On Lie Algebras Defined by Tangent Directions to Homogeneous Projective Varieties

Let $X$ be an embedded projective variety. The Lie algebra $\mathfrak L$ defined by the tangent directions to $X$ at smooth points is an interesting algebraic invariant of $X$. In some cases, this algebra is isomorphic to the symbol algebra of a filtered system of distributions on a Fano manifold, which plays an important role in the theory of these manifolds. In addition, algebras defined by tangent directions are interesting on their own right. In this paper, we study the Lie algebra $\mathfrak L$ corresponding to a variety $X$ that is the projectivization of the orbit of the lowest weight vector of an irreducible representation of a complex semisimple Lie group. We describe these algebras in terms of generators and relations. In many cases, we can describe their structure completely.