Recent zbMATH articles in MSC 17https://www.zbmath.org/atom/cc/172021-02-27T13:50:00+00:00Unknown authorWerkzeugOn canonical bases for the Letzter algebra \(\mathbf{U}^\imath(\mathfrak{sl}_2)\).https://www.zbmath.org/1453.170122021-02-27T13:50:00+00:00"Li, Yiqiang"https://www.zbmath.org/authors/?q=ai:li.yiqiangThe main result of this paper says that the canonical basis of (the modified form of) Letzter's coideal subalgebra of quantum \({sl}_2\), constructed via a geometric approach in [Bull. Inst. Math., Acad. Sin. (N.S.) 13, No. 2, 143-198 (2018; Zbl 1440.17012)] by the author and \textit{W. Wang}, coincides with the algebraic basis conjectured by \textit{H. Bao} and \textit{W. Wang} [A new approach to Kazhdan-Lusztig theory of type \(B\) via quantum symmetric pairs. Paris: SMF (2018; Zbl 1411.17001)] and proved in [J. Pure Appl. Algebra 222, No. 9, 2667--2702 (2018; Zbl 1388.17004)] by \textit{C. Berman} and \textit{W. Wang}.
Reviewer: Sonia Natale (Córdoba)Modified trace from pivotal Hopf \(G\)-coalgebras.https://www.zbmath.org/1453.160342021-02-27T13:50:00+00:00"Ha, Ngoc Phu"https://www.zbmath.org/authors/?q=ai:ha.ngoc-phuSummary: In a recent paper the authors \textit{A. Beliakova} et al. [``Modified trace is a symmetrised integral'', Preprint, \url{arXiv:1801.00321}] have shown that the modified trace on the category \(H\)-pmod of the projective modules corresponds to the symmetrised integral on the finite dimensional pivotal Hopf algebra \(H\). We generalize this fact to the context of \(G\)-graded categories and Hopf \(G\)-coalgebra studied by \textit{V. Turaev} [Homotopy quantum field theory. With appendices by Michael Müger and Alexis Virelizier. Zürich: European Mathematical Society (EMS) (2010; Zbl 1243.81016)] and
\textit{A. Virelizier} [J. Pure Appl. Algebra 171, No. 1, 75--122 (2002; Zbl 1011.16023)]. We show that the symmetrised \(G\)-integral on a finite type pivotal Hopf \(G\)-coalgebra induces a modified trace in the associated \(G\)-graded category.Symmetries of supersymmetric CKP hierarchy and its reduction.https://www.zbmath.org/1453.370612021-02-27T13:50:00+00:00"Li, Chuanzhong"https://www.zbmath.org/authors/?q=ai:li.chuanzhong"Ge, Ruiling"https://www.zbmath.org/authors/?q=ai:ge.ruilingThe aim of this paper is to construct a supersymmetric C-type Kadomtsev-Petviashvili (CKP) hierarchy and a six-reduced supersymmetric CKP hierarchy with its additional symmetries. These additional flows of the supersymmetric CKP hierarchy
span a C-type \(SW_{1+\infty}\) Lie algebra. Further, the authors prove that the six-reduced supersymmetric CKP hierarchy contains the super-Sawada-Kotera equation as a primary equation. Finally, they prove that the additional C-type \(SW_{1+\infty}\) Lie algebraic structure is preserved after performing the six-reduction from the supersymmetric CKP hierarchy.
This paper is organized as follows. Section 1 is an introduction to the subject. In Section 2, the authors recall all the relevant facts about supersymmetric integrable theory and they define the supersymmetric CKP hierarchy. In Section 3, the authors construct additional symmetries of the supersymmetric CKP hierarchy, using Orlov-Schulman operators. Section 4 deals with the six-reduced supersymmetric CKP hierarchy and its connection with the super Sawada-Kotera equation. Section 5 presents the additional symmetries for the six-reduced supersymmetric CKP hierarchy.
