1 Lax operator and the loop algebra 2 Further generalization

Generalizations of the Lax pair

#### 1Lax operator and the loop algebra

One can interpret , , , as generators of the twisted loop superalgebra .

The word ``twisted'' means that the power of the spectral parameter mod 4 should correlate with the -grading of the generators

This observation allows us to rewrite (30) and (31) as follows:
 (34)
 (35)
where replaces etc.; operators are generators of the twisted loop superalgebra. Withe these new notations, the spectral parameter is not present in . Instead of entering explicitly in , it now parametrizes a representation of the generators .

#### 2Further generalization

The basic relations (23) imply:
 (36)
 (37)
and similar equations for the commutators of .

Eq. (36) we already discussed; it encodes the SUGRA constraints and at the same time defines . Eq. (37) is a theorem-definition: the theorem says that the left hand side is proportional to , and the definition is of

It turns out, that there is the following generalization of the Lax pair:
 (38)
 (39)