arXiv subjects

Arup Kumar Pal

Publications and source records attributed to Arup Kumar Pal.

7 recordsLinked to original sources

arXiv2025

A triangular decomposition for the crystal lattice of quantized function algebras

We prove a triangular decomposition theorem for the lower crystal lattice $\mathcal{O}_{t}^{A_{0}}(G)$ of the quantized function algebra $\mathcal{O}_{t}^{}(G)$, where $G$ is a connected simply connected complex Lie group with Lie algebra $\mathfrak {g}$ of type $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$, $E_{6}$ or $E_{7}$. As a consequence, we prove the inclusion $\mathcal{O}_{t}^{A_{0}}(G)\subseteq\mathcal{O}_{t}^{A_{0}}(K)$ conjectured by Matassa \& Yuncken in these cases. Using this inclusion, we then prove that the notions of crystallized quantized function algebra given by Matassa \& Yuncken coincide with that of Giri \& Pal.

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↗

arXiv2022

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↗

arXiv2022

An approximate equivalence for the GNS representation of the Haar state of $SU_{q}(2)$

We use the crystallised $C^*$-algebra $C(SU_{q}(2))$ at $q=0$ to obtain a unitary that gives an approximate equivalence involving the GNS representation on the $L^{2}$ space of the Haar state of the quantum $SU(2)$ group and the direct integral of all the infinite dimensional irreducible representations of the $C^{*}$-algebra $C(SU_{q}(2))$ for nonzero values of the parameter $q$. This approximate equivalence gives a $KK$ class via the Cuntz picture in terms of quasihomomorphisms as well as a Fredholm representation of the dual quantum group $\widehat{SU_q(2)}$ with coefficients in a $C^*$-algebra in the sense of Mishchenko.

math.OA↗

arXiv2014

An invariant for homogeneous spaces of compact quantum groups

The central notion in Connes' formulation of non commutative geometry is that of a spectral triple. Given a homogeneous space of a compact quantum group, restricting our attention to all spectral triples that are `well behaved' with respect to the group action, we construct a certain dimensional invariant. In particular, taking the (quantum) group itself as the homogeneous space, this gives an invariant for a compact quantum group. Computations of this invariant in several cases, including all type A quantum groups, are given.

math.QA↗

arXiv2011

Design of Image Cryptosystem by Simultaneous VQ-Compression and Shuffling of Codebook and Index Matrix

The popularity of Internet usage although increases exponentially, it is incapable of providing the security for exchange of confidential data between the users. As a result, several cryptosystems for encryption of data and images have been developed for secured transmission over Internet. In this work, a scheme for Image encryption/decryption based on Vector Quantization (VQ) has been proposed that concurrently encodes the images for compression and shuffles the codebook and the index matrix using pseudorandom sequences for encryption. The processing time of the proposed scheme is much less than the other cryptosystems, because it does not use any traditional cryptographic operations, and instead it performs swapping between the contents of the codebook with respect to a random sequence, which resulted an indirect shuffling of the contents of the index matrix. It may be noted that the security of the proposed cryptosystem depends on the generation and the exchange of the random sequences used. Since the generation of truly random sequences are not practically feasible, we simulate the proposed scheme using MATLAB, where its operators like rand(method, seed), randperm(n) has been used to generate pseudorandom sequences and it has been seen that the proposed cryptosystem shows the expected performance.

cs.CR↗

arXiv2011

An Efficient Codebook Initialization Approach for LBG Algorithm

In VQ based image compression technique has three major steps namely (i) Codebook Design, (ii) VQ Encoding Process and (iii) VQ Decoding Process. The performance of VQ based image compression technique depends upon the constructed codebook. A widely used technique for VQ codebook design is the Linde-Buzo-Gray (LBG) algorithm. However the performance of the standard LBG algorithm is highly dependent on the choice of the initial codebook. In this paper, we have proposed a simple and very effective approach for codebook initialization for LBG algorithm. The simulation results show that the proposed scheme is computationally efficient and gives expected performance as compared to the standard LBG algorithm.

cs.CV↗