Reviewer: Ahmed Lesfari (El Jadida)Polyhedral parametrizations of canonical bases \& cluster duality.https://www.zbmath.org/1453.130652021-02-27T13:50:00+00:00"Genz, Volker"https://www.zbmath.org/authors/?q=ai:genz.volker"Koshevoy, Gleb"https://www.zbmath.org/authors/?q=ai:koshevoy.gleb-a"Schumann, Bea"https://www.zbmath.org/authors/?q=ai:schumann.beaSummary: We establish the relation of Berenstein-Kazhdan's decoration function and Gross-Hacking-Keel-Kontsevich's potential on the open double Bruhat cell in the base affine space \(\text{G} / \mathcal{N}\) of a simple, simply connected, simply laced algebraic group \(G\). As a byproduct we derive explicit identifications of polyhedral parametrization of canonical bases of the ring of regular functions on \(\text{G} / \mathcal{N}\) arising from the tropicalizations of the potential and decoration function with the classical string and Lusztig parametrizations. In the appendix we construct maximal green sequences for the open double Bruhat cell in \(\text{G} / \mathcal{N}\) which is a crucial assumption for Gross-Hacking-Keel-Kontsevich's construction.Associated binomial inversion for unified Stirling numbers and counting subspaces generated by subsets of a root system.https://www.zbmath.org/1453.110372021-02-27T13:50:00+00:00"Kamiyoshi, Tomohiro"https://www.zbmath.org/authors/?q=ai:kamiyoshi.tomohiro"Nagura, Makoto"https://www.zbmath.org/authors/?q=ai:nagura.makoto"Otani, Shin-Ichi"https://www.zbmath.org/authors/?q=ai:otani.shinichiSummary: We introduce an associated version of the binomial inversion for unified Stirling numbers defined by \textit{L. C. Hsu} and \textit{P. J. S. Shiue} [Adv. Appl. Math. 20, No. 3, 366--384 (1998; Zbl 0913.05006)]. This naturally appears when we count the number of subspaces generated by subsets of a root system. We count such subspaces of any dimension by using associated unified Stirling numbers, and then we will also give a combinatorial interpretation of our inversion formula. In particular, the well-known explicit formula for classical Stirling numbers of the second kind can be understood as a special case of our formula.Nonassociativity of the Norton algebras of some distance regular graphs.https://www.zbmath.org/1453.051372021-02-27T13:50:00+00:00"Huang, Jia"https://www.zbmath.org/authors/?q=ai:huang.jiaSummary: A Norton algebra is an eigenspace of a distance regular graph endowed with a commutative nonassociative product called the Norton product, which is defined as the projection of the entrywise product onto this eigenspace. The Norton algebras are useful in finite group theory as they have interesting automorphism groups. We provide a precise quantitative measurement for the nonassociativity of the Norton product on the eigenspace of the second largest eigenvalue of the Johnson graphs, Grassman graphs, Hamming graphs, and dual polar graphs, based on the formulas for this product established in previous work of \textit{F. Levstein} et al. [Eur. J. Comb. 27, No. 1, 1--10 (2006; Zbl 1078.05091)]. Our result shows that this product is as nonassociative as possible except for two cases, one being the trivial vanishing case while the other having connections with the integer sequence A000975 on OEIS and the so-called double minus operation studied recently by \textit{J. Huang} et al. [J. Integer Seq. 20, No. 10, Article 17.10.3, 11 p. (2017; Zbl 1384.05026)].The Drinfeld Yangian of the queer Lie superalgebra. Defining relations.https://www.zbmath.org/1453.320072021-02-27T13:50:00+00:00"Stukopin, V. A."https://www.zbmath.org/authors/?q=ai:stukopin.vladimir-alekseevichSummary: Drinfeld Yangian of a queer Lie superalgebra is defined as the quantization of a Lie bisuperelgebra of twisted polynomial currents. An analogue of the new system of generators of Drinfeld is being constructed. It is proved for the partial case of Lie superalgebra \(sq_1\) that this so defined Yangian and the Yangian, introduced earlier by M. Nazarov using the Faddeev-Reshetikhin-Takhtadzhjan approach, are isomorphic.Retracted: Eight-dimensional octonion-like but associative normed division algebra.https://www.zbmath.org/1453.160172021-02-27T13:50:00+00:00"Christian, Joy"https://www.zbmath.org/authors/?q=ai:christian.joySummary: We present an eight-dimensional even sub-algebra of the \(2^4=16\)-dimensional associative Clifford algebra \(Cl_{4,0}\) and show that its eight-dimensional elements denoted as \(\mathbf X\) and \(\mathbf Y\) respect the norm relation \(\|\mathbf X \mathbf Y\| = \|\mathbf X\| \|\mathbf Y\|\) thus forming an octonion-like but associative normed division algebra, where the norms are calculated using the fundamental geometric product instead of the usual scalar product. The corresponding 7-sphere has a topology that differs from that of octonionic 7-sphere.
