In this paper, we construct six families of conformal superalgebras of infinite type, motivated from free quadratic fermionic fields with derivatives and prove their simplicity. The Lie superalgebras generated by these conformal superalgebras are proved to be simple except a few special cases, in which the Lie superalgebras have one-dimensional centers and the quotient Lie superalgebras modulo the centers are simple. Certain natural central extensions of these families of conformal superalgebras are also given. Moreover, we prove that these conformal superalgebras are generated by their finite-dimensional subspaces of minimal weight in a certain sense. We show that a conformal superalgebra is simple if and only if its generated Lie superalgebra does not contain a proper nontrivial ideal...[ Read more ]

In this paper, we construct six families of conformal superalgebras of infinite type, motivated from free quadratic fermionic fields with derivatives and prove their simplicity. The Lie superalgebras generated by these conformal superalgebras are proved to be simple except a few special cases, in which the Lie superalgebras have one-dimensional centers and the quotient Lie superalgebras modulo the centers are simple. Certain natural central extensions of these families of conformal superalgebras are also given. Moreover, we prove that these conformal superalgebras are generated by their finite-dimensional subspaces of minimal weight in a certain sense. We show that a conformal superalgebra is simple if and only if its generated Lie superalgebra does not contain a proper nontrivial ideal with one-variable structure. The theory of mixed quadratic bosonic fermionic fields can be realized by pure fermionic fields. Some of our results are generalizations of Xu's results on simple conformal superalgebras of finite growth. Furthermore, we give a classification of the 2-cocycles of these Lie superalgebras. These 2-cocycles correspond to anomalies in quantum field theory. Lastly, we construct certain irreducible modules of classical Lie algebras over the left ideals of the algebra of differential operators on the circle.