NOTTINGHAM ALGEBRA WITH DIAMOND OF FINITE AND INFINITE TYPE - Agung Shahputra, Rizky
Thin Lie algebras are infinite-dimensional graded Lie algebras L = L? i=1, where Li’s is a subspace of L called a homogeneous component. The first homogeneous component is two-dimensional and equipped with a covering property, which results in its homogeneous components having no more than two dimensions. The twodimensional homogeneous component is called a diamond. Hence the first homogeneous component is the first diamond. It has been well-known that the degree of the second diamond can only be 3, 5, q, or 2q?1, where q is a power of the field characteristic. We will give more straightforward proof regarding the possible values for the degree of the second diamond. Nottingham algebras are thin Lie algebras whose degree of the second diamond is in the form q. In Nottingham algebras, every diamond except the first can be assigned to a type that belongs to the underlying field or is equal to?. We will give general structure to the Nottingham algebras, such as the pattern of the types on the diamonds and the distance between the diamonds. We will also construct a class of Nottingham algebras as loop algebras of Hamiltonian algebras H(2 : (s+1, n); ?0) of Cartan type. One can show that the exponential of a nilpotent derivation gives new grading for a graded Lie algebras, such grading switching plays an important role in our construction.