Editorial remark. This article has been retracted by the Editors and Publisher. The error is obvious both from the title and abstract, which claims to construct an 8-dimensional normed division algebra over \(\mathbb R\), which doesn't exist due to Hurwitz's theorem [\textit{A. Hurwitz}, Math. Ann. 88, 1--25 (1922; JFM 48.1164.03)].Macdonald's formula for Kac-Moody groups over local fields.https://www.zbmath.org/1453.200662021-02-27T13:50:00+00:00"Bardy-Panse, Nicole"https://www.zbmath.org/authors/?q=ai:bardy-panse.nicole"Gaussent, Stéphane"https://www.zbmath.org/authors/?q=ai:gaussent.stephane"Rousseau, Guy"https://www.zbmath.org/authors/?q=ai:rousseau.guySummary: For an almost split Kac-Moody group \(G\) over a local non-Archimedean field, the last two authors constructed a spherical Hecke algebra \(^s\mathcal{H}\) (over \(\mathbb{C}\), say) and its Satake isomorphism \(\mathcal S\) with the commutative algebra \(\mathbb{C}[[Y]]^{W^v}\) of Weyl invariant elements in some formal series algebra \(\mathbb{C}[[Y]]\). In this article, we prove a Macdonald's formula, that is, an explicit formula for the image \(\mathcal S(c_\lambda)\) of a basis element of \(^s\mathcal{H}\). The proof involves geometric arguments in the masure associated to G and algebraic tools, including the Cherednik's representation of the Bernstein-Lusztig-Hecke algebra (introduced in a previous article) and the Cherednik's identity between some symmetrizers.Homomorphisms and rigid isomorphisms of twisted group doubles.https://www.zbmath.org/1453.170112021-02-27T13:50:00+00:00"Keilberg, Marc"https://www.zbmath.org/authors/?q=ai:keilberg.marcIf \(G\) is a finite group, and \(\omega \in Z^3(G,U(1))\) is a 3-cocycle of \(G\) with values in the circle group \(U(1)\), \textit{R. Dijkgraaf} et al. [Nucl. Phys. B, Proc. Suppl. 18B, No. 2, 60--72 (1991; Zbl 0957.81670)] introduced a quasi-Hopf algebra \(\mathcal{D}^{\omega}(G)\) over the complex numbers, called the twisted double, which has proved to be of great interest in mathematical physics. If \(\omega\) is the trivial 3-cocycle, \(\mathcal{D}^{\omega}(G)\) is the usual Drinfeld double \(\mathcal{D}(G)\).
