Extension Of Algebra Research Articles

Exact solutions of Klein–Gordon equations in external electromagnetic fields on 3D de Sitter background

We study symmetry properties and the possibility of exact integration of Klein–Gordon equations in external electromagnetic fields on 3D de Sitter background dS3. We present an algorithm for constructing the first-order symmetry algebra and describe its structure in terms of Lie algebra extensions. Based on the well-known classification of the subalgebras of the algebra so(1,3), we classify all electromagnetic fields on dS3 for which the corresponding Klein–Gordon equations admit first-order symmetry algebras. Then, we select the integrable cases, and for each of them, we construct exact solutions using the noncommutative integration method developed by Shapovalov and Shirokov [Theor. Math. Phys. 104, 921–934 (1995)]. We also propose an original algebraic method for constructing the special local coordinates on de Sitter space dS3, in which basis vector fields for subalgebras of the Lie algebra so(1,3) have the simplest form.

Symmetry criteria for Hochschild extensions

We give two sufficient and necessary conditions for the Hochschild extension of a finite-dimensional algebra by its dual bimodule and a Hochschild 2-cocycle to be a symmetric algebra.

Torsion points of Drinfeld modules over large algebraic extensions of finitely generated function fields

Geyer and Jarden proved several results for torsion points of elliptic curves defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated field over the prime field in its algebraic closure. As an analogue of these results, this paper studies torsion points of Drinfeld modules defined over the fixed field by finitely many elements in the absolute Galois group of a finitely generated function field in its algebraic closure. We prove some results which are similar to those of Geyer and Jarden.

Onto extensions of free groups

An extension of subgroups $H\leqslant K\leqslant F_A$ of the free group of rank $|A|=r\geqslant 2$ is called onto when, for every ambient free basis $A'$, the Stallings graph $\Gamma_{A'}(K)$ is a quotient of $\Gamma_{A'}(H)$. Algebraic extensions are onto and the converse implication was conjectured by Miasnikov-Ventura-Weil, and resolved in the negative, first by Parzanchevski-Puder for rank $r=2$, and recently by Kolodner for general rank. In this note we study properties of this new type of extension among free groups (as well as the fully onto variant), and investigate their corresponding closure operators. Interestingly, the natural attempt for a dual notion -- into extensions -- becomes trivial, making a Takahasi type theorem not possible in this setting.

Algebraic Extensions

Summary In this article we further develop field theory in Mizar [1], [2], [3] towards splitting fields. We deal with algebraic extensions [4], [5]: a field extension E of a field F is algebraic, if every element of E is algebraic over F. We prove amongst others that finite extensions are algebraic and that field extensions generated by a finite set of algebraic elements are finite. From this immediately follows that field extensions generated by roots of a polynomial over F are both finite and algebraic. We also define the field of algebraic elements of E over F and show that this field is an intermediate field of E|F.

Soft Int-Field Extension

The relation between soft element-wise field and soft int-field has been established and then some properties of soft int-field are studied. We define the notions of soft algebraic element and soft purely inseparable element of a soft int-field extension. Some characterizations of soft algebraic and soft purely inseparable int-field extensions are given. Lastly, we define soft separable algebraic int-field extension and study some of its properties.

Nakayama automorphisms of graded Ore extensions of Koszul Artin-Schelter regular algebras

Let A be a Koszul Artin-Schelter regular algebra, σ a graded automorphism of A and δ a degree-one σ-derivation of A. We introduce an invariant for δ called the σ-divergence of δ. We describe the Nakayama automorphism of the graded Ore extension B=A[z;σ,δ] explicitly using the σ-divergence of δ, and construct a twisted superpotential ωˆ for B so that it is a derivation quotient algebra defined by ωˆ. We also determine all graded Ore extensions of noetherian Artin-Schelter regular algebras of dimension 2 and compute their Nakayama automorphisms.

The Roger–Yang skein algebra and the decorated Teichmüller space

Based on hyperbolic geometric considerations, Roger and Yang introduced an extension of the Kauffman bracket skein algebra that includes arcs. In particular, their skein algebra is a deformation quantization of a certain commutative curve algebra, and there is a Poisson algebra homomorphism between the curve algebra and the algebra of smooth functions on decorated Teichmüller space. In this paper, we consider surfaces with punctures which are not the 3-holed sphere and which have an ideal triangulation without self-folded edges or triangles. For those surfaces, we prove that Roger and Yang’s Poisson algebra homomorphism is injective, and the skein algebra has no zero divisors. A section about generalized corner coordinates for normal arcs may be of independent interest.

A matrix viewpoint for various algebraic extensions

A matrix viewpoint for various algebraic extensions

A Category of Restricted Modules for the Ovisenko-Roger Algebra

In this paper, we study a category of restricted modules for the Ovsienko-Roger algebra, which is an extension of the Virasoro algebra of its tensor density module of degree one. We construct and characterize simple modules in this category and give natural free field realizations of certain restricted modules using the Weyl vertex algebra.

