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isLieAlgebraRepresentation -- checks whether a list of matrices defines a Lie algebra representation

Description

Let LAB be a basis of $\mathfrak{g}$, and let $L$ be a list of $n \times n$ matrices with $\#L = \#LAB$. Let $\rho: \mathfrak{g} \rightarrow \mathfrak{gl}_n$ be the linear transformation defined by mapping $B_i$ in LAB to $L_i$. This function checks whether $\rho$ preserves the Lie bracket; that is, for each pair of indices $i,j$, if $[B_i,B_j] = \sum c_{ijk} B_k$, then is $[\rho(B_i),\rho(B_j)] = \sum c_{ijk} \rho(B_k)$?

In the example below, we compute the adjoint representation of $sl_3$ directly, and check that the list of matrices we obtain defines a Lie algebra representation.

i1 : sl3 = simpleLieAlgebra("A",2);
 
i2 : LAB = lieAlgebraBasis("A",2);
 
i3 : br = LAB#"Bracket";
 
i4 : writeInBasis = LAB#"WriteInBasis";
 
i5 : B = LAB#"BasisElements"

o5 = {| 1 0  0 |, | 0 0 0  |, | 0 1 0 |, | 0 0 0 |, | 0 0 1 |, | 0 0 0 |, | 0
      | 0 -1 0 |  | 0 1 0  |  | 0 0 0 |  | 0 0 1 |  | 0 0 0 |  | 1 0 0 |  | 0
      | 0 0  0 |  | 0 0 -1 |  | 0 0 0 |  | 0 0 0 |  | 0 0 0 |  | 0 0 0 |  | 0
     ------------------------------------------------------------------------
     0 0 |, | 0 0 0 |}
     0 0 |  | 0 0 0 |
     1 0 |  | 1 0 0 |

o5 : List
 
i6 : ad = X -> transpose matrix apply(B, Y -> writeInBasis br(X,Y))

o6 = ad

o6 : FunctionClosure
 
i7 : L1 = apply(B, X -> ad X)

o7 = {| 0 0 0 0  0 0  0 0  |, | 0 0 0  0 0 0 0  0  |, | 0  0 0 0 0 1 0 0  |,
      | 0 0 0 0  0 0  0 0  |  | 0 0 0  0 0 0 0  0  |  | 0  0 0 0 0 0 0 0  | 
      | 0 0 2 0  0 0  0 0  |  | 0 0 -1 0 0 0 0  0  |  | -2 1 0 0 0 0 0 0  | 
      | 0 0 0 -1 0 0  0 0  |  | 0 0 0  2 0 0 0  0  |  | 0  0 0 0 0 0 0 0  | 
      | 0 0 0 0  1 0  0 0  |  | 0 0 0  0 1 0 0  0  |  | 0  0 0 1 0 0 0 0  | 
      | 0 0 0 0  0 -2 0 0  |  | 0 0 0  0 0 1 0  0  |  | 0  0 0 0 0 0 0 0  | 
      | 0 0 0 0  0 0  1 0  |  | 0 0 0  0 0 0 -2 0  |  | 0  0 0 0 0 0 0 -1 | 
      | 0 0 0 0  0 0  0 -1 |  | 0 0 0  0 0 0 0  -1 |  | 0  0 0 0 0 0 0 0  | 
     ------------------------------------------------------------------------
     | 0 0  0  0 0 0 0 0 |, | 0  0  0 0 0 0  0 1 |, | 0 0  -1 0 0 0 0  0 |, |
     | 0 0  0  0 0 0 1 0 |  | 0  0  0 0 0 0  0 1 |  | 0 0  0  0 0 0 0  0 |  |
     | 0 0  0  0 0 0 0 0 |  | 0  0  0 0 0 0  1 0 |  | 0 0  0  0 0 0 0  0 |  |
     | 1 -2 0  0 0 0 0 0 |  | 0  0  0 0 0 -1 0 0 |  | 0 0  0  0 1 0 0  0 |  |
     | 0 0  -1 0 0 0 0 0 |  | -1 -1 0 0 0 0  0 0 |  | 0 0  0  0 0 0 0  0 |  |
     | 0 0  0  0 0 0 0 1 |  | 0  0  0 0 0 0  0 0 |  | 2 -1 0  0 0 0 0  0 |  |
     | 0 0  0  0 0 0 0 0 |  | 0  0  0 0 0 0  0 0 |  | 0 0  0  0 0 0 0  0 |  |
     | 0 0  0  0 0 0 0 0 |  | 0  0  0 0 0 0  0 0 |  | 0 0  0  0 0 0 -1 0 |  |
     ------------------------------------------------------------------------
     0  0 0 0  0  0 0 0 |, | 0 0 0 0  -1 0 0 0 |}
     0  0 0 -1 0  0 0 0 |  | 0 0 0 0  -1 0 0 0 |
     0  0 0 0  -1 0 0 0 |  | 0 0 0 0  0  0 0 0 |
     0  0 0 0  0  0 0 0 |  | 0 0 0 0  0  0 0 0 |
     0  0 0 0  0  0 0 0 |  | 0 0 0 0  0  0 0 0 |
     0  0 0 0  0  0 0 0 |  | 0 0 0 -1 0  0 0 0 |
     -1 2 0 0  0  0 0 0 |  | 0 0 1 0  0  0 0 0 |
     0  0 0 0  0  1 0 0 |  | 1 1 0 0  0  0 0 0 |