The aim of the paper is to discuss quasi-bialgebra morphisms between two such twisted doubles \(\mathcal{D}^{\omega}(G)\) and \(\mathcal{D}^{\eta}(H)\). The idea of the investigation is to associate to a morphism \(\mathcal{D}^{\omega}(G)\rightarrow \mathcal{D}^{\eta}(H)\) a quadruple of morphisms of certain kinds. The properties of the components of this quadruple are discussed, and then used to classify those quasi-bialgebra isomorphisms which also naturally determine an isomorphism of Hopf algebras \(\mathcal{D}(G)\rightarrow \mathcal{D}(H)\). As an application, \(\Aut(\mathcal{D}^{\omega}(G))\) is described in the case where \(\omega \in Z^3(G/Z(G),U(1))\).
Reviewer: Sorin Dascalescu (Bucureşti)Degenerations of nilpotent associative commutative algebras.https://www.zbmath.org/1453.170042021-02-27T13:50:00+00:00"Kaygorodov, Ivan"https://www.zbmath.org/authors/?q=ai:kaigorodov.i-b"Lopes, Samuel A."https://www.zbmath.org/authors/?q=ai:lopes.samuel-a"Popov, Yury"https://www.zbmath.org/authors/?q=ai:popov.yuryThe algebraic classification of 5-dimensional nilpotent associative and commutative algebras \(A\) over algebraic closed field \(k\) (of characteristic \(\ne 2\)) was given by G. Mazzola as a particular case of his classification of all 5-dimensional algebras -- there are 24 such algebras \(A\). Also he proved that the variety of all 5-dimensional nilpotent associative commutative algebras over \(k\) has only one irreducible component. The main result of this article is the following: the graph of all degenerations in the variety of 5-dimensional complex nilpotent associative commutative algebras is drawn. The proof is given by presenting in two tables all primary degenerations (by explicitly specifying one-parameter families of matrices that transform bases of algebras) and nondegenerations, using special properties of aforementioned 24 algebras.
Then 5-dimensional commutative Leibniz algebras are considered (they all are 2-step nilpotent associative and commutative algebras). All non-trivial complex 5-dimensional commutative Leibniz algebras are found (there are 13 such algebras) and it is proved that variety of such algebras has three irreducible components ree rigid algebras. Also two series of rigid \(n\)-dimensional Leibniz algebras are found.
Reviewer: V. V. Gorbatsevich (Moskva)Some properties of the Schur multiplier and stem covers of Leibniz crossed modules.https://www.zbmath.org/1453.170032021-02-27T13:50:00+00:00"Casas, José Manuel"https://www.zbmath.org/authors/?q=ai:casas-miras.jose-manuel"Ravanbod, Hajar"https://www.zbmath.org/authors/?q=ai:ravanbod.hajarLeibniz crossed modules are usual generalization of Leibniz algebras and also modules and ideals over Leibniz algebras. So it is a natural to search for similar concepts and results of Leibniz algebras in a higher level.
In this paper, the well-known notions related to the Schur multiplier of groups, Lie algebras, Leibniz algebras are studied in the category \textbf{XLb} of Leibniz algebras crossed modules. At first, authors recalled some basic categorical concepts such as the commutator of two ideals, the center and central extensions of Leibniz crossed modules. Moreover, they showed that the category of Leibniz crossed modules has enough projective objects. Hence, the Baer invariant associated with a projective presentation of \((\mathfrak{n},\mathfrak{q},\delta)\) can be defined and is called the Schur multiplier of the Leibniz crossed module \((\mathfrak{n},\mathfrak{q},\delta)\) which is denoted by \(\mathcal{M}(\mathfrak{n},\mathfrak{q},\delta)\).
As an interesting description of the Scur multiplier, authors proved that
\[\mathcal{M}(\mathfrak{n},\mathfrak{q},\delta)\cong \mathrm{Ker}((\mathfrak{q}\curlywedge\mathfrak{n}, \mathfrak{q}\curlywedge\mathfrak{q}, id\curlywedge\delta)\to (\mathfrak{n}, \mathfrak{q}, \delta)), \]
where the symbol \(\curlywedge\) is the non-abelian exterior product of Leibniz algebras. They used this result to construct a Stallings type of six-term exact sequence of Leibniz crossed modules associated with a central extension of Leibniz crossed modules.
