Pan-graded Lie Superalgebras

For a given negatively graded lie superalgebra K-, the universal graded Lie su-peralgebra U of type K- is defined and its existence proved. By posing additional conditions on K- , other types of universal graded Lie superalgebras are defined and discussed. The concept of universal graded Lie superalgebra leads naturally to the graded Cartan type Lie superalgrebras and it is proved that the graded Cartan type Lie superalgrebras K(m, n, O>A), S(m, n) and H(m, n) can be characterized as certain types of universal graded Lie superalgrebras.The main results in this paper are the following:Theorem2.1: For any given negatively graded Lie Superalgrebra K~ the universal graded Lie superalgrebra of type K~ is the unique maximal homogeneous subalge-bral of co(m, n, K ) with negative part P(K ) .Therorem 2.2: W(m, n), with the ordinary gradation, is the universal graded Lie superalgrebra of depth 1 and breadth m+ n(s).Therorem 3.1: U(K~ ) =Pw(m, n,K~ )(p(K~ ))0Therorem 4.2: The universal graded Lie is superalgrebra of even Heisenberg type is exactly the contact Cartan type algebra. K(m, n, O>A) : = 1x6 W(m, n, A") I xo)AG A (m, n)coA,m=2r+ 1[ .Therorem 5.3: The Cartan type superalgebras S(m, n) and H(m, n) are the u-niveral graded Lie superalgebra of type sl(m, n) and osp(m, n), respectively.