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Snehashis Mukherjee

Publications and source records attributed to Snehashis Mukherjee.

12 recordsLinked to original sources
arXiv2025

Simple Modules and PI Structure of the Two-Parameter Quantized Algebra $U^+_{r,s}(B_2)$

We study the two-parameter quantized enveloping algebra $U^+_{r,s}(B_2)$ at roots of unity and investigate its structure and representations. We first show that when $r$ and $s$ are roots of unity, the algebra becomes a PI algebra, and we compute its PI degree explicitly using De Concini-Procesi method. We construct and classify finite-dimensional simple modules for $U^+_{r,s}(B_2)$ by analyzing a subalgebra $B\subset U^+_{r,s}(B_2)$. Simple modules are categorized into torsion-free and torsion types with respect to a distinguished normal element. We classify all torsion-free simple $B$-modules and lift them to $U^+_{r,s}(B_2)$. The remaining simple modules are constructed in the nilpotent case. This work provides a complete classification of simple $U^+_{r,s}(B_2)$-modules at roots of unity and contributes to the understanding of two-parameter quantum groups in type $B_2$.

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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 $K_{n,\Gamma}^{P,Q}(\mathbb{K})$ 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 $K_{n,\Gamma}^{P,Q}(\mathbb{K})$ are computed.

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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

Classification of right nilpotent $\mathbb{F}_p$-braces of cardinality $p^5$

In this article the right nilpotent $\mathbb{F}_p$-braces of cardinality $p^5$ has been classified. We use the connection between nilpotent $\mathbb{F}_p$-braces of cardinality $p^5$ and nilpotent pre-Lie algebras of the same order, building on the known relationship between pre-Lie algebras and braces. Leveraging insights from the classification of nilpotent pre-Lie algebras over $\mathbb{F}_p$, we aim to provide a comprehensive classification of right nilpotent $\mathbb{F}_p$-braces of cardinality $p^5$.

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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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arXiv2020

Azumaya Loci Of The Quantum Euclidean $2n$-Space

This article delves into the Azumaya loci of quantum euclidean 2n-space, providing a comprehensive exploration. We introduce a significant class of maximal-dimensional simple modules associated with this algebra. Moreover, we establish a proof asserting that the maximal-dimensional simple modules are exactly the ones we have constructed. The computation of the algebra's center, coupled with the necessary and sufficient conditions derived for maximal-dimensional simple modules, enables us to determine the Azumaya locus of the algebra.

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