Borel subalgebras of restricted Cartan -Type Lie algebras.

It is still an open problem to determine the conjugacy classes of Borel subalgebras of non-classical type Lie algebras. In this paper, we prove that there are at least 2 conjugacy classes of Borel subalgebras as well as maximal triangulable subalgebras of restricted Cartan type Lie algebras of type W, S and H. We are particularly interested in maximal triangulable subalgebras of W (n) under some conditions which is called B -subalgebras (Definition 3.1). We classify the conjugacy classes of B -subalgebras for W (n) and determine their representatives. This paper and its sequel [Z. Lin, K. Ou and B. Shu, Geometric Setting of Jacobson–Witt Algebras, preprint] attempt to establish both algebraic and geometric setting for geometric representation theory of W (n). [ABSTRACT FROM AUTHOR]