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How to define using the GAP package, with the Lie algebra package included, some classical Lie algebras like sl(n,C), so(n,R), su(n) etc. without using commands referring to the type A_n?

  • Possibly one has first to define the corresponding associative algebras.

  • Or one has to define first the classical groups SL(n,C), SO(R,n), SU(n) and then apply a command which gives the Lie algebra of a Lie group? Which command?

Jo Wehler
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  • It is unclear to me what your desired input and output should be. Why don't you want to use the Dynkin diagram? Also be aware that GAP does implement cyclotomic fields (and maybe algebraic extensions thereof) but does not work with real or complex numbers. – ahulpke Mar 22 '18 at 16:16
  • In the most simple case the input should be: Consider the set of all nxn-matrices A with coefficients from the field K and with trace A = 0, i.e. the Lie algebra sl(n,K). The output should be the same GAP-object as the Lie algebra A_n-1 over K. In more refined versions the input should be the set of skew-symmetric matrices etc. For the reason, why I don't want to start with A_n etc., see the discussion with Dietrich. – Jo Wehler Mar 22 '18 at 16:47
  • I fear the answer then simply is that there is no such command. GAP has no object that would represent the structureless set of matrices $A$. It also in general does not have functionality that would "upgrade" a set of objects to an algebraic structure by imposing a new multiplication on these objects. If you want a group, you need to make the object as Group etc. Instead structures are represented in a computationally convenient way -- this might not always be the most natural for every user. So, for better or worse, the Lie algebras are new structures. – ahulpke Mar 22 '18 at 21:47

1 Answers1

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Use the user-contributed Lie algebra package by Willem de Graaf and Thomas Breuer for GAP, see here.

Dietrich Burde
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  • My question refers to the use of exactly this package. It is included in the standard GAP-installation. Being a beginner with GAP I did not find the answer to my question in the reference manual, § 64. – Jo Wehler Mar 22 '18 at 12:21
  • Thanks, I made an addition to my question. I know the difference between Lie groups, associative algebras and Lie algebras. But the Lie algebra package has a command which converts an associative algebra to the corresponding Lie algebra. – Jo Wehler Mar 22 '18 at 12:28
  • You can always just define your Lie algebras using "LieAlgebraByStructureConstants". And why not use type $A_n,B_n$ etc.? – Dietrich Burde Mar 22 '18 at 14:39
  • Due to pedagogical reasons: 1. Type A_n, B_n etc. are the result at the end of a course on the structure of semsimple Lie algebras. 2. Definition by structure constants is possible. 3. IMO the most natural way to introduce Lie algebras seems the linearization of the classical groups. – Jo Wehler Mar 22 '18 at 15:06