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In this paper, we associate the TKK algebra G ˆ (J) with vertex algebras through twisted modules. Firstly, we prove that for any complex number ℓ , the category of restricted G ˆ (J) -modules of level ℓ is canonically isomorphic to the category of σ -twisted V C g ˆ (ℓ , 0) -modules, where V C g ˆ (ℓ , 0) is a vertex algebra arising from the toroidal Lie algebra of type C 2 and σ is an isomorphism of V C g ˆ (ℓ , 0) induced from the involution of this toroidal Lie algebra. Secondly, we prove that for any nonnegative integer ℓ , the integrable restricted G ˆ (J) -modules of level ℓ are exactly the σ -twisted modules for the quotient vertex algebra L C g ˆ (ℓ , 0) of V C g ˆ (ℓ , 0). Finally, we classify the irreducible 1 2 N -graded σ -twisted L C g ˆ (ℓ , 0) -modules. [ABSTRACT FROM AUTHOR]

We describe transposed Poisson algebra structures on Block Lie algebras B (q) and Block Lie superalgebras S (q) , where q is an arbitrary complex number. Specifically, we show that the transposed Poisson structures on B (q) are trivial whenever q ∉ Z , and for each q ∈ Z there is only one (up to an isomorphism) non-trivial transposed Poisson structure on B (q). The superalgebra S (q) admits only trivial transposed Poisson superalgebra structures for q ≠ 0 and two non-isomorphic non-trivial transposed Poisson superalgebra structures for q = 0. As a consequence, new Lie algebras and superalgebras that admit non-trivial Hom-Lie algebra structures are found. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we study some non-weight modules over the loop-Virasoro algebra L and Block type Lie algebras B(q) , where q is a nonzero complex number. Those modules when restricted to the Cartan subalgebra (modulo center) are free of rank one. We provide a sufficient and necessary condition for such modules to be simple, and determine their isomorphism classes. Moreover, we obtain the simplicity of modules over loop-Virasoro algebras by taking tensor products of some irreducible modules mentioned above with irreducible highest weight modules or Whittaker modules. [ABSTRACT FROM AUTHOR]

The conventional research topic in operator algebras involves exploring the structure of algebras and using homomorphic mappings to study the classification of algebras. In this study, a new invariant is developed based on the characteristics of the operator using the linear preserving method. The results show that the isomorphic mapping is used for preserving this invariant, which provides the classification information of operator algebra from a new perspective. Let H and K be Hilbert spaces with dimensions greater than two, and let B (H) and B (K) be the set of all bounded linear operators on H and K , respectively. For A , B ∈ B (H) , the ∗, ∗-Lie, and ∗-Jordan products are defined by A ∗ B , A ∗ B − B ∗ A , and A ∗ B + B ∗ A , respectively. Let Φ : B (H) → B (K) be an additive unital surjective map. It is confirmed that if Φ preserves zero ∗, ∗-Lie, and ∗-Jordan products, then Φ is unitary or conjugate unitary isomorphisms. [ABSTRACT FROM AUTHOR]

An endo-commutative algebra is a nonassociative algebra in which the square mapping preserves multiplication. In this paper, we give a complete classification of endo-commutative curled algebras of dimension 2 over an arbitrary nontrivial field, where a curled algebra satisfies the condition that the square of any element is a scalar multiple of that element. We list all multiplication tables of the algebras up to isomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we study the link between finite-dimensional Zinbiel algebras and combinatorial structures or (pseudo)digraphs determining which configurations are associated with those algebras. Some properties of Zinbiel algebras that can be read from their associated combinatorial structures are studied. We also analyze the isomorphism classes for each configuration associated with these algebras providing a new method to classify them and we compare our results with the current classifications of 2- and 3-dimensional Zinbiel algebras. We also obtain the 3-vertices combinatorial structures associated with such algebras. In order to complement the theoretical study, we have designed and performed the implementation of an algorithm which constructs and draws the (pseudo)digraph associated with a given Zinbiel algebra and, conversely, another procedure to test if a given combinatorial structure is associated with some Zinbiel algebra. [ABSTRACT FROM AUTHOR]

In this paper, a link between oriented CW complexes (that will also be mentioned as configurations) and alternative algebras is studied, determining which configurations are associated with those algebras. Moreover, the isomorphism classes of each two‐dimensional configuration associated with these algebras is analyzed, providing a new method to classify them. In order to complement the theoretical study, two algorithmic methods are implemented: The first one checks if a given algebra is alternative, while the second one constructs and draws the (pseudo)digraph associated with a given alternative algebra. [ABSTRACT FROM AUTHOR]

