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Sanu Bera

Publications and source records attributed to Sanu Bera.

11 recordsLinked to original sources
arXiv2025

$U_q^+(B_2)$ and its representations

In this article we investigate the algebra $U_q^+(B_2)$. Assume that $q$ is a primitive $m$-th root of unity with $m \geq 5$. We prove that $U_q^+(B_2)$ becomes a Polynomial Identity (PI) algebra. It was previously known that for such algebras the simple modules are finite-dimensional with dimension at most the PI degree. We determine the PI degree of $U_q^+(B_2)$ and we classify up to isomorphism the simple $U_q^+(B_2)$-modules. We also find the center of $U_q^+(B_2)$.

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arXiv2025

On Simple Modules over the Quantum Matrix algebra at roots of unity

This article investigates the two-parameter quantum matrix algebra at roots of unity. In the roots of unity setting, this algebra becomes a Polynomial Identity (PI) algebra and it is known that simple modules over such algebra are finite-dimensional with dimension at most the PI degree. We determine the center, compute the PI degree, and classify simple modules for two-parameter quantum matrix algebra, up to isomorphism, over an algebraically closed field of arbitrary characteristics.

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arXiv2025

Bernstein-type inequalities for quantum algebras

We establish Bernstein-type inequalities for the quantum algebras $K_{n,Γ}^{P,Q}(\mathbb{K})$ introduced by K. L. Horton that include the graded quantum Weyl algebra, the quantum symplectic space, the quantum Euclidean space, and quantum Heisenberg algebra etc., obtaining new results and as well as simplified proofs of previously known results. The Krull and global dimensions of certain further localizations of $K_{n,Γ}^{P,Q}(\mathbb{K})$ are computed.

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arXiv2024

Simple Modules over Second Quantum Weyl Algebra

In this article, we study the multiparameter second quantum Weyl algebra at roots of unity. In this setting, the algebra is a polynomial identity (PI) algebra, and the dimension of its simple modules is bounded above by its PI degree. We explicitly determine the PI degree and provide a complete classification of simple modules. This classification offers a comprehensive solution to $\href{https://doi.org/10.1007/978-3-030-19486-4_23}{\text{Problem 2: C, Walton (2019)- An Invitation to Noncommutative Algebra}}$ for the second quantum Weyl algebra.

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arXiv2024

Simple modules over 3-cyclic quantum Weyl Algebra at roots of unity

This article undertakes an exploration of simple modules of 3-cyclic quantum Weyl algebra at roots of unity. Under the roots of unity assumption, the algebra becomes a Polynomial Identity algebra and the vector space dimension of the simple modules is bounded above by its PI degree. The article systematically classifies all potential simple modules and computes the algebra's center.

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arXiv2024

Quantum Heisenberg Enveloping Algebra

In this article, the two-parameter quantum Heisenberg enveloping algebra, which serves as a model for certain quantum generalized Heisenberg algebras, have been studied at roots of unity. In this context, the quantum Heisenberg enveloping algebra becomes a polynomial identity algebra, and the dimension of simple modules is bounded by its PI degree. The PI degree, center, and complete classification of simple modules up to isomorphism are explicitly presented. We work over a field of arbitrary characteristic, although our results concerning the representations require that it is algebraically closed.

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arXiv2021

Construction of Simple Modules over the Quantum Affine Space

The coordinate ring $\mathcal{O}_{\mathbf{q}}(\mathbb{K}^n)$ of quantum affine space is the $\mathbb{K}$-algebra presented by generators $x_1,\cdots ,x_n$ and relations $x_ix_j=q_{ij}x_jx_i$ for all $i,j$. We construct simple $\mathcal{O}_{\mathbf{q}}(\mathbb{K}^n)$-modules in a more general setting where the entries $q_{ij}$ lie in a torsion subgroup of $\mathbb{K}^*$ and show analogous results hold as in single parameter case.

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