This article focuses on the representation theory of algebras associated with $\mathfrak {sl}_2$, including the affine Lie algebra $\widehat {\mathfrak {sl}_2}$, the affine Kac–Moody algebra $\widetilde {\mathfrak {sl}_2}$, and the affine-Virasoro algebra $\mathfrak {Vir}\ltimes \widehat {\mathfrak {sl}_2}$. First, we classify certain modules over these algebras, which are free of rank one when restricted to some specific subalgebras. We demonstrate a connection between these modules and modules over the Weyl algebras, which allows us to construct large families of modules that are free of arbitrary finite rank when restricted to the Cartan subalgebra. We then investigate the simplicity of these modules. For reducible modules, we fully characterize their composition factors. Through a comparison with existing simple modules in the literature, we have identified a novel family of simple modules over the affine Kac–Moody algebra $\widetilde {\mathfrak {sl}_2}$. Finally, we turn our attention to a class of tensor product modules over the affine-Virasoro algebra $\mathfrak {Vir}\ltimes \widehat {\mathfrak {sl}_2}$. We derive a necessary and sufficient condition for the simplicity of these modules and determine their isomorphism classes.

A categorical axiomatic theory of creation/annihilation operators on symmetric Fock space is introduced, and the combinatorial model that motivated it is presented. Commutation relations and coherent states are considered in both frameworks.

Consider a pair of elements f and g in a commutative ring Q. Given a matrix factorization of f and another of g, the tensor product of matrix factorizations, which was first introduced by Knörrer and later generalized by Yoshino, produces a matrix factorization of the sum $f+g$. We will study the tensor product of d-fold matrix factorizations, with a particular emphasis on understanding when the construction has a non-trivial direct sum decomposition. As an application of our results, we construct indecomposable maximal Cohen–Macaulay and Ulrich modules over hypersurface domains of a certain form.

This paper presents a classification of all simple Harish-Chandra modules for the $N=1$ Heisenberg–Virasoro superalgebra, which turn out to be highest weight modules, lowest weight modules, and evaluation modules of the intermediate series (all weight spaces are one dimensional). Moreover, a characterization of the tensor product of highest weight modules with intermediate series modules is obtained.

We study symmetric and antisymmetric tensor products of Hilbert-space operators, focusing on norms and spectra for some well-known classes favored by function-theoretic operator theorists. We pose many open questions that should interest the field.

Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex matrices and $\|\cdot \|_{(p,k)}$ the $(p,k)$ norm on $M_{mn}$ with a positive integer $k\leq mn$ and a real number $p>2$. We show that a linear map $\phi :M_{mn}\rightarrow M_{mn}$ satisfies

$$ \begin{align*}\|\phi(A\otimes B)\|_{(p,k)}=\|A\otimes B\|_{(p,k)} \mathrm{\quad for~ all\quad}A\in M_m\ \mathrm{and}\ B\in M_n\end{align*} $$

if and only if there exist unitary matrices $U,V\in M_{mn}$ such that

$$ \begin{align*}\phi(A\otimes B)=U(\varphi_1(A)\otimes \varphi_2(B))V \mathrm{\quad for~ all\quad}A\in M_m\ \mathrm{and}\ B\in M_n,\end{align*} $$

where $\varphi _s$ is the identity map or the transposition map $X\mapsto X^T$ for $s=1,2$. The result is also extended to multipartite systems.

We introduce self-divisible ultrafilters, which we prove to be precisely those $w$ such that the weak congruence relation $\equiv _w$ introduced by Šobot is an equivalence relation on $\beta {\mathbb Z}$. We provide several examples and additional characterisations; notably we show that $w$ is self-divisible if and only if $\equiv _w$ coincides with the strong congruence relation $\mathrel {\equiv ^{\mathrm {s}}_{w}}$, if and only if the quotient $(\beta {\mathbb Z},\oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$ is a profinite group. We also construct an ultrafilter $w$ such that $\equiv _w$ fails to be symmetric, and describe the interaction between the aforementioned quotient and the profinite completion $\hat {{\mathbb Z}}$ of the integers.

A $C^{*}$-algebra A is said to detect nuclearity if, whenever a $C^{*}$-algebra B satisfies $A\otimes _{\mathrm{min}} B = A\otimes _{\mathrm{max}} B,$ it follows that B is nuclear. In this note, we survey the main results associated with this topic and present the background and tools necessary for proving the main results. In particular, we show that the $C^{*}$-algebra $A = C^{*}(\mathbb {F}_{\infty })\otimes _{\mathrm{min}} B(\ell ^{2})/K(\ell ^{2})$ detects nuclearity. This result is known to experts, but has never appeared in the literature.

