Non-combinatorial involutive braidings: the quantum algebra ${\mathfrak{g}\mathfrak{l}}_{k,m}$

We investigate involutive, non-combinatorial solutions of the braid equation, viewing them as special deformations of the permutation map. Utilizing these solutions, we identify the associated quantum algebra and introduce it as the ${\mathfrak{g}\mathfrak{l}}_{k,m}$ Yangian. This newly derived Yangian is distinct from the known Yangian of the general linear Lie superalgebra; crucially, as a Hopf algebra, it possesses the standard tensor product algebra structure. The underlying algebra ${\mathfrak{g}\mathfrak{l}}_{k,m}$ is also introduced as a novel structure and constitutes a subalgebra of the Yangian. We then construct specific highest-weight modules of ${\mathfrak{g}\mathfrak{l}}_{k,m}$ that simultaneously yield the eigenstates of spin-chain-like ``Hamiltonians’', which are defined as the sum of the generators of the $A$-type braid group. Furthermore, we study the highest-weight representations and the corresponding combinatorial bases for ${\mathfrak{g}\mathfrak{l}}_{1,1}$, explicitly linking them to specific shapes of Young tableaux.