18.747: Problem set 2; due Thursday, February 23

1. (a) Check directly that the positive part n_{+} of Lie algebra sl(3) is defined by the standard generators e_{i} and the Serre relations.

(b) Do the same for sp(4). Hint: Try to write down the basis of the Lie algebra generated by e_{i} and the Serre relations, and compare its cardinality with dim(n_{+}).

1. Establish Serre relations for the affine Lie algebra which is the 1-dimensional central extension of the loop algebra Lg of a simple Lie algebra g.

Kac’s book, exersices 7.1, 7.2, 7.3, 7.4, 7.6, 7.7.