Non-combinatorial involutive braidings: the quantum algebra $\mathfrak{gl}_{k,m}$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911740925575168 |
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| author | Doikou, Anastasia |
| author_facet | Doikou, Anastasia |
| contents | 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 $\mathfrak{gl}_{k,m}$ 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 $\mathfrak{gl}_{k,m}$ is also introduced as a novel structure and constitutes a subalgebra of the Yangian. We then construct specific highest-weight modules of $\mathfrak{gl}_{k,m}$ 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 $\mathfrak{gl}_{1,1}$, explicitly linking them to specific shapes of Young tableaux. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16121 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-combinatorial involutive braidings: the quantum algebra $\mathfrak{gl}_{k,m}$ Doikou, Anastasia Quantum Algebra Mathematical Physics Representation Theory 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 $\mathfrak{gl}_{k,m}$ 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 $\mathfrak{gl}_{k,m}$ is also introduced as a novel structure and constitutes a subalgebra of the Yangian. We then construct specific highest-weight modules of $\mathfrak{gl}_{k,m}$ 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 $\mathfrak{gl}_{1,1}$, explicitly linking them to specific shapes of Young tableaux. |
| title | Non-combinatorial involutive braidings: the quantum algebra $\mathfrak{gl}_{k,m}$ |
| topic | Quantum Algebra Mathematical Physics Representation Theory |
| url | https://arxiv.org/abs/2605.16121 |