Quantum upper triangular matrix algebras
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912783132524544 |
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| author | Fornaroli, Érica Z. Khrypchenko, Mykola Lopes, Samuel A. Santulo Jr, Ednei A. |
| author_facet | Fornaroli, Érica Z. Khrypchenko, Mykola Lopes, Samuel A. Santulo Jr, Ednei A. |
| contents | Following the ideas in~\cite{yM88} and some inspiration from~\cite{KO24}, we construct a bialgebra $T_q(n)$ and a pointed Hopf algebra $UT_q(n)$ which quantize the coordinate rings of the algebra of upper triangular matrices and of the group of invertible upper triangular matrices of size $n\geq 2$, respectively, where $q$ is a nonzero parameter. The resulting structure on $UT_q(n)$ is neither commutative nor cocommutative, so we obtain a quantum group. The motivation comes from the idea of quantizing the incidence algebra of a finite poset, as the latter can be embedded as a subalgebra of the algebra of upper triangular matrices. After defining the bialgebra $T_q(n)$ and the Hopf algebra $UT_q(n)$, we study and compare their Lie algebras of derivations, their automorphism groups and their low degree Hochschild cohomology, in case $n=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19664 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantum upper triangular matrix algebras Fornaroli, Érica Z. Khrypchenko, Mykola Lopes, Samuel A. Santulo Jr, Ednei A. Quantum Algebra Rings and Algebras 16T20, 16S36, 16W20, 16W25, 16E40 Following the ideas in~\cite{yM88} and some inspiration from~\cite{KO24}, we construct a bialgebra $T_q(n)$ and a pointed Hopf algebra $UT_q(n)$ which quantize the coordinate rings of the algebra of upper triangular matrices and of the group of invertible upper triangular matrices of size $n\geq 2$, respectively, where $q$ is a nonzero parameter. The resulting structure on $UT_q(n)$ is neither commutative nor cocommutative, so we obtain a quantum group. The motivation comes from the idea of quantizing the incidence algebra of a finite poset, as the latter can be embedded as a subalgebra of the algebra of upper triangular matrices. After defining the bialgebra $T_q(n)$ and the Hopf algebra $UT_q(n)$, we study and compare their Lie algebras of derivations, their automorphism groups and their low degree Hochschild cohomology, in case $n=2$. |
| title | Quantum upper triangular matrix algebras |
| topic | Quantum Algebra Rings and Algebras 16T20, 16S36, 16W20, 16W25, 16E40 |
| url | https://arxiv.org/abs/2512.19664 |