Nonlinear nonisospectral differential coverings for the hyper-CR equation of Einstein–Weyl structures and the Gibbons–Tsarev equation

We apply the technique based on twisted extensions of symmetry algebras to construct new nonlinear four-dimensional differential coverings for the hyper-CR equation of Einstein–Weyl structures and for the associated integrable hierarchy. We expose related multi-component three-dimensional covering. By the symmetry reduction of the hyper-CR equation of Einstein–Weyl structures we derive nonlinear three-dimensional differential covering for the Gibbons-Tsarev equation.

A categorical approach to operator semigroups

The aim of this paper is to exploit the structure of strongly continuous operator semigroups in order to formulate a categorical framework in which a fresh perspective can be applied to past operator theoretic results. In particular, we investigate the inverse-producing Arens extension for Banach algebras (Trans. Am. Math. Soc. 88:536–548, 1958) adapted for operators and operator semigroups by Batty and Geyer (J. Oper. Theory 78(2):473–500, 2017) in this new framework, asking and answering questions using categorical language. We demonstrate that the Arens extension defines an extension functor in this setting and that it forms an adjunction with the suitably defined forgetful functor. As a by-product of this categorical framework, we also revisit the work on Banach direct sums by Lachowicz and Moszyński (Semigroup Forum 93(1):34–70, 2016). This paper can be considered as a brief exploration of the triple interface between operator semigroups, Banach algebras, and category theory.

The Vertex Algebras $$\mathcal {R}^{(p)}$$ and $$\mathcal {V}^{({p})}$$

The vertex algebras $$\mathcal {V}^{(p)}$$ and $$\mathcal R^{(p)}$$ introduced in Adamović (Transform Groups 21(2):299–327, 2016) are very interesting relatives of the well-known triplet algebras of logarithmic CFT. The algebra $$\mathcal {V}^{(p)}$$ (respectively, $$\mathcal {R}^{(p)}$$ ) is a large extension of the simple affine vertex algebra $$L_{k}(\mathfrak {sl}_{2})$$ (respectively, $$L_{k}(\mathfrak {sl}_{2})$$ times a Heisenberg algebra), at level $$k=-2+1/p$$ for positive integer p. Particularly, the algebra $$\mathcal {V}^{(2)}$$ is the simple small $$N=4$$ superconformal vertex algebra with $$c=-9$$ , and $$\mathcal {R}^{(2)}$$ is $$L_{-3/2}(\mathfrak {sl}_3)$$ . In this paper, we derive structural results of these algebras and prove various conjectures coming from representation theory and physics. We show that SU(2) acts as automorphisms on $$\mathcal {V}^{({p})}$$ and we decompose $$\mathcal {V}^{({p})}$$ as an $$L_{k}(\mathfrak {sl}_{2})$$ -module and $$\mathcal {R}^{({p})}$$ as an $$L_k(\mathfrak {gl}_2)$$ -module. The decomposition of $$\mathcal {V}^{({p})}$$ shows that $$\mathcal {V}^{({p})}$$ is the large level limit of a corner vertex algebra appearing in the context of S-duality. We also show that the quantum Hamiltonian reduction of $$\mathcal {V}^{({p})}$$ is the logarithmic doublet algebra $$\mathcal {A}^{({p})}$$ introduced in Adamović and Milas (Contemp Math 602:23–38, 2013), while the reduction of $$\mathcal {R}^{({p})}$$ yields the $$\mathcal {B}^{({p})}$$ -algebra of Creutzig et al. (Lett Math Phys 104(5):553–583, 2014). Conversely, we realize $$\mathcal {V}^{({p})}$$ and $$\mathcal {R}^{({p})}$$ from $$\mathcal {A}^{({p})}$$ and $$\mathcal {B}^{({p})}$$ via a procedure that deserves to be called inverse quantum Hamiltonian reduction. As a corollary, we obtain that the category $$KL_{k}$$ of ordinary $$L_{k}(\mathfrak {sl}_{2})$$ -modules at level $$k=-2+1/p$$ is a rigid vertex tensor category equivalent to a twist of the category $$\text {Rep}(SU(2))$$ . This finally completes rigid braided tensor category structures for $$L_{k}(\mathfrak {sl}_{2})$$ at all complex levels k. We also establish a uniqueness result of certain vertex operator algebra extensions and use this result to prove that both $$\mathcal {R}^{({p})}$$ and $$\mathcal {B}^{({p})}$$ are certain non-principal $$\mathcal {W}$$ -algebras of type A at boundary admissible levels. The same uniqueness result also shows that $$\mathcal {R}^{({p})}$$ and $$\mathcal {B}^{({p})}$$ are the chiral algebras of Argyres-Douglas theories of type $$(A_1, D_{2p})$$ and $$(A_1, A_{2p-3})$$ .

