A Class of Representations of the \({\Bbb Z}_2\times{\Bbb Z}_2\)-Graded Special Linear Lie Superalgebra \(\small\mathfrak{sl}(m_1+1,m_2|n_1,n_2)\) and Quantum Statistics

The description of the $\ZZ_2\times\ZZ_2$-graded special linear Lie superalgebra is carried out via a set of generators that satisfy triple relations and are called creation and annihilation operators. With respect to these generators, a class of Fock type representations of the algebra is constructed. The properties of the underlying statistics are discussed and its Pauli principle is formulated.