Affine.m - Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

Abstract In this paper we present Affine.m —a program for computations in representation theory of finite-dimensional and affine Lie algebras and describe implemented algorithms. The algorithms are based on the properties of weights and Weyl symmetry. Computation of weight multiplicities in irreducible and Verma modules, branching of representations and tensor product decomposition are the most important problems for us. These problems have numerous applications in physics and we provide some examples of these applications. The program is implemented in the popular computer algebra system Mathematica and works with finite-dimensional and affine Lie algebras. Program summary Program title: Affine.m Catalogue identifier: AENA_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AENB_v1_0.html Program obtainable from: CPC Program Library, Queen’s University, Belfast, UK Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 24 844 No. of bytes in distributed program, including test data, etc.: 1 045 908 Distribution format: tar.gz Programming language: Mathematica. Computer: i386–i686, x86_64. Operating system: Linux, Windows, Mac OS, Solaris. RAM: 5–500 Mb Classification: 4.2, 5. Nature of problem: Representation theory of finite-dimensional Lie algebras has many applications in different branches of physics, including elementary particle physics, molecular physics, nuclear physics. Representations of affine Lie algebras appear in string theories and two-dimensional conformal field theory used for the description of critical phenomena in two-dimensional systems. Also Lie symmetries play a major role in a study of quantum integrable systems. Solution method: We work with weights and roots of finite-dimensional and affine Lie algebras and use Weyl symmetry extensively. Central problems which are the computations of weight multiplicities, branching and fusion coefficients are solved using one general recurrent algorithm based on generalization of Weyl character formula. We also offer alternative implementation based on the Freudenthal multiplicity formula which can be faster in some cases. Restrictions: Computational complexity grows fast with the rank of an algebra, so computations for algebras of ranks greater than 8 are not practical. Unusual features: We offer the possibility of using a traditional mathematical notation for the objects in representation theory of Lie algebras in computations if Affine.m is used in the Mathematica notebook interface. Running time: From seconds to days depending on the rank of the algebra and the complexity of the representation.

[1]  A. Kent,et al.  Virasoro algebras and coset space models , 1985 .

[2]  AUTOMATA TO PERFORM BASICCALCULATIONS IN COXETER , .

[3]  John B. Shoven,et al.  I , Edinburgh Medical and Surgical Journal.

[4]  M. Walton.  Affine Kac-Moody Algebras and the Wess-Zumino-Witten Model , 1999, hep-th/9911187.

[5]  Roger W. Carter,et al.  Lie Algebras of Finite and Affine Type , 2005 .

[6]  I. Gel'fand,et al.  Structure of representations generated by vectors of highest weight , 1971 .

[7]  J. Humphreys.  Reflection groups and coxeter groups , 1990 .

[8]  V. Lyakhovsky,et al.  Recursive algorithm and branching for nonmaximal embeddings , 2010, 1007.0318.

[9]  Affine Lie Algebras , 2000 .

[10]  D. Kazhdan,et al.  Tensor structures arising from affine Lie algebras. III , 1993 .

[11]  Pierre Mathieu,et al.  Conformal Field Theory , 1999 .

[12]  J. Humphreys.  Representations of Semisimple Lie Algebras in the BGG Category O , 2008 .

[13]  W. Soergel.  Book Review: Representations of semisimple Lie algebras in the BGG category $\mathcal O$ , 2009 .

[14]  J. Fuchs,et al.  UvA-DARE ( Digital Academic Repository ) A matrix S for all simple current extensions , 2003 .

[15]  B. Kolman,et al.  A survey of Lie groups and Lie algebras with applications and computational methods , 1972 .

[16]  V. Lyakhovsky,et al.  String Functions for Affine Lie Algebras Integrable Modules , 2008, 0812.2381.

[17]  Bernard Kolman,et al.  A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods. , 1973 .

[18]  J. Humphreys.  Introduction to Lie Algebras and Representation Theory , 1973 .

[19]  W. Casselman.  Machine calculations in Weyl groups , 1994 .

[20]  V. Kac,et al.  Modular and conformal invariance constraints in representation theory of affine algebras , 1988 .

[21]  V. Lyakhovsky,et al.  Folded fans and string functions , 2010 .

[22]  Joe Harris,et al.  Representation Theory: A First Course , 1991 .

[23]  I. Gel'fand,et al.  Category ofg modules , 1976 .

[24]  Thomas Fischbacher,et al.  Introducing LambdaTensor1.0 - A package for explicit symbolic and numeric Lie algebra and Lie group calculations , 2002, ArXiv.

[25]  M. V. Leeuwen,et al.  LiE, a software package for Lie group computations , 1994 .

[26]  Roman E. Maeder.  Computer Science with MATHEMATICA ®: Theory and Practice for Science, Mathematics, and Engineering , 2000 .

[27]  Algorithms for affine Kac-Moody algebras , 2001, hep-th/0106123.

[28]  E C G Sudarshan,et al.  Lie Groups and Lie Algebras , 2010 .

[29]  M. Walton,et al.  Conformal branching rules and modular invariants , 1989 .

[30]  V. Kac,et al.  Infinite-dimensional Lie algebras, theta functions and modular forms , 1984 .

[31]  John R. Stembridge,et al.  A Maple Package for Symmetric Functions , 1995, J. Symb. Comput..

[32]  W. Ledermann.  INTRODUCTION TO LIE ALGEBRAS AND REPRESENTATION THEORY , 1974 .

[33]  J. Patera,et al.  Fast recursion formula for weight multiplicities , 1982 .

[34]  V. Lyakhovsky,et al.  Recursive properties of branching and BGG resolution , 2011, 1102.1702.