined by the quadratic forms:

= [x, y] = [x, Z] = [z, u] = [u, v] = [u, v] = 0,

(13) (26) (14)

cy,

21- [x, ul = 0,

6’7)

(15)

[z,z] +2[x,v]

(16)

CY,

ul - cu,xl = 0,

(29)

(17)

cz, ul - CY,VI = 0.

(30)

(18) (19)

H(t) = 5t + St2 + 3t3 + 3t4 + 5t5 + 6t6 + 3t’ + 3t8 + 5tg + 6t’” + ...

H(t) =

(24) (25) and y is odd where xi, x2, x3 are even generators generator; p and q are arbitrary scalar parameters. The

.

By observing the periodicity in coefficients rewritten as the rational function

(22) (23)

(28)

Here all generators x, y, x, u and v are odd. If one sets all weights for generators equal to unit, the program generates (presumably) the infinite-dimensional algebra and prints out the initial part of its Hilbert series:

(20) (21)

=o,

this series is

t(5 + 5t + 3t2 + 3t3 + t5) 1 - t4

In more complicated cases one can guess a rational function presenting a Hilbert series using the following recipe [9]. Taking more and more terms in the series for H(t) compute the Pade approximants (i.e. best rational function approximation of the given polynomial) until these approximants are getting to be stable. However,

V.P. Gerdt, V.% Kornyak/Nucl.

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insfr. and Meth. in Phys. Res. A 389 (19973 370-373

the obtained function is only prediction which must be proved.

4. Conclusion

Our C code can be easily modified for working over finite fields and generalized to handle color Lie superalgebras with finite grading groups [lo]. More important development could be adding of additional means for analyzing the structure of obtained algebras, because it is not clear what should be done with thousands of output relations. The program can be also modified for working with jniteIy generated but injinitely defined algebras in the case when infinite set of defining relations can be successively created in some regular way. Such problems arise in physics, e.g., in the investigation [11] of the algebra of the invariant charges in the Nambu-Goto string theory.

Acknowledgements

We are grateful to A. Cohen, D. Leites, D. Pasechnik and J.-E. Roos for fruitful discussions. This work was supported by Grant No. INTAS-93-1180.

References Cl1 V.P. Gerdt and V.V. Kornyak, J. Symb. Comput. 21(1996) 337. PI P.K.H. Gragert. Acta Appl. Math. 16 (1989) 231. c31 I.R. Akselrod, V.P. Gerdt, V.E. Kovtun and V.N. Robuk, in: Comput. Algebra in Phys. Res., ed. D.V. Shirkov et al. (World Scientific, Singapore, 1991) p. 306. On conc41 V.P. Gerdt, V.N. Robuk and V.M. Severyanov, struction of finitely presented Lie algebras, Preprint JINR E5-94-302, Dubna 1994. Roelofs, The LIESUPER Package for c51 G.H.M. REDUCE, Memorandum 943, Univ. of Twente. Netherlands, 1991). C61 V.P. Gerdt and V.V. Kornyak, A program for constructing a complete set of relations, basis elements and their commutator table for finitely presented Lie algebras and superalgebras, Preprint JlNR El l-96-103. Dubna 1996. structures of supersymmec71 G.H.M. Roelofs, Prolongation tric systems, Ph.D. Thesis. Univ. of Twente. Enschede, Netherlands, 1993. PI J.-E. Roos, private communication. c91 J.-E. Roos, J. Pure Appl. Algebra 91 (1994) 255. and rw YuA. Bahturin, A.A. Mikhalev, V.M. Petrogradsky M.V. Zaicev, Infinite Dimensional Lie Superalgebras (Walter de Gruyter, Berlin, New York, 1992). Cl11 K. Pohlmeyer, Comm. Math. Phys. 163 (1994) 629.

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