o7 : List
 
i8 : isLieAlgebraRepresentation(LAB,L1)

o8 = true

Next, we present an example where the linear transformation $\rho: sl_3 \rightarrow \mathfrak{gl}(\mathbb{C}^8)$ does not preserve the Lie bracket.

i9 : L2 = apply(#L1, i -> if i==6 then -2*L1_i else L1_i)

o9 = {| 0 0 0 0  0 0  0 0  |, | 0 0 0  0 0 0 0  0  |, | 0  0 0 0 0 1 0 0  |,
      | 0 0 0 0  0 0  0 0  |  | 0 0 0  0 0 0 0  0  |  | 0  0 0 0 0 0 0 0  | 
      | 0 0 2 0  0 0  0 0  |  | 0 0 -1 0 0 0 0  0  |  | -2 1 0 0 0 0 0 0  | 
      | 0 0 0 -1 0 0  0 0  |  | 0 0 0  2 0 0 0  0  |  | 0  0 0 0 0 0 0 0  | 
      | 0 0 0 0  1 0  0 0  |  | 0 0 0  0 1 0 0  0  |  | 0  0 0 1 0 0 0 0  | 
      | 0 0 0 0  0 -2 0 0  |  | 0 0 0  0 0 1 0  0  |  | 0  0 0 0 0 0 0 0  | 
      | 0 0 0 0  0 0  1 0  |  | 0 0 0  0 0 0 -2 0  |  | 0  0 0 0 0 0 0 -1 | 
      | 0 0 0 0  0 0  0 -1 |  | 0 0 0  0 0 0 0  -1 |  | 0  0 0 0 0 0 0 0  | 
     ------------------------------------------------------------------------
     | 0 0  0  0 0 0 0 0 |, | 0  0  0 0 0 0  0 1 |, | 0 0  -1 0 0 0 0  0 |, |
     | 0 0  0  0 0 0 1 0 |  | 0  0  0 0 0 0  0 1 |  | 0 0  0  0 0 0 0  0 |  |
     | 0 0  0  0 0 0 0 0 |  | 0  0  0 0 0 0  1 0 |  | 0 0  0  0 0 0 0  0 |  |
     | 1 -2 0  0 0 0 0 0 |  | 0  0  0 0 0 -1 0 0 |  | 0 0  0  0 1 0 0  0 |  |
     | 0 0  -1 0 0 0 0 0 |  | -1 -1 0 0 0 0  0 0 |  | 0 0  0  0 0 0 0  0 |  |
     | 0 0  0  0 0 0 0 1 |  | 0  0  0 0 0 0  0 0 |  | 2 -1 0  0 0 0 0  0 |  |
     | 0 0  0  0 0 0 0 0 |  | 0  0  0 0 0 0  0 0 |  | 0 0  0  0 0 0 0  0 |  |
     | 0 0  0  0 0 0 0 0 |  | 0  0  0 0 0 0  0 0 |  | 0 0  0  0 0 0 -1 0 |  |
     ------------------------------------------------------------------------
     0 0  0 0 0 0  0 0 |, | 0 0 0 0  -1 0 0 0 |}
     0 0  0 2 0 0  0 0 |  | 0 0 0 0  -1 0 0 0 |
     0 0  0 0 2 0  0 0 |  | 0 0 0 0  0  0 0 0 |
     0 0  0 0 0 0  0 0 |  | 0 0 0 0  0  0 0 0 |
     0 0  0 0 0 0  0 0 |  | 0 0 0 0  0  0 0 0 |
     0 0  0 0 0 0  0 0 |  | 0 0 0 -1 0  0 0 0 |
     2 -4 0 0 0 0  0 0 |  | 0 0 1 0  0  0 0 0 |
     0 0  0 0 0 -2 0 0 |  | 1 1 0 0  0  0 0 0 |

o9 : List
 
i10 : isLieAlgebraRepresentation(LAB,L2)
Brackets not compatible on basis elements {2, 7}

o10 = false

Ways to use isLieAlgebraRepresentation:

  • isLieAlgebraRepresentation(LieAlgebraBasis,List)

For the programmer

The object isLieAlgebraRepresentation is a method function.

The source of this document is in LieAlgebraRepresentations/documentation.m2:2594:0.