Diagonal reduction algebra for $$\mathfrak{osp}(1|2)$$

Theoretical and Mathematical Physics - The problem of providing complete presentations of reduction algebras associated to a pair of Lie algebras $$( \mathfrak{G} , \mathfrak{g} )$$ has previously...     

$$|End \mathfrak{A}| = |Con \mathfrak{A}| = |End \mathfrak{A}| = 2^{|\mathfrak{A}|}$$ for any uncountable 1-unary algebra $$\mathfrak{A}$$

Algebra universalis -     

Deformation of \mathfrak{sl}\, (2) and \mathfrak{osp}\, (1|2)-modules of symbols

We consider the $\mathfrak{sl}\, (2)$ -module structure on the spaces of symbols of differential operators acting on the spaces of weighted densities. We compute the necessary and sufficient integrability conditions of a given infinitesimal deformation of this structure and prove that any formal deformation is equivalent to its infinitesimal part. We study also the super analogue of this problem getting the same results.     

Topological invariants from quantum group $$\mathcal {U}_{\xi }\mathfrak {sl}(2|1)$$ at roots of unity

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with the Lie superalgebra \(\mathfrak {sl}(2|1)\). The construction is based on nilpotent irreducible finite dimensional representations of quantum group \(\mathcal {U}_{\xi }\mathfrak {sl}(2|1)\) where \(\xi \) is a root of unity of odd order. These constructions use the notion of modified trace and relative \( G \)-modular category of previous authors.     

Toward logarithmic extensions of $$\widehat{s\ell }(2)_k $$ conformal field models

For positive integers p = k + 2, we construct a logarithmic extension of the conformal field theory of integrable representations by taking the kernel of two fermionic screening operators in a butterfly resolution of a three-boson realization of . The currents W⁻(z) and W⁺(z) of a W-algebra acting in the kernel are determined by a highest-weight state of dimension 4p − 2 and charge 2p − 1 and by a (θ=1)-twisted highest-weight state of the same dimension 4p − 2 and opposite charge −2p+1. We construct 2p W-algebra representations, evaluate their characters, and show that together with the p−1 integrable representation characters, they generate a modular group representation whose structure is described as a deformation of the (9p−3)-dimensional representation R _p+1⊕ℂ²⊗R _p+1ʕR _p−1⊕ℂ² R _p−1⊕ℂ³ R _p−1, where R _p−1 is the SL(2, ℤ)-representation on integrable-representation characters and R _p+1 is a (p+1)-dimensional SL(2, ℤ)-representation known from the logarithmic (p, 1) model. The dimension 9p − 3 is conjecturally the dimension of the space of torus amplitudes, and the ℂⁿ with n = 2 and 3 suggest the Jordan cell sizes in indecomposable W-algebra modules. We show that under Hamiltonian reduction, the W-algebra currents map into the currents of the triplet W-algebra of the logarithmic (p, 1) model. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 153, No. 3, pp. 291–346, December, 2007.     

Klingen $${\mathfrak {p}}^2$$ p 2 vectors for $$\mathrm{GSp}(4)$$ GSp ( 4 )

The Ramanujan Journal - We calculate the dimensions of the spaces of invariant vectors under the Klingen congruence subgroup of level $${\mathfrak {p}}^2$$ for all irreducible, admissible...     

Matrix-valued orthogonal polynomials related to the quantum analogue of $$(\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})$$

Matrix-valued spherical functions related to the quantum symmetric pair for the quantum analogue of \((\mathrm{SU}(2) \times \mathrm{SU}(2), \mathrm{diag})\) are introduced and studied in detail. The quantum symmetric pair is given in terms of a quantised universal enveloping algebra with a coideal subalgebra. The matrix-valued spherical functions give rise to matrix-valued orthogonal polynomials, which are matrix-valued analogues of a subfamily of Askey–Wilson polynomials. For these matrix-valued orthogonal polynomials, a number of properties are derived using this quantum group interpretation: the orthogonality relations from the Schur orthogonality relations, the three-term recurrence relation and the structure of the weight matrix in terms of Chebyshev polynomials from tensor product decompositions, and the matrix-valued Askey–Wilson type q-difference operators from the action of the Casimir elements. A more analytic study of the weight gives an explicit LDU-decomposition in terms of continuous q-ultraspherical polynomials. The LDU-decomposition gives the possibility to find explicit expressions of the matrix entries of the matrix-valued orthogonal polynomials in terms of continuous q-ultraspherical polynomials and q-Racah polynomials.     

