Description of the simple Lie algebras

If F is a finite field, then, up to isomorphism, there is precisely one simple Lie algebra with dimension 3, and another one with dimension 6; these can be accessed by calling NonSolvableLieAlgebra(F,[3,1]) and NonSolvableLieAlgebra(F,[6,2]) (see NonSolvableLieAlgebra for the details). Over a field of characteristic 5, there is an additional simple Lie algebra with dimension 5, namely NonSolvableLieAlgebra(F,[5,3]). These are the only isomorphism types of simple Lie algebras over finite fields up to dimension 6.

In addition to the algebras above the package contains the simple Lie algebras of dimension between 7 and 9 over GF(2). These Lie algebras were determined by and can be described as follows.

There are two isomorphism classes of 7-dimensional Lie algebras over GF(2). In a basis b1,\ldots,b7 the non-trivial products in the first algebra are

[b1,b2]=b3, [b1,b3]=b4, [b1,b4]=b5, [b1,b5]=b6 [b1,b6]=b7, [b1,b7]=b1, [b2,b7]=b2, [b3,b6]=b2, [b4,b5]=b2, [b4,b6]=b3, [b4,b7]=b4, [b6,b7]=b6;

and those in the second are [b1,b2]=b3, [b1,b3]=b1+b4, [b1,b4]=b5, [b1,b5]=b6, [b1,b6]=b7, [b2,b3]=b2, [b2,b5]=b2+b4, [b2,b6]=b5, [b2,b7]=b1+b4, [b3,b4]=b2+b4, [b3,b5]=b3, [b3,b6]=b1+b4+b6, [b3,b7]=b5, [b4,b7]=b6, [b5,b6]=b6, [b5,b7]=b7.

Over GF(2) there are two isomorphism types of simple Lie algebras with dimension 8. In the basis b1,\ldots,b8 the non-trivial products for the first one are

[b1,b3]=b5, [b1,b4]=b6, [b1,b7]=b2, [b1,b8]=b1, [b2,b3]=b7, [b2,b4]=b5+b8, [b2,b5]=b2, [b2,b6]=b1, [b2,b8]=b2, [b3,b6]=b4, [b3,b8]=b3, [b4,b5]=b4, [b4,b7]=b3, [b4,b8]=b4, [b5,b6]=b6, [b5,b7]=b7, [b6,b7]=b8;

and for the second one they are [b1,b2]=b3, [b1,b3]=b2+b5, [b1,b4]=b6, [b1,b5]=b2, [b1,b6]=b1+b4+b8, [b1,b8]=b4, [b2,b3]=b4, [b2,b4]=b1, [b2,b5]=b6, [b2,b6]=b2+b7, [b2,b7]=b2+b5, [b3,b4]=b2+b7, [b3,b5]=b1+b4+b8, [b3,b6]=b1, [b3,b7]=b2+b3, [b3,b8]=b1, [b4,b5]=b3, [b4,b6]=b2+b4, [b4,b7]=b1+b4+b8, [b4,b8]=b3, [b5,b6]=b1+b2+b5, [b5,b7]=b3, [b5,b8]=b2+b7, [b6,b7]=b4+b6, [b6,b8]=b2+b5, [b7,b8]=b6.

The non-trivial products for the unique simple Lie algebra with dimension 9 over GF(2) are as follows:

[b1,b2]=b3, [b1,b3]=b5, [b1,b5]=b6, [b1,b6]=b7, [b1,b7]=b6+b9, [b1,b9]=b2, [b2,b3]=b4, [b2,b4]=b6, [b2,b6]=b8, [b2,b8]=b6+b9, [b2,b9]=b1, [b3,b4]=b7, [b3,b5]=b8, [b3,b7]=b1+b8, [b3,b8]=b2+b7, [b4,b5]=b6+b9, [b4,b6]=b2+b7, [b4,b7]=b3+b6+b9, [b4,b9]=b5, [b5,b6]=b1+b8, [b5,b8]=b3+b6+b9, [b5,b9]=b4, [b6,b7]=b1+b4+b8, [b6,b8]=b2+b5+b7, [b7,b8]=b3+b9, [b7,b9]=b8, [b8,b9]=b7.