Related papers: Higher extensions between modules for SL_2

For an abelian category $\mathcal{A}$, we establish the relation between its derived and extension dimensions. Then for an artin algebra $\Lambda$, we give the upper bounds of the extension dimension of $\Lambda$ in terms of the radical…

We introduce and fully analyze a new commutation relation $\overline{K} L_1 = L_2 K$ between finite convolution integral operator $K$ and differential operators $L_1$ and $L_{2}$, that has implications for spectral properties of $K$. This…

This paper continues the study of highest weight categorical sl_2-actions started in part I. We start by refining the definition given there and showing that all examples considered in part I are also highest weight categorifications in the…

Assume that $X$ is a connected $(q+1)$-regular undirected graph of finite order $n$. Let $A$ denote the adjacency matrix of $X$. Let $\lambda_1=q+1>\lambda_2\geq \lambda_3\geq \ldots \geq \lambda_n$ denote the eigenvalues of $A$. The…

We derive the universal R-matrix of the quantum-deformed enveloping algebra of centrally extended sl(2|2) using Drinfeld's quantum double construction. We are led to enlarging the algebra by additional generators corresponding to an sl(2)…

Let $\lambda$ be a primitive root of unity of order $\ell$. We introduce a family of finite-dimensional algebras $\{\mathcal{D}_{\lambda,N}(\mathfrak{sl}_2)\}_{N\in\mathbb{N}_0}$ over the complex numbers, such that…

In this paper, we study the elliptic spectral sequence computing $tmf_*(\mathbb{R} P^2)$ and $tmf_* (\mathbb{R} P^2 \wedge \mathbb{C} P^2)$. Specifically, we compute all differentials and resolve exotic extensions by 2, $\eta$, and $\nu$.…

Let $L$ be a link in a thickened annulus. Grigsby-Licata-Wehrli showed that the annular Khovanov homology of $L$ is equipped with an action of $sl_2(\wedge)$, the exterior current algebra of the Lie algebra $sl_2$. In this paper, we upgrade…

Let $G$ be a Hermitian type Lie group with maximal compact subgroup $K$. Let $L(\lambda)$ be a highest weight Harish-Chandra module of $G$ with the infinitesimal character $\lambda$. By using some combinatorial algorithm, we obtain a…

Let K be an algebraically closed field of characteristic p>0 and let Sp(2m) be the symplectic group of rank m over K. The main theorem of this article gives the character of the rational simple Sp(2m)-modules with fundamental highest weight…

We consider the decomposition into irreducible components of the exterior algebra $\bigwedge\left(\mathbb{C}^{n}\otimes \left(\mathbb{C}^{k}\right)^{*}\right)$ regarded as a $GL_{n}\times GL_{k}$ module. Irreducible $GL_{n}\times GL_{k}$…

The modular discriminant $\Delta$ is known to structure the sequence of modular forms $(M_{2k}(SL_2(\mathbb{Z})))_{k\in \; \mathbb{N}^*}$ at level $1$.\\ For all positive integer $N$, we define a strong modular unit $\Delta_N$ at level $N$…

Let $(\lambda_f(n))_{n\geqslant1}$ be the Hecke eigenvalues of a holomorphic cusp form $f$. We prove that the exponent of distribution of $\lambda_f*1$ in arithmetic progressions is as large as $\frac{1}{2}+\frac{1}{46}$ when the modulus…

Let $\mathbf{S}_k$ denote a maximal torus in the complex Lie group $\mathbf{G} = \mathrm{SL}_n(\mathbb{C})/C_k$ and let $T_k$ denote a maximal torus in its compact real form $\mathrm{SU}_n(\mathbb{C})/C_k$, where $k$ divides $n$. Let $W$…

Operating just once with the naive Foldy-Wouthuysen-Tani transformation on the relativistic Fermi-Yang equation for $Q\bar q$ bound states described by the semi-relativistic Hamiltonian which includes Coulomb-like as well as confining…

We develop explicit variational formulas for the $p(\cdot)$-modulus of curve families in symmetric domains of $\mathbb{R}^n$, under a log-H\"older continuous exponent $p\colon\Omega\to(1,\infty)$, where $\Omega$ is an open set. For annuli…

Let $\frak F_{\l}$ be the space of tensor densities of degree $\lambda$ on the supercircle $S^{1|1}$. We consider the space $\mathfrak{D}_{\lambda,\mu}^k$ of k-th order linear differential operators from $\frak F_{\l}$ to $\frak F_{\mu}$ as…

Let $L(\lambda)$ be a highest weight Harish-Chandra module with highest weight $\lambda$. When the associated variety of $L(\lambda)$ is not maximal, that is, not equal to the nilradical of the corresponding parabolic subalgebra, we prove…

Let $L((n-\tfrac 3 2)\Lambda_0)$, $n \in \Bbb N$, be a vertex operator algebra associated to the irreducible highest weight module $L((n-\tfrac 3 2)\Lambda_0)$ for a symplectic affine Lie algebra. We find a complete set of irreducible…

The quantized version of a discrete Knizhnik-Zamolodchikov system is solved by an extension of the generalized Bethe Ansatz. The solutions are constructed to be of highest weight which means they fully reflect the internal quantum group…