Non-relativistic three-dimensional supergravity theories and semigroup expansion method

In this work we present an alternative method to construct diverse non-relativistic Chern-Simons supergravity theories in three spacetime dimensions. To this end, we apply the Lie algebra expansion method based on semigroups to a supersymmetric extension of the Nappi-Witten algebra. Two different families of non-relativistic superalgebras are obtained, corresponding to generalizations of the extended Bargmann superalgebra and extended Newton-Hooke superalgebra, respectively. The expansion method considered here allows to obtain known and new non-relativistic supergravity models in a systematic way. In particular, it immediately provides an invariant tensor for the expanded superalgebra, which is essential to construct the corresponding Chern-Simons supergravity action. We show that the extended Bargmann supergravity and its Maxwellian generalization appear as particular subcases of a generalized extended Bargmann supergravity theory. In addition, we demonstrate that the generalized extended Bargmann and generalized extended Newton-Hooke supergravity families are related through a contraction process.

An Application of a Theorem of Sheila Brenner for Hochschild Extension Algebras of a Truncated Quiver Algebra

Let A be a truncated quiver algebra over an algebraically closed field such that any oriented cycle in the ordinary quiver of A is zero in A. We give the number of the indecomposable direct summands of the middle term of an almost split sequence for a class of Hochschild extension algebras of A by the standard duality module D(A).

Biprojectivity of generalized module extension and second dual of Banach algebras

In this paper, we study biprojectivity of generalized module extension Banach algebras and second dual of Banach algebras. As a main result, we prove that if [Formula: see text] is a contractible closed ideal of [Formula: see text] such that [Formula: see text] is biprojective, then [Formula: see text] is biprojective. As a consequence, we give some results on biprojectivity of generalized module extension Banach algebras. Indeed, this is a generalization of results in [A. Ebaddian and A. Jabbari, Biprojectivity and biflatness of amalgamated duplication of Banach algebras, J. Algebra Its Appl. 19(7) (2020) 2050132 (15 pp.)]. Moreover, we show that if [Formula: see text] is biprojective and [Formula: see text] is a two-sided ideal of [Formula: see text] then [Formula: see text] is biprojective. This fact generalizes some known results in [M. S. Moslehian and A. Niknam, Biflatness and biprojectivity of second dual of Banach algebras, Southeast Asian Bull. Math 27(1) (2003) 129–133].

Two Tilts of Higher Spherical Algebras

We introduce and study two exotic families of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher spherical algebra studied by Erdmann and Skowroński (Arch. Math. 114, 25–39, 2020), and hence that it is a tame symmetric periodic algebra of period 4. This together with the results of Erdmann and Skowroński (Algebr. Represent. Theor. 22, 387–406, 2019; Arch. Math. 114, 25–39, 2020) shows that every trivial extension algebra of a tubular algebra of type (2,2,2,2) admits a family of periodic symmetric higher deformations which are tame of non-polynomial growth and have the same Gabriel quiver, answering the question recently raised by Skowroński.

On isoclinic extensions of Lie algebras and Nilpotent Lie algebras

On isoclinic extensions of Lie algebras and Nilpotent Lie algebras

Higher Kac–Moody algebras and symmetries of holomorphic field theories

We introduce a higher dimensional generalization of the affine Kac-Moody algebra using the language of factorization algebras. In particular, on any complex manifold there is a factorization algebra of currents associated to any Lie algebra. We classify local cocycles of these current algebras, and compare them to central extensions of higher affine algebras recently proposed by Faonte-Hennion-Kapranov. A central goal of this paper is to witness higher Kac-Moody algebras as symmetries of a class of holomorphic quantum field theories. In particular, we prove a generalization of the free field realization of an affine Kac-Moody algebra and also develop the theory of q-characters for this class of algebras in terms of factorization homology. Finally, we exhibit the large N behavior of higher Kac-Moody algebras and their relationship to symmetries of non-commutative field theories.

Infinite linear and algebraic independence of values of F-series at polyadic Liouvillea points

This paper proves infinite linear and algebraic independence of the values of 𝐹-series at polyadic Liouville points using a modification of the generalised Siegel-Shidlovskii method. 𝐹- series have form 𝑓𝑛 =Σ∞𝑛=0 𝑎𝑛𝑛!𝑧𝑛 whose coefficients 𝑎𝑛 satisfy some arithmetic properties. These series converge in the field Q𝑝 of 𝑝-adic numbers and their algebraic extensions K𝑣. Polyadic number is a series of the form Σ∞𝑛=0 𝑎𝑛𝑛!, 𝑎𝑛 ∈ Z. Liouville number is a real number x with the property that, for every positive integer n, there exist infinitely many pairs of integers (𝑝, 𝑞) with 𝑞 > 1 such that 0 <|𝑥 − 𝑝/𝑞|< 1/𝑞^𝑛 . The polyadic Liouville number 𝛼 has the property that for any numbers 𝑃,𝐷 there exists an integer |𝐴| such that for all primes 𝑝 ≤ 𝑃 the inequality |𝛼−𝐴|𝑝 < 𝐴−𝐷. Infinite linear (algebraic) independence means that for any nonzero linear form (any nonzero polynomial) there are infinitely many primes 𝑝 and valuations 𝑣 extending 𝑝-adic valuation to an algebraic number field K with the following property: the result of substitution in the considered linear form (polynomial) of the values of 𝐹 - of series instead of variables is a nonzero element of the field. Previously, only the existence of at least one prime number 𝑝 with the properties listed above was proved.