Consider a diagonal matrix $A$ with entries $e^{s_1},e^{s_2},\cdot\cdot\cdot$ on the diagonal s.t. $\sum_{i \ge1} s_i=0.$ From what I understand this is a subgroup of $SL_n(\Bbb R).$ This is because the determinant of a diagonal matrix is the product of the diagonal entries, and when we do this computation we get that the determinant equals $1$. The special linear group is the space of matrices with determinant $1$.
Does $A$ have a name? What is the motivation for studying this subgroup of $SL_n(\Bbb R)?$