Also, the authors study basic properties of stem covers of Leibniz crossed modules. In particular, they prove the existence of stem covers for an arbitrary Leibniz crossed module and determined the common structure of all stem covers of a Leibniz crossed modules whose Schur multiplier is finite dimensional.
Finally, in the last section, they study the connections between the stem covers of a Lie crossed module in the categories of Lie and Leibniz crossed modules.
Reviewer: Behrouz Edalatzadeh (Kermanshah)Nonassociative rings satisfying \(a(bc)=b(ca)\) and \((a,a,b)=(b,a,a)\).https://www.zbmath.org/1453.170012021-02-27T13:50:00+00:00"Samanta, Dhabalendu"https://www.zbmath.org/authors/?q=ai:samanta.dhabalendu"Hentzel, Irvin Roy"https://www.zbmath.org/authors/?q=ai:hentzel.irvin-roySummary: In this paper, we study nonassociative rings satisfying the polynomial identities \(a(bc)=b(ca)\) and \((a,a,b)=(b,a,a)\). We prove that nonassociative rings satisfying the identities \(a(bc)=b(ca)\) and \((a,a,b)=(b,a,a)\) with characteristic \(\neq 2\) are associative and commutative of degree five.Quantizations of regular functions on nilpotent orbits.https://www.zbmath.org/1453.170092021-02-27T13:50:00+00:00"Losev, Ivan"https://www.zbmath.org/authors/?q=ai:losev.ivan-vIn the paper under review the author studies a quantizations of an algebras of regular functions on a nilpotent orbits. The main result of the paper is the following one. ``Such a quantization always exists and is unique if the orbit is birationally rigid. For special birationally rigid orbits, the quantization has integral central character in all cases but four
Reviewer: Dmitry Artamonov (Moskva)Corrigendum to: ``A new approach to the representation theory of the symmetric groups. IV. \(\mathbb{Z}_2\)-graded groups and algebras''.https://www.zbmath.org/1453.200232021-02-27T13:50:00+00:00"Vershik, A."https://www.zbmath.org/authors/?q=ai:vershik.anatoli-m"Sergeev, A."https://www.zbmath.org/authors/?q=ai:sergeev.alexander-n|sergeev.armen-glebovichCorrigendum to the authors' paper [ibid. 8, No. 4, 813--842 (2004; Zbl 1196.20017)].Torsion free sheaves on Weierstrass cubic curves and the classical Yang-Baxter equation.https://www.zbmath.org/1453.140902021-02-27T13:50:00+00:00"Burban, Igor"https://www.zbmath.org/authors/?q=ai:burban.igor"Galinat, Lennart"https://www.zbmath.org/authors/?q=ai:galinat.lennartSummary: This work deals with an algebro-geometric theory of solutions of the classical Yang-Baxter equation based on torsion free coherent sheaves of Lie algebras on Weierstraß cubic curves.Wildness of the problem of classifying nilpotent Lie algebras of vector fields in four variables.https://www.zbmath.org/1453.170142021-02-27T13:50:00+00:00"Bondarenko, Vitalij M."https://www.zbmath.org/authors/?q=ai:bondarenko.vitalij-m"Petravchuk, Anatoliy P."https://www.zbmath.org/authors/?q=ai:petravchuk.anatolii-pSummary: Let $\mathbb{F}$ be a field of characteristic zero. Let $W_n(\mathbb{F})$ be the Lie algebra of all $\mathbb{F}$-derivations with the Lie bracket $[D_1, D_2] : = D_1 D_2 - D_2 D_1$ on the polynomial ring $\mathbb{F} [x_1, \ldots, x_n]$. The problem of classifying finite dimensional subalgebras of $W_n(\mathbb{F})$ was solved if $n \leqslant 2$ and $\mathbb{F} = \mathbb{C}$ or $\mathbb{F} = \mathbb{R}$. We prove that this problem is wild if $n \geqslant 4$, which means that it contains the classical unsolved problem of classifying matrix pairs up to similarity. The structure of finite dimensional subalgebras of $W_n(\mathbb{F})$ is interesting since each derivation in case $\mathbb{F} = \mathbb{R}$ can be considered as a vector field with polynomial coefficients on the manifold $\mathbb{R}^n$.Strange duality revisited.https://www.zbmath.org/1453.810552021-02-27T13:50:00+00:00"Pauly, Christian"https://www.zbmath.org/authors/?q=ai:pauly.christianSummary: We give a proof of the strange duality or rank-level duality of the WZW models of conformal blocks by extending the genus-\(0\) result, obtained by \textit{T. Nakanishi} and \textit{A. Tsuchiya} [Commun. Math. Phys. 144, No. 2, 351--372 (1992; Zbl 0751.17024)], to higher genus curves via the sewing procedure. The new ingredient of the proof is an explicit use of the branching rules of the conformal embedding of affine Lie algebras \(\widehat{\mathfrak{sl}(r)} \times \widehat{\mathfrak{sl}(l)} \subset \widehat{\mathfrak{sl}(rl)}\). We recover the strange duality of spaces of generalized theta functions obtained by Belkale, Marian-Oprea, as well as by Oudompheng in the parabolic case.Formal Abel-Jacobi maps.https://www.zbmath.org/1453.140272021-02-27T13:50:00+00:00"Fiorenza, Domenico"https://www.zbmath.org/authors/?q=ai:fiorenza.domenico"Manetti, Marco"https://www.zbmath.org/authors/?q=ai:manetti.marcoSummary: We realize the infinitesimal Abel-Jacobi map as a morphism of formal deformation theories, realized as a morphism in the homotopy category of differential graded Lie algebras. The whole construction is carried out in a general setting, of which the classical Abel-Jacobi map is a special example.Analysis in noncommutative algebras and modules.https://www.zbmath.org/1453.160302021-02-27T13:50:00+00:00"Zharinov, V. V."https://www.zbmath.org/authors/?q=ai:zharinov.victor-vSummary: In a previous paper [ibid. 301, 98--108 (2018; Zbl 1448.16038); translation from Tr. Mat. Inst. Steklova 301, 108--118 (2018)], we developed an analysis in associative commutative algebras and in modules over them, which may be useful in problems of contemporary mathematical and theoretical physics. Here we work out similar methods in the noncommutative case.Takiff algebras with polynomial rings of symmetric invariants.https://www.zbmath.org/1453.170102021-02-27T13:50:00+00:00"Panyushev, Dmitri I."https://www.zbmath.org/authors/?q=ai:panyushev.dmitri-i"Yakimova, Oksana S."https://www.zbmath.org/authors/?q=ai:yakimova.oksana-sLet \(Q\) be an algebraic group over an algebraically closed field \(k\) of characteristic \(0\). Let \(q\) be it's Lie algebra. There exist examples of non-reductive \(Q\)'s such that the invariant algebra \(k[q^*]^Q\) is polynomial. In the paper under review new such examples are found. It is proved that under mild restrictions on \(q\) the passage from \(a\) to the \(m\)-th Takiff algebra \(q< m > =q\otimes k[T]/(T^{m+1})\) preserves the polynomiality of symmetric invariants.
Reviewer: Dmitry Artamonov (Moskva)The geometry of gaussoids.https://www.zbmath.org/1453.130762021-02-27T13:50:00+00:00"Boege, Tobias"https://www.zbmath.org/authors/?q=ai:boege.tobias"D'Alì, Alessio"https://www.zbmath.org/authors/?q=ai:dali.alessio"Kahle, Thomas"https://www.zbmath.org/authors/?q=ai:kahle.thomas"Sturmfels, Bernd"https://www.zbmath.org/authors/?q=ai:sturmfels.berndThis work arises from trying to answer Problem 4 presented in [\textit{B. Sturmfels}, IMA Vol. Math. Appl. 149, 351--363 (2009; Zbl 1158.13300)]: ``General problem: Study the geometry of conditional independence models for multivariate Gaussian random variables.'' Thus, this survey develops the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. Throughout the 30 pages, the authors introduce and classify oriented gaussoids, connect valuated gaussoids to tropical geometry, addresses the realizability problem for gaussoids and oriented gaussoids and so on. The reader can find additional materials on the web \url{www.gaussoids.de}.