On $$\ell $$-regular bipartitions modulo $$\ell $$

Let \(B_\ell (n)\) denote the number of \(\ell \)-regular bipartitions of n. In this paper, we prove several infinite families of congruences satisfied by \(B_\ell (n)\) for \(\ell \in {\{5,7,13\}}\). For example, we show that for all \(\alpha >0\) and \(n\ge 0\),

$$\begin{aligned} B_5\left( 4^\alpha n+\frac{5\times 4^\alpha -2}{6}\right)\equiv & {} 0 \ (\text {mod}\ 5),\\ B_7\left( 5^{8\alpha }n+\displaystyle \frac{5^{8\alpha }-1}{2}\right)\equiv & {} 3^\alpha B_7(n)\ (\text {mod}\ 7) \end{aligned}$$

and

$$\begin{aligned} B_{13}\left( 5^{12\alpha }n+5^{12\alpha }-1\right) \equiv B_{13}(n)\ (\text {mod}\ 13). \end{aligned}$$

     

A Clifford Algebraic Framework for $$\mathfrak{sp}(m)$$-Invariant Differential Operators

We introduce a framework for studying differential operators which are invariant with respect to the real (complex) symplectic Lie algebra ( ), associated to a quaternionic structure on a vector space . To do so, these algebras are realized within the orthogonal Lie algebra . This leads in a natural way to a refinement of the recently introduced notion of complex Hermitean Clifford analysis, in which four variations of the classical Dirac operator play a dominant role. David Eelbode: Postdoctoral fellow supported by the F.W.O. Vlaanderen (Belgium).     

On the diophantine equation $$\varvec{y^{2} = \prod _{i \le 8}(x + k_i)}$$

This paper improves the result of Tengely (Periodica Math. Hung., 72(1) (2016) 23–28).     

Quantum algebras with representation ring of $$ \mathfrak{s}\mathfrak{l}_2 $$ type

A reducible representation of the Temperley-Lieb algebra is constructed on a tensor product of n-dimensional spaces. As a centralizer of this action, we obtain a quantum algebra (quasi-triangular Hopf algebra) with the representation ring that is equivalent to the representation ring of the Lie algebra. Bibliography: 23 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 347, 2007, pp. 167–177.     

On the generalized $$\text {SO}(2n,{{\mathbb {C}}})$$ SO ( 2 n,C ) -opers

Annals of Global Analysis and Geometry - Since their introduction by Beilinson–Drinfeld (Opers, 1993. arXiv math/0501398; Quantization of Hitchin’s integrable system and Hecke...     

$$\mathfrak{h}$$ -platitude des Structures de contact

Monatshefte für Mathematik - For most of the usual structures in differential geometry (foliations, complex structures, symplectic structures, ...) the condition of integrability, that is to...     

On a Notion of Averaged Mappings in $$\operatorname{CAT}(0)$$ Spaces

Functional Analysis and Its Applications - We introduce a notion of averaged mappings in the broader class of $$\operatorname{CAT}(0)$$ spaces. We call these mappings $$\alpha$$ -firmly...     

On the reflexivity of $$\mathcal {P}_{w}(^{n}E;F)$$

In this paper we prove that if E and F are reflexive Banach spaces and G is a closed linear subspace of the space \(\mathcal {L}_{K}(E;F)\) of all compact linear operators from E into F, then G is either reflexive or non-isomorphic to a dual space. This result generalizes (Israel J Math 21:38-49, 1975, Theorem 2) and gives the solution to a problem posed by Feder (Ill J Math 24:196-205, 1980, Problem 1). We also prove that if E and F are reflexive Banach spaces, then the space \(\mathcal {P}_{w}(^{n}E;F)\) of all n-homogeneous polynomials from E into F which are weakly continuous on bounded sets is either reflexive or non-isomorphic to a dual space.     

Stability of $$\mathcal{P}(S)$$ under Finite Perturbation

Abstract. T. Kato [9] found an important property of semi-Fredholm pencils, now called the Kato decomposition. Few years later, M.A. Kaashoek [7] introduced operators having the property as a generalization of semi-Fredholm operators. In [4], it is proved that these two notions are linked. The aim of this work is to study the stability of the property under finite perturbation.     

Dimension-Free Harnack Inequalities on $$\hbox {RCD}(K, \infty )$$ Spaces

The dimension-free Harnack inequality for the heat semigroup is established on the \(\mathrm{{RCD}}(K,\infty )\) space, which is a non-smooth metric measure space having the Ricci curvature bounded from below in the sense of Lott–Sturm–Villani plus the Cheeger energy being quadratic. As its applications, the heat semigroup entropy-cost inequality and contractivity properties of the semigroup are studied, and a strong-enough Gaussian concentration implying the log-Sobolev inequality is also shown as a generalization of the one on the smooth Riemannian manifold.