Graded contractions of the Pauli graded \(\text{sl}(3,\mathbb C)\). (English) Zbl 1161.17305
Summary: The Lie algebra \(\text{sl}(3,\mathbb C)\) is considered in the basis of generalized Pauli matrices. Corresponding grading is the Pauli grading here. It is one of the four gradings of the algebra which cannot be further refined.
The set \(\mathcal S\) of 48 contraction equations for 24 contraction parameters is solved. Our main tools are the symmetry group of the Pauli grading of \(\text{sl}(3,\mathbb C)\), which is essentially the finite group \(\text{SL}(2,\mathbb Z_3)\), and the induced symmetry of the system \(\mathcal S\). A list of all equivalence classes of solutions of the contraction equations is provided. Among the solutions, 175 equivalence classes are non-parametric and 13 solutions depend on one or two continuous parameters, providing a continuum of equivalence classes and subsequently continuum of non-isomorphic Lie algebras. Solutions of the contraction equations of Pauli graded \(\text{sl}(3,\mathbb C)\) are identified here as specific solvable Lie algebras of dimensions up to 8. Earlier algorithms for identification of Lie algebras, given by their structure constants, had to be made more efficient in order to distinguish non-isomorphic Lie algebras encountered here.
The resulting Lie algebras are summarized in tabular form. There are 88 indecomposable solvable Lie algebras of dimension 8, 77 of them being nilpotent. There are 11 infinite sets of parametric Lie algebra which still deserve further study.
The set \(\mathcal S\) of 48 contraction equations for 24 contraction parameters is solved. Our main tools are the symmetry group of the Pauli grading of \(\text{sl}(3,\mathbb C)\), which is essentially the finite group \(\text{SL}(2,\mathbb Z_3)\), and the induced symmetry of the system \(\mathcal S\). A list of all equivalence classes of solutions of the contraction equations is provided. Among the solutions, 175 equivalence classes are non-parametric and 13 solutions depend on one or two continuous parameters, providing a continuum of equivalence classes and subsequently continuum of non-isomorphic Lie algebras. Solutions of the contraction equations of Pauli graded \(\text{sl}(3,\mathbb C)\) are identified here as specific solvable Lie algebras of dimensions up to 8. Earlier algorithms for identification of Lie algebras, given by their structure constants, had to be made more efficient in order to distinguish non-isomorphic Lie algebras encountered here.
The resulting Lie algebras are summarized in tabular form. There are 88 indecomposable solvable Lie algebras of dimension 8, 77 of them being nilpotent. There are 11 infinite sets of parametric Lie algebra which still deserve further study.
MSC:
| 17B20 | Simple, semisimple, reductive (super)algebras |
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
Keywords:
Lie algebra; \(\text{sl}(3,\mathbb C)\); Pauli grading; graded contraction; Lie algebra identificationReferences:
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