- : We start with , the Cartan matrix of which is Equation (4.4). The defining relations are then: The commutator is not killed by the defining relations and hence is not equal to zero (the defining relations are all the relations). All the commutators with three (or more) ’s are however zero. A similar phenomenon occurs on the negative side. Hence, the algebra is eight-dimensional and one may take as basis . The vector corresponds to the positive root .
- : The algebra , the Cartan matrix of which is Equation (4.5), is defined by the same set of
generators, but the Serre relations are now and (and similar
relations for the ’s). The algebra is still finite-dimensional and contains, besides the generators, the
commutators , , their negative counterparts and , and
nothing else. The triple commutator vanishes by the Serre relations. The other
triple commutator vanishes also by the Jacobi identity and the Serre
relations,
(Each term on the right-hand side is zero: The first by antisymmetry of the bracket and the second because .) The algebra is 10-dimensional and is isomorphic to .

- : We now turn to , the Cartan matrix of which is Equation (4.8). This algebra is defined by
the same set of generators as , but with Serre relations given by
(and similar relations for the ’s). This innocent-looking change in the Serre relations has
dramatic consequences because the corresponding algebra is infinite-dimensional. (We analyze
here the algebra generated by the ’s, ’s and ’s, which is in fact the derived
Kac–Moody algebra – see Section 4.5 on affine Kac–Moody algebras. The derived algebra is
already infinite-dimensional.) To see this, consider the current algebra, defined by
where , are the structure constants of and where is the invariant
metric on which we normalize here so that . The subalgebra with is
isomorphic to ,
The current algebra (4.22) is generated by , , and since any element can be written as a multi-commutator involving them. The map

preserves the defining relations of the Kac–Moody algebra and defines an isomorphism of the (derived) Kac–Moody algebra with the current algebra. The Kac–Moody algebra is therefore infinite-dimensional. One can construct non-vanishing infinite multi-commutators, in which and alternate: The Serre relations do not cut the chains of multi-commutators to a finite number.

We see from these examples that the exact consequences of the Serre relations might be intricate to derive explicitly. This is one of the difficulties of the theory.

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