A class of representations of the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded special linear Lie superalgebra $\mathfrak{s}\mathfrak{l}(m_1+1,m_2\mathrm{\mid }{n}_1,n_2)$ and quantum statistics

The description of the ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$-graded special linear Lie superalgebra $\mathfrak{s}\mathfrak{l}(m_1+1,m_2\mathrm{\mid }{n}_1,n_2)$ is carried out via generators $a_1^{\pm },\dots ,a_{m_1+m_2+n_1+n_2}^{\pm }$ that satisfy triple relations and are called creation and annihilation operators. With respect to these generators, a class of Fock type representations of $\mathfrak{s}\mathfrak{l}(m_1+1,m_2\mathrm{\mid }{n}_1,n_2)$ is constructed. The properties of the underlying statistics are discussed and its Pauli principle is formulated.