2025/04/28 by Sharma, Heerak, Shirokov, Dmitry · 1 citation
#15A66 #15A69 #Commutative Algebra (math.AC) #FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2504.19763
We investigate commutative analogues of Clifford algebras -- algebras whose generators square to ±1 but commute, instead of anti-commuting as they do in Clifford algebras. We observe that commutativity allows for elegant results. We note that these algebras generalise multicomplex spaces -- we show that a commutative analogue of Clifford algebra is either isomorphic to a multicomplex space or to `multi split-complex space' (space defined just like multicomplex numbers but uses split-complex numbers instead of complex numbers). We do a general study of commutative analogues of Clifford algebras and use tools like operations of conjugation and idempotents to give a tensor product decomposition and a direct sum decomposition for them. Tensor product decomposition follows relatively easily from the definition. For the direct sum decomposition, we give explicit basis using new techniques.