Representation theory of the reflection equation algebra I: A quantization of Sylvester's law of inertia
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910398560600064 |
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| author | De Commer, Kenny Moore, Stephen T. |
| author_facet | De Commer, Kenny Moore, Stephen T. |
| contents | We prove a version of Sylvester's law of inertia for the Reflection Equation Algebra (=REA). We will only be concerned with the REA constructed from the $R$-matrix associated to the standard $q$-deformation of $GL(N,\mathbb{C})$. For $q$ positive, this particular REA comes equipped with a natural $*$-structure, by which it can be viewed as a $q$-deformation of the $*$-algebra of polynomial functions on the space of self-adjoint $N$-by-$N$-matrices. We will show that this REA satisfies a type $I$-condition, so that its irreducible representations can in principle be classified. Moreover, we will show that, up to the adjoint action of quantum $GL(N,\mathbb{C})$, any irreducible representation of the REA is determined by its \emph{extended signature}, which is a classical signature vector extended by a parameter in $\mathbb{R}/\mathbb{Z}$. It is this latter result that we see as a quantized version of Sylvester's law of inertia. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03640 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Representation theory of the reflection equation algebra I: A quantization of Sylvester's law of inertia De Commer, Kenny Moore, Stephen T. Representation Theory Operator Algebras Quantum Algebra We prove a version of Sylvester's law of inertia for the Reflection Equation Algebra (=REA). We will only be concerned with the REA constructed from the $R$-matrix associated to the standard $q$-deformation of $GL(N,\mathbb{C})$. For $q$ positive, this particular REA comes equipped with a natural $*$-structure, by which it can be viewed as a $q$-deformation of the $*$-algebra of polynomial functions on the space of self-adjoint $N$-by-$N$-matrices. We will show that this REA satisfies a type $I$-condition, so that its irreducible representations can in principle be classified. Moreover, we will show that, up to the adjoint action of quantum $GL(N,\mathbb{C})$, any irreducible representation of the REA is determined by its \emph{extended signature}, which is a classical signature vector extended by a parameter in $\mathbb{R}/\mathbb{Z}$. It is this latter result that we see as a quantized version of Sylvester's law of inertia. |
| title | Representation theory of the reflection equation algebra I: A quantization of Sylvester's law of inertia |
| topic | Representation Theory Operator Algebras Quantum Algebra |
| url | https://arxiv.org/abs/2404.03640 |