Reviewer: Gema Maria Diaz Toca (Murcia)Essential bases and toric degenerations arising from birational sequences.https://www.zbmath.org/1453.170052021-02-27T13:50:00+00:00"Fang, Xin"https://www.zbmath.org/authors/?q=ai:fang.xin"Fourier, Ghislain"https://www.zbmath.org/authors/?q=ai:fourier.ghislain"Littelmann, Peter"https://www.zbmath.org/authors/?q=ai:littelmann.peterSummary: We present a new approach to construct \(T\)-equivariant flat toric degenerations of flag varieties and spherical varieties, combining ideas coming from the theory of Newton-Okounkov bodies with ideas originally stemming from PBW-filtrations. For each pair \((S, >)\) consisting of a birational sequence and a monomial order, we attach to the affine variety \(G / / U\) a monoid \(\Gamma = \Gamma(S, >)\). As a side effect we get a vector space basis \(\mathbb{B}_{\Gamma}\) of \(\mathbb{C} [G / / U]\), the elements being indexed by \(\Gamma\). The basis \(\mathbb{B}_{\Gamma}\) has multiplicative properties very similar to those of the dual canonical basis. This makes it possible to transfer the methods of \textit{V. Alexeev} and \textit{M. Brion} [Sel. Math., New Ser. 10, No. 4, 453--478 (2004; Zbl 1078.14075)] to this more general setting, once one knows that the monoid \(\Gamma\) is finitely generated and saturated.Leibniz algebras associated with representations of Euclidean Lie algebra.https://www.zbmath.org/1453.170022021-02-27T13:50:00+00:00"Adashev, J. Q."https://www.zbmath.org/authors/?q=ai:adashev.jobir-q"Omirov, B. A."https://www.zbmath.org/authors/?q=ai:omirov.bakhrom-a"Uguz, S."https://www.zbmath.org/authors/?q=ai:uguz.selmanGiven a Lie algebra \((G, [-,-])\) and a right \(G\)-module \(M\), then \(G \oplus M\) can be endowed with a structure of Leibniz algebra by means of the bracket \((-,-)\) defined by \[(g_1,g_2)=[g_1,g_2],~(g,m)=0; ~(m,g) = m^g; ~[m_1,m_2]=0;\]
for all \( ~g, g_1, g_2 \in G;~ m, m_1, m_2 \in M\).
When \(\mathfrak{g}\) is a Leibniz algebra, then the above construction can be reproduced with \(G = \mathfrak{g}_{\mathrm{Lie}}=\mathfrak{g}/\mathfrak{g}^{\mathrm{ann}}\) and \(M=\mathfrak{g}^{\mathrm{ann}}=\langle\{[\mathrm{x},\mathrm{x}]:\mathrm{x} \in \mathfrak{g} \} \rangle\).
The paper is devoted to study the structure of the above construction for Leibniz algebras \(\mathfrak{g}\) such that \(\mathfrak{g}_{\mathrm{Lie}}\) is the Euclidean Lie algebra \(\mathfrak{e}(n)\) and some of its modules, in particular, those that arise from the matrix realization of \(\mathfrak{e}(n), n \geq 3\).
Moreover, the construction \(\mathfrak{D}_{\mathrm{k}}\oplus I\), where \(I\) is the Fock module over the diamond Lie algebra \(\mathfrak{D}_{\mathrm{k}}\), is analyzed.
