The action of certain simple lie algebras on some of their modules

Abstract

Let L be one of the simple Lie algebras S1 (3, C) or ๐(5, C), V a finite-dimensional irreducible L-module, v+a maximal vector of V with highest weight . Set  (h1) = (m1) and (h2) = m2. Let {x_1, x_2, y_1, y_2} be a set of Chevalley generators of L and let h_1 = [x_1, y_1] for all iε {1, 2} For L = ๐ (5, C) a Verma basis of V consists of all elements of the form y_1 a3y2 a2y1 a1.v+where a_1 , a_2 , a_3 , a_4ε z_0^+ and 0 ≤ a_1≤ m_1 0 ≤a_2 ≤ m_2 + a_1 0 ≤ a_3 ≤ min {m_2, a_2}For L = ๐ (5, C), a Verma basis of V consists of all elements of the form y_1a4y2 a3y1 a2y2a1.v+ where a_1 , a_2 , a_3 , a_4ε z_0^+ and 0 ≤ a_1≤ m_20 ≤a_2 ≤ m_1 + a_1 0 ≤ a_3 ≤ min {m_1, a_2, 2a_2} 0 ≤ a_4 ≤ min {m_1, La_3/2J} The main results of this research are as follows : For L = S1 (3, C), we have found closed-form formulas for the actions of〖 x〗_1, x_2, y_1, and y_2 on the elements of the Verma basis of V given above. For L = ๐(5, C) similar results have been obtained, except that in some cases it has not been possible to find closed-form formulas for the actions. In these cases algorithms are given for calculating the desired actions.