The Scope of this Document

Lie groups and algebras

Sage can be used to do standard computations for Lie groups and Lie algebras. The following categories of representations are equivalent:

  • Complex representations of a compact, semisimple simply connected Lie group \(G\).
  • Complex representations of its Lie algebra \(\mathfrak{g}\). This is a real Lie algebra, so representations are not required to be complex linear maps.
  • Complex representations of its complexified Lie algebra \(\mathfrak{g}_{\mathbf{C}} = \mathbf{C} \otimes \mathfrak{g}\). This is a complex Lie algebra and representations are required to be complex linear transformations.
  • The complex analytic representations of the semisimple simply-connected complex analytic group \(G_{\mathbf{C}}\) having \(\mathfrak{g}_{\mathbf{C}}\) as its Lie algebra.
  • Modules of the universal enveloping algebra \(U(\mathfrak{g}_{\mathbf{C}})\).
  • Modules of the quantized enveloping algebra \(U_q(\mathfrak{g}_{\mathbf{C}})\).

For example, we could take \(G = SU(n)\), \(\mathfrak{g} = \mathfrak{sl}(n, \mathbf{R})\), \(\mathfrak{g}_{\mathbf{C}} = \mathfrak{sl}(n, \mathbf{C})\) and \(G_{\mathbf{C}} = SL(n, \mathbf{C})\). Because these categories are the same, their representations may be studied simultaneously. The above equivalences may be expanded to include reductive groups like \(U(n)\) and \(GL(n)\) with a bit of care.

Here are some typical problems that can be solved using Sage:

  • Decompose a module in any one of these categories into irreducibles.
  • Compute the Frobenius-Schur indicator of an irreducible module.
  • Compute the tensor product of two modules.
  • If \(H\) is a subgroup of \(G\), study the restriction of modules for \(G\) to \(H\). The solution to this problem is called a branching rule.
  • Find the multiplicities of the weights of the representation.

In addition to its representations, which we may study as above, a Lie group has various related structures. These include:

  • The Weyl Group \(W\).
  • The Weight Lattice.
  • The Root System
  • The Cartan Type.
  • The Dynkin diagram.
  • The extended Dynkin diagram.

Sage contains methods for working with these structures.

If there is something you need that is not implemented, getting it added to Sage will likely be possible. You may write your own algorithm for an unimplemented task, and if it is something others will be interested in, it is probably possible to get it added to Sage.

Combinatorics

Sage supports a great many related mathematical objects. Some of these properly belong to combinatorics. It is beyond the scope of these notes to cover all the combinatorics in Sage, but we will try to touch on those combinatorial methods which have some connection with Lie groups and representation theory. These include:

  • The affine Weyl group, an infinite group containing \(W\).
  • Kashiwara crystals, which are combinatorial analogs of modules in the above categories.
  • Coxeter group methods applicable to Weyl groups and the affine Weyl group, such as Bruhat order.
  • The Iwahori Hecke algebras, which are deformations of the group algebras of \(W\) and the affine Weyl group.
  • Kazhdan-Lusztig polynomials.