Z_2^n grading of the classical Lie algebras

TL;DR

This paper explores the Z_2^n gradings of classical Lie algebras by expressing them through matrix and Clifford algebras, revealing higher gradings linked to matrix dimensions divisible by powers of two.

Contribution

It provides a detailed description of Z_2^n gradings of classical Lie algebras using matrix and Clifford algebra frameworks, including higher gradings for specific dimensions.

Findings

Classical Lie algebras can be expressed with matrix and Clifford algebras carrying Z_2^n gradings.

Higher gradings are identified when matrix dimensions are divisible by powers of two.

The gradings range from zero to Z_2^3, depending on the algebra and dimension.

Abstract

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.