arXiv subjects

Manabendra Giri

Publications and source records attributed to Manabendra Giri.

4 recordsLinked to original sources

arXiv2026

Quantum unitary group $U_{q,\Theta}(3)$ for complex deformation parameters

In this article, we consider a particular Hayashi $R$-matrix that satisfies the Yang-Baxter equation $R_{12}R_{13}R_{23}=R_{23}R_{13}R_{12}$. Using the FRT-bialgebra technique and Woronowicz's method of construction, we construct a concrete compact quantum group $U_{q,\Theta}(3)$ for non zero real $q$ and modulus one complex deformation parameters $\theta_{ij}$. We then study in detail the irreducible $*$-representations of the underlying $C^*$-algebra $C(U_{q,\Theta}(3))$, using the representations of the three dimensional noncommutative torus. Also, a monomial basis for the dense Hopf *-algebra $\mathbb{C}[U_{q,\Theta}(3)]$ is obtained.

math.OA↗

arXiv2026

Compact Quantum Group Extensions of $USp_q(2n)$, $O_q(n)$ and $SO_q(2n)$

I introduce compact quantum group extensions associated with the $q$-deformations of the classical compact groups $USp(2n)$, $O(n,\mathbb{R})$ and $SO(2n,\mathbb{R})$. Motivated by the relationship between $SU_q(n)$ and $U_q(n)$, I study the problem of constructing compact quantum groups $Z_{q,n}$ extending the standard compact quantum groups $A_{q,n}\in\{ {USp_q(2n), O_q(N), SO_q(2n)}\}$ through an additional central unitary element.

math.OA↗

arXiv2024

Quantized function algebras at $q=0$: type $A_{n}$ case

We define the notion of quantized function algebras at $q=0$ or crystallization of the $q$ deformations of the type $A_{n}$ compact Lie groups at the $C^*$-algebra level. The $C^{*}$-algebra $A_{n}(0)$ is defined as a universal $C^*$-algebra given by a finite set of generators and relations. We obtain these relations by looking at the irreducible representations of the quantized function algebras for $q>0$ and taking limit as $q\to 0+$ after rescaling the generating elements appropriately. We then prove that in the $n=2$ case the irreducible representations $A_{2}(0)$ are precisely the $q\to 0+$ limits of the irreducible representations of the $C^*$-algebras $A_{2}(q)$.

math.QA↗

arXiv2024

Irreducible representations of the crystallization of the quantized function algebras $C(SU_{q}(n+1))$

Crystallization of the $C^*$-algebras $C(SU_{q}(n+1))$ was introduced by Giri \& Pal as a $C^*$-algebra $C(SU_{0}(n+1))$ given by a finite set of generators and relations. Here we study representations of the $C^*$-algebra $C(SU_{0}(n+1))$ and prove a factorization theorem for its irreducible representations. This leads to a complete classification of all irreducible representations of this $C^*$-algebra. As an important consequence, we prove that all the irreducible representations of $C(SU_{0}(n+1))$ arise exactly as $q\to 0+$ limits of irreducible representations of $C(SU_{q}(n+1))$. We also present a few other important corollaries of the classification theorem.

math.OA↗