arXiv subjects

Debabrata Jana

Publications and source records attributed to Debabrata Jana.

4 recordsLinked to original sources

arXiv2026

Quantum unitary group $U_{q,Θ}(3)$ 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 $U_{q,Θ}(3)$ for non zero real $q$ and modulus one complex deformation parameters $θ_{ij}$. We then study in detail the irreducible $*$-representations of the underlying $C^*$-algebra $C(U_{q,Θ}(3))$, using the representations of the three dimensional noncommutative torus. Also, a monomial basis for the dense Hopf *-algebra $\mathbb{C}[U_{q,Θ}(3)]$ is obtained.

math.OA↗

arXiv2025

Overlap distribution of spherical spin glass models with general eigenvalue distribution of the interaction matrix

In this paper, we show that the replica symmetry of the Gibbs measure of spherical spin systems is a property of the eigenvalue spacing at the edge of the interaction matrix. In particular, our interaction matrix has \textbf{two} large outlier eigenvalues with mutual distance $\frac{c}{n}$. The empirical measure of the rest of the eigenvalues is close to the semicircular law with some rigidity conditions. We prove that in this scenario the overlap distribution of two independent samples from the Gibbs measure has a continuous density at a low enough temperature. Hence, the model is a full replica symmetry-breaking model. One might compare this result with only one outlier eigenvalue. This model comes for the Sherrington-Kirkpatrick model with Curie-Weiss interaction in the ferromagnetic case. Here, it is well known that the model is replica symmetric, although the free energy limit of this model is the same as the free energy limit of our model. In our limited understanding, we believe that this kind of phenomenon cannot be explained by the Parisi approach.

math.PR↗

arXiv2025

A few remarks on intermediate subalgebras of an inclusion of C*-algebras

We show that the angle between intermediate $C^*$-subalgebras of an inclusion of simple $C^*$-algebras with finite Watatani index is stable. The notion of angle is instrumental in providing a bound for the cardinality of the lattice of intermediate subalgebras for an irreducible inclusion of simple $C^*$-algebras. We improve the existing upper bound for the cardinality of this set.

math.OA↗

arXiv2024

On some algebraic and geometric aspects of the quantum unitary group

Consider the compact quantum group $U_q(2)$, where $q$ is a non-zero complex deformation parameter such that $|q|\neq 1$. Let $C(U_q(2))$ denote the underlying $C^*$-algebra of the compact quantum group $U_q(2)$. We prove that if $q$ is a non-real complex number and $q^\prime$ is real, then the underlying $C^*$-algebras $C(U_q(2))$ and $C(U_{q^\prime}(2))$ are non-isomorphic. This is in sharp contrast with the case of braided $SU_q(2)$, introduced earlier by Woronowicz et al., where $q$ is a non-zero complex deformation parameter. In another direction, on a geometric aspect of $U_q(2)$, we introduce torus action on the $C^*$-algebra $C(U_q(2))$ and obtain a $C^*$-dynamical system $(C(U_q(2)),\mathbb{T}^3,α)$. We construct a $\mathbb{T}^3$-equivariant spectral triple for $U_q(2)$ that is even and $3^+$-summable. It is shown that the Dirac operator is K-homologically nontrivial.

math.OA↗