Let p be a prime and let $J_r$ denote a full $r \times r$ Jordan block matrix with eigenvalue $1$ over a field F of characteristic p. For positive integers r and s with $r \leq s$, the Jordan canonical form of the $r s \times r s$ matrix $J_{r} \otimes J_{s}$ has the form $J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _{r}}$. This decomposition determines a partition $\lambda (r,s,p)=(\lambda _1,\lambda _2,\ldots , \lambda _{r})$ of $r s$. Let $n_1, \ldots , n_k$ be the multiplicities of the distinct parts of the partition and set $c(r,s,p)=(n_1,\ldots ,n_k)$. Then $c(r,s,p)$ is a composition of r. We present a new bottom-up algorithm for computing $c(r,s,p)$ and $\lambda (r,s,p)$ directly from the base-p expansions for r and s.

Recent work by Baum et al. [‘Expanders, exact crossed products, and the Baum–Connes conjecture’, Ann. K-Theory 1(2) (2016), 155–208], further developed by Buss et al. [‘Exotic crossed products and the Baum–Connes conjecture’, J. reine angew. Math. 740 (2018), 111–159], introduced a crossed-product functor that involves tensoring an action with a fixed action $(C,\unicode[STIX]{x1D6FE})$, then forming the image inside the crossed product of the maximal-tensor-product action. For discrete groups, we give an analogue for coaction functors. We prove that composing our tensor-product coaction functor with the full crossed product of an action reproduces their tensor-crossed-product functor. We prove that every such tensor-product coaction functor is exact, and if $(C,\unicode[STIX]{x1D6FE})$ is the action by translation on $\ell ^{\infty }(G)$, we prove that the associated tensor-product coaction functor is minimal, thereby recovering the analogous result by the above authors. Finally, we discuss the connection with the $E$-ization functor we defined earlier, where $E$ is a large ideal of $B(G)$.

An ergodic probability measure preserving (p.m.p.) equivalence relation ${\mathcal{R}}$ is said to be stable if ${\mathcal{R}}\cong {\mathcal{R}}\times {\mathcal{R}}_{0}$ where ${\mathcal{R}}_{0}$ is the unique hyperfinite ergodic type $\text{II}_{1}$ equivalence relation. We prove that a direct product ${\mathcal{R}}\times {\mathcal{S}}$ of two ergodic p.m.p. equivalence relations is stable if and only if one of the two components ${\mathcal{R}}$ or ${\mathcal{S}}$ is stable. This result is deduced from a new local characterization of stable equivalence relations. The similar question on McDuff $\text{II}_{1}$ factors is also discussed and some partial results are given.

For a C*-algebra A, determining the Cuntz semigroup Cu(A ⊗) in terms of Cu(A) is an important problem, which we approach from the point of view of semigroup tensor products in the category of abstract Cuntz semigroups by analysing the passage of significant properties from Cu(A) to Cu(A)⊗Cu Cu(). We describe the effect of the natural map Cu(A) → Cu(A)⊗Cu Cu() in the order of Cu(A), and show that if A has real rank 0 and no elementary subquotients, Cu(A)⊗Cu Cu() enjoys the corresponding property of having a dense set of (equivalence classes of) projections. In the simple, non-elementary, real rank 0 and stable rank 1 situation, our investigations lead us to identify almost unperforation for projections with the fact that tensoring with is inert at the level of the Cuntz semigroup.

In this paper, we study condition $C_{\hat{\ }}^{'}$ , which is a projective tensor product analogue of condition $C'$ . We show that the finite-dimensional $\text{OLLP}$ operator spaces have condition $C_{\hat{\ }}^{'}$ , and ${{M}_{n}}\left( n\,>\,2 \right)$ does not have that property.

Let A and B be operators acting on infinite-dimensional complex Banach spaces. We say that the Weyl spectral identity holds for the tensor product A⊗B if σ w(A⊗B) = σ w(A)·σ(B)∪σ(A)·σw(B), where σ(·) and σ w(·) stand for the spectrum and the Weyl spectrum, respectively. Conditions on A and B for which the Weyl spectral identity holds are investigated. Especially, it is shown that if A and B are biquasitriangular (in particular, if the spectra of A and B have empty interior), then the Weyl spectral identity holds. It is also proved that if A and B are biquasitriangular, then the tensor product A ⊗ B is biquasitriangular.