### References & Citations

# Mathematical Physics

# Title: Classification of minimal ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie (super)algebras and some applications

(Submitted on 7 Mar 2021 (v1), last revised 8 Jun 2021 (this version, v2))

Abstract: This paper presents the classification, over the fields of real and complex numbers, of the minimal ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Lie algebras and Lie superalgebras spanned by $4$ generators and with no empty graded sector. The inequivalent graded Lie (super)algebras are obtained by solving the constraints imposed by the respective graded Jacobi identities. A motivation for this mathematical result is to systematically investigate the properties of dynamical systems invariant under graded (super)algebras. Recent works only paid attention to the special case of the one-dimensional ${\mathbb Z}_2\times{\mathbb Z}_2$-graded Poincar\'e superalgebra. As applications, we are able to extend certain constructions originally introduced for this special superalgebra to other listed ${\mathbb Z}_2\times{\mathbb Z}_2$-graded (super)algebras. We mention, in particular, the notion of ${\mathbb Z_2}\times{\mathbb Z}_2$-graded superspace and of invariant dynamical systems (both classical worldline sigma models and quantum Hamiltonians). As a further byproduct we point out that, contrary to ${\mathbb Z}_2\times{\mathbb Z}_2$-graded superalgebras, a theory invariant under a ${\mathbb Z}_2\times{\mathbb Z}_2$-graded algebra implies the presence of ordinary bosons and three different types of exotic bosons, with exotic bosons of different types anticommuting among themselves.

## Submission history

From: Francesco Toppan [view email]**[v1]**Sun, 7 Mar 2021 16:01:43 GMT (21kb)

**[v2]**Tue, 8 Jun 2021 15:10:17 GMT (23kb)

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