The Z_2^n gradings of the classical Lie algebras are described. To elucidate the grading, the classical Lie algebras are expressed in terms of matrix algebras over one of eight fields or Clifford algebras which carry gradings ranging from zero to Z_2^3. When the matrix dimension is divisible by a power of two, the Lie algebras have higher gradings which may be expressed in terms of higher Clifford algebras: these are also described.
We note that large classes of contractions of algebras that arise in physics can be understood purely algebraically, via identifying appropriate $\mathbb{Z}_m$-gradings (and their generalizations) on the parent algebra. This includes various types of flat space/Carroll limits of finite and infinite dimensional (A)dS algebras, as well as Galilean and Galilean Conformal algebras. Our observations can be regarded as providing a natural context for the Grassmann approach of arXiv:1312.2941. We also introduce a related notion, which we call partial grading, that arises naturally in this context.
We give two constructions of grading-restricted vertex (super)algebras. We first give a new construction of a class of grading-restricted vertex (super)algebras originally obtained by Meurman and Primc using a different method. This construction is based on a new definition of vertex operators and a new method. Our second construction is a generalization of the author's construction of the moonshine module vertex operator algebra and a related vertex operator superalgebra. This construction needs properties of intertwining operators formulated and proved by the author.
We continue the study of Exceptional Periodicity and Magic Star algebras, which provide non-Lie, countably infinite chains of finite dimensional generalizations of exceptional Lie algebras. We analyze the graded algebraic structures arising in the Magic Star projection, as well as the Hermitian part of rank-3 Vinberg's matrix algebras (which we dub HT-algebras), occurring on each vertex of the Magic Star.
