Restricted Lie algebras of bounded type

Richard D. Pollack
1968 Bulletin of the American Mathematical Society  
Introduction. It is known [13] that a Lie algebra over a modular field has indecomposable representations of arbitrarily high dimensionalities. If, however, the Lie algebra and its representations are required to be restricted (see [6, Chapter 5] for definitions), this need no longer be the case. A restricted Lie algebra for which the degrees of its (restricted) indecomposable representations are bounded by some constant is said to be of bounded type; one for which this is not the case is said
more » ... t the case is said to be of unbounded type. The simple three-dimensional Lie algebra, A x . Let A x be the split simple three-dimensional Lie algebra over the field K of characteristic p>3. Then Ai has a basis e, ƒ, h with [e, ƒ]=&, [e, h]~2e, [ƒ, h]= -2/ and with £-power mapping given by e p =f p = 0, h p = h. There are p inequivalent irreducible (restricted) modules for Au classified by their highest weight. Let M\, OgX^£-1, be the irreducible ^U-module with highest weight X, so that [M\: K] = X + 1 [5]. Let U be the w-algebra [ô] of Ai and U= 23?-I © U$ its decomposition into its principal indecomposable modules (p.i.m.). Since U is a symmetric algebra [9] each Uj has a unique top and bottom composition factor, these are isomorphic, and each M\ is isomorphic to the top composition factor of some Uj [2]. If M is an -4i-module, denote by M~M\ V M\ v • • • , M\ t the fact that the M\ p in the given order, are the composition factors of some composition series for M.
doi:10.1090/s0002-9904-1968-11943-3 fatcat:vtsvtxayqve6ddwixkdgl6p4ge