We define a product between quantum superoperators which is preserved under the Choiâ€"JamioĹ‚kowskiâ€"Krausâ€"Sudarshan channel-state isomorphism. We then identify the product as the convolution on the space of superoperators, with respect to which the channel-state duality is also an algebra isomorphism. We find that any witness operator for detecting nonseparability of quantum operations on separated parties can be written entirely within the space of superoperators with the help of the convolution product. [ABSTRACT FROM AUTHOR]

Let |${\mathfrak{g}}$| be a symmetrizable Kac–Moody algebra with associated Yangian |$Y_\hbar{\mathfrak{g}}$| and Yangian double |$\textrm{D}Y_\hbar{\mathfrak{g}}$|⁠. An elementary result of fundamental importance to the theory of Yangians is that, for each |$c\in{\mathbb{C}}$|⁠ , there is an automorphism |$\tau _c$| of |$Y_\hbar{\mathfrak{g}}$| corresponding to the translation |$t\mapsto t+c$| of the complex plane. Replacing |$c$| by a formal parameter |$z$| yields the so-called formal shift homomorphism |$\tau _z$| from |$Y_\hbar{\mathfrak{g}}$| to the polynomial algebra |$Y_\hbar{\mathfrak{g}}[z]$|⁠. We prove that |$\tau _z$| uniquely extends to an algebra homomorphism |$\Phi _z$| from the Yangian double |$\textrm{D}Y_\hbar{\mathfrak{g}}$| into the |$\hbar $| -adic closure of the algebra of Laurent series in |$z^{-1}$| with coefficients in the Yangian |$Y_\hbar{\mathfrak{g}}$|⁠. This induces, via evaluation at any point |$c\in{\mathbb{C}}^\times $|⁠ , a homomorphism from |$\textrm{D}Y_\hbar{\mathfrak{g}}$| into the completion of the Yangian with respect to its grading. We show that each such homomorphism gives rise to an isomorphism between completions of |$\textrm{D}Y_\hbar{\mathfrak{g}}$| and |$Y_\hbar{\mathfrak{g}}$| and, as a corollary, we find that the Yangian |$Y_\hbar{\mathfrak{g}}$| can be realized as a degeneration of the Yangian double |$\textrm{D}Y_\hbar{\mathfrak{g}}$|⁠. Using these results, we obtain a Poincaré–Birkhoff–Witt theorem for |$\textrm{D}Y_\hbar{\mathfrak{g}}$| applicable when |${\mathfrak{g}}$| is of finite type or of simply laced affine type. [ABSTRACT FROM AUTHOR]

In Schweigert and Yang (Methods Appl Symmetry Integr Geom, 2021) it was shown how string-net spaces for the Cardy bulk algebra in the Drinfeld center Z (C) of a modular tensor category C give rise to a consistent set of correlators. We extend their results to include open-closed world sheets and allow for more general field algebras, which come in the form of (C | Z (C)) -Cardy algebras. To be more precise, we show that a set of fundamental string-nets with input data from a (C | Z (C)) -Cardy algebra gives rise to a solution of the sewing constraints formulated in Kong et al. (Adv Math 262:604–681, 2014) and that any set of fundamental string-nets solving the sewing constraints determine a (C | Z (C)) -Cardy algebra up to isomorphism. Hence we give an alternative proof of the results in Kong et al. (2014) in terms of string-nets. [ABSTRACT FROM AUTHOR]

In this paper, we construct and study a new class of modules over the Block algebra B (q) , where q is a nonzero complex number. We determine the irreducibilities of these modules and the isomorphisms among them. We also show that these modules exhaust all U (h) -free B (q) -modules of rank-1. [ABSTRACT FROM AUTHOR]

We present a quantization of an isomorphism of Mirković and Vybornov which relates the intersection of a Slodowy slice and a nilpotent orbit closure in gln to a slice between spherical Schubert varieties in the affine Grassmannian of PGLn (with weights encoded by the Jordan types of the nilpotent orbits). A quantization of the former variety is provided by a parabolic W-algebra and of the latter by a truncated shifted Yangian. Building on earlier work of Brundan and Kleshchev, we define an explicit isomorphism between these non-commutative algebras and show that its classical limit is a variation of the original isomorphism of Mirković and Vybornov. As a corollary, we deduce that the W-algebra is free as a left (or right) module over its Gelfand-Tsetlin subalgebra, as conjectured by Futorny, Molev, and Ovsienko. [ABSTRACT FROM AUTHOR]

