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Douglas Algebras That Have No Maximal Subalgebra and No Minimal Superalgebra

Carroll Guillory

Abstract

We give several examples of Douglas Algebras that do not have any maximal subalgebra. We find a condition on these algebras that guarantees that some do not have any minimal superalgebra. We also show that if $A$ is the only maximal subalgebra of a Douglas algebra $B$, then the algebra $A$ does not have any maximal subalgebra.

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Key concepts: Subalgebra, Superalgebra, Mathematics, Cartan subalgebra, Pure mathematics, Algebra over a field, Lie conformal algebra, Adjoint representation of a Lie algebra