Graded identities for algebras with elementary gradings over an infinite field

Let $F$ be an infinite field of positive characteristic $p > 2$ and let $G$ be a group. In this paper, we study the graded identities satisfied by an associative algebra equipped with an elementary $G$ -grading. Let $E$ be the infinite-dimensional Grassmann algebra. For every $a$ , $b\in \mathbb {N}$ , we provide a basis for the graded polynomial identities, up to graded monomial identities, for the verbally prime algebras $M_{a,b}(E)$ , as well as their tensor products, with their elementary gradings. Moreover, we give an alternative proof of the fact that the tensor product $M_{a,b}(E)\otimes M_{r,s}(E)$ and $M_{ar+bs,as+br}(E)$ are $F$ -algebras which are not PI equivalent. Actually, we prove that the $T_{G}$ -ideal of the former algebra is contained in the $T$ -ideal of the latter. Furthermore, the inclusion is proper. Recall that it is well known that these algebras satisfy the same multilinear identities and hence in characteristic 0 they are PI equivalent.

Controlled generalized fusion frame in the tensor product of Hilbert spaces

We present controlled by operators generalized fusion frame in the tensor product of Hilbert spaces and discuss some of its properties. We also describe the frame operator for a pair of controlled $g$-fusion Bessel sequences in the tensor product of Hilbert spaces.

∗-K-Operator Frame for Hom∗A(X)

In this work, we introduce the concept of ∗-K-operator frames in Hilbert pro-C∗-modules, which is a generalization of K-operator frame. We present the analysis operator, the synthesis operator and the frame operator. We also give some properties and we study the tensor product of ∗-K-operator frame for Hilbert pro-C ∗ -modules.

Quantum mereology in finite quantum mechanics

Any Hilbert space with composite dimension can be factored into a tensor product of smaller Hilbert spaces. This allows us to decompose a quantum system into subsystems. We propose a model based on finite quantum mechanics for a constructive study of such decompositions.

On the dimension of $H^{*}((\mathbb Z_2)^{\times t}, \mathbb Z_2)$ as a module over Steenrod ring

We write $\mathbb P$ for the polynomial algebra in one variable over the finite field $\mathbb Z_2$ and $\mathbb P^{\otimes t} = \mathbb Z_2[x_1, \ldots, x_t]$ for its $t$-fold tensor product with itself. We grade $\mathbb P^{\otimes t}$ by assigning degree $1$ to each generator. We are interested in determining a minimal set of generators for the ring of invariants $(\mathbb P^{\otimes t})^{G_t}$ as a module over Steenrod ring, $\mathscr A_2.$ Here $G_t$ is a subgroup of the general linear group $GL(t, \mathbb Z_2).$ An equivalent problem is to find a monomial basis of the space of "unhit" elements, $\mathbb Z_2\otimes_{\mathscr A_2} (\mathbb P^{\otimes t})^{G_t}$ in each $t$ and degree $n\geq 0.$ The structure of this tensor product is proved surprisingly difficult and has been not yet known for $t\geq 5,$ even for the trivial subgroup $G_t = \{e\}.$ In the present paper, we consider the subgroup $G_t = \{e\}$ for $t \in \{5, 6\},$ and obtain some new results on $\mathscr A_2$-generators of $(\mathbb P^{\otimes t})^{G_t}$ in some degrees. At the same time, some of their applications have been proposed. We also provide an algorithm in MAGMA for verifying the results. This study can be understood as a continuation of our recent works in [23, 25].

$$\infty $$-Operads via symmetric sequences

We construct a generalization of the Day convolution tensor product of presheaves that works for certain double $$\infty $$ ∞ -categories. Using this construction, we obtain an $$\infty $$ ∞ -categorical version of the well-known description of (one-object) operads as associative algebras in symmetric sequences; more generally, we show that (enriched) $$\infty $$ ∞ -operads with varying spaces of objects can be described as associative algebras in a double $$\infty $$ ∞ -category of symmetric collections.