Physics NotePublished
Classification of minimal -graded Lie (super)algebras and some applications
F. Toppan; Zhanna Kuznetsova
Publication details
- Authors
- F. Toppan; Zhanna Kuznetsova
- Type
- Physics Note
- Status
- Published
- Publication date
- 11/03/2021
- Description
- 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.\par 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. \par 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 a quantum Hamiltonian).\par 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.
- Year
- 2021
- Number
- CBPF-NF-001/21
- Linked to a thesis
- No
- Linked to a grant
- No