It is proved that on nilmanifolds with abelian complex structure, there exists a canonically constructed non-trivial holomorphic Poisson structure. We identify the necessary and sufficient condition for its associated cohomology to be isomorphic to the cohomology associated to trivial (zero) holomorphic Poisson structure. We also identify a sufficient condition for this isomorphism to be at the level of Gerstenhaber algebras. [ABSTRACT FROM AUTHOR]

This paper studies the link between isomorphic digraphs and isomorphic Leibniz algebras, determining in detail this fact when using (psuedo)digraphs of 2 and 3 vertices associated with Leibniz algebras according to their isomorphism classes. Moreover, we give the complete list with all the combinatorial structures of 3 vertices associated with Leibniz algebras, studying their isomorphism classes. We also compare our results with the current classifications of 2‐ and 3‐dimensional Leibniz algebras. Finally, we introduce and implement the algorithmic procedure used for our goals and devoted to decide if a given combinatorial structure is associated or not with a Leibniz algebra. [ABSTRACT FROM AUTHOR]

The known descriptions of the groups of order automorphisms of operator intervals are very simple with only one exception: there are two known results describing the general form of order automorphisms of the effect algebra and they both look quite complicated. It is the aim of this paper to show that the group of order automorphisms of the effect algebra is isomorphic to the group of order automorphisms of any other proper operator interval. After proving this statement we will present a new description of order automorphisms of the effect algebra explaining better that the case of the effect algebra is not more complicated than the other operator intervals. [ABSTRACT FROM AUTHOR]

Let G be a simple complex group of adjoint type. In his unpublished work, Z. Yun associated to each θ-group (G0, g1) and a vector X ∈ g1 a flat G-connection ∇X on the trivial G-bundle on P1−{0,∞}, generalizing the construction of Frenkel and Gross in [5]. In this paper, we study the local monodromies of the flat G-connection ∇X and compute the de Rham cohomology of ∇X with value in the adjoint representation of G. In particular, we show that in many cases the connection ∇X is cohomologically rigid. [ABSTRACT FROM AUTHOR]

Let be a finite dimensional simple real algebra with a division grading by a finite abelian group . In this paper, we provide a finite basis for the -ideal of graded polynomial identities for and a finite basis for the -space of graded central polynomials for . [ABSTRACT FROM AUTHOR]

The goals of this paper fall into two closely related areas. First, we develop a formal framework for deterministic unital quantities in which measurement unitization is understood to be a built-in feature of quantities rather than a mere annotation of their numerical values with convenient units. We introduce this idea within the setting of certain ordered semigroups of physical-geometric states of classical physical systems. States are assumed to serve as truth makers of metrological statements about quantity values. A unital quantity is presented as an isomorphism from the target system's ordered semigroup of states to that of positive reals. This framework allows us to include various derived and variable quantities, encountered in engineering and the natural sciences. For illustration and ease of presentation, we use the classical notions of length, time, electric current and mean velocity as primordial examples. The most important application of the resulting unital quantity calculus is in dimensional analysis. Second, in evaluating measurement uncertainty due to the analog-to-digital conversion of the measured quantity's value into its measuring instrument's pointer quantity value, we employ an ordered semigroup framework of pointer states. Pointer states encode the measuring instrument's indiscernibility relation, manifested by not being able to distinguish the measured system's topologically proximal states. Once again, we focus mainly on the measurement of length and electric current quantities as our motivating examples. Our approach to quantities and their measurement is strictly state-based and algebraic in flavor, rather than that of a representationalist-style structure-preserving numerical assignment. [ABSTRACT FROM AUTHOR]

The Jones-Wenzl idempotents are elements of the Temperley-Lieb (TL) planar algebra that are important, but complicated to write down. We will present a new planar algebra, the pop-switch planar algebra (PSPA), which contains the TL planar algebra. It is motivated by Jones' idea of the graph planar algebra (GPA) of type An. In the tensor category of idempotents of the PSPA, the nth Jones-Wenzl idempotent is isomorphic to a direct sum of n + 1 diagrams consisting of only vertical strands. [ABSTRACT FROM AUTHOR]

A problem that has been open for a long time is whether an automorphism of the center of the integral group ring of a finite group permutes the class sums. As a generalization, the similar problem for integral table algebras has also been studied in a few papers. In this paper we further the investigation of the problem for integralC-algebras and table algebras. We will first present some necessary and sufficient conditions under which an algebra isomorphism between integralC-algebras is monomial, and as an application, obtain a conceptual proof of the class sum correspondence theorem of group rings of finite groups. Then we prove that an algebra isomorphism compatible with degree maps between standard integral nilpotent table algebras over the ring of algebraic integers is exact. As a direct consequence, we get Hertweck's result [6] for finite nilpotent groups. An application to association schemes is also presented. [ABSTRACT FROM AUTHOR]