Reviewer: José Manuel Casas Mirás (Pontevedra)Centrally essential rings.https://www.zbmath.org/1453.160372021-02-27T13:50:00+00:00"Tuganbaev, A. A."https://www.zbmath.org/authors/?q=ai:tuganbaev.askar-aSummary: This paper is a review of recent results on centrally essential rings.Calculations on Lie algebra of the group of affine symplectomorphisms.https://www.zbmath.org/1453.170132021-02-27T13:50:00+00:00"Altawallbeh, Zuhier"https://www.zbmath.org/authors/?q=ai:altawallbeh.zuhierThis paper analyses a technical question concerning the realization of the affine symplectic Lie algebra, which is mapped from the Leibniz homology space to the Lie algebra homology space. It turns out that the image is of relevance in the context of the Hochschild cohomology, as well as in applications to the topology of strings.
Reviewer: Rutwig Campoamor-Stursberg (Madrid)A new symmetry-based method for constructing nonlocally related PDE systems from admitted multi-parameter groups.https://www.zbmath.org/1453.350102021-02-27T13:50:00+00:00"Bluman, George W."https://www.zbmath.org/authors/?q=ai:bluman.george-w"de la Rosa, Rafael"https://www.zbmath.org/authors/?q=ai:dela-rosa.rafael-s"Bruzón, María Santos"https://www.zbmath.org/authors/?q=ai:bruzon.maria-santos"Gandarias, María Luz"https://www.zbmath.org/authors/?q=ai:gandarias.maria-luzSummary: Nonlocally related partial differential equation (PDE) systems can play an important role in the analysis of a given PDE system. In this paper, a new systematic method for obtaining nonlocally related PDE systems is developed. In particular, it is shown that if a PDE system admits \(q\) point symmetries whose infinitesimal generators form a \(q\)-dimensional solvable Lie algebra, then, for each resulting \(q\)-dimensional solvable algebra chain, one can obtain systematically \(q\) nonlocally related PDE systems. Such nonlocally related systems are obtained for a general class of nonlinear reaction-diffusion equations admitting two- to four-dimensional solvable algebras.
{\copyright 2020 American Institute of Physics}The duality of \(\mathfrak{gl}_{m | n}\) and \(\mathfrak{gl}_k\) Gaudin models.https://www.zbmath.org/1453.820082021-02-27T13:50:00+00:00"Huang, Chenliang"https://www.zbmath.org/authors/?q=ai:huang.chenliang"Mukhin, Evgeny"https://www.zbmath.org/authors/?q=ai:mukhin.evgenySummary: We establish a duality of the non-periodic Gaudin model associated with superalgebra \(\mathfrak{gl}_{m | n}\) and the non-periodic Gaudin model associated with algebra \(\mathfrak{gl}_k\). The Hamiltonians of the Gaudin models are given by expansions of a Berezinian of an \((m + n) \times(m + n)\) matrix in the case of \(\mathfrak{gl}_{m | n}\) and of a column determinant of a \(k \times k\) matrix in the case of \(\mathfrak{gl}_k\). We obtain our results by proving Capelli type identities for both cases and comparing the results.Symplectic Lie algebras with degenerate center.https://www.zbmath.org/1453.170072021-02-27T13:50:00+00:00"Fischer, Mathias"https://www.zbmath.org/authors/?q=ai:fischer.mathiasThis paper is devoted to the analysis of Lie algebras endowed with a symplectic 2-form and possessing a degenerate center. It is proved that Lie algebras of this type are determined by a quadratic extension of symplectic Lie algebras. This allows a description of equivalence classes of algebras in terms of quadratic cohomology sets. These tools are used to propose a generic scheme to classify symplectic algebras with degenerate center, as well as nilpotent symplectic algebras. In particular, amendments to previously published lists of the latter type are made, from which a complete list in the six dimensional case is deduced.
Reviewer: Rutwig Campoamor-Stursberg (Madrid)Building vertex algebras from parts.https://www.zbmath.org/1453.170152021-02-27T13:50:00+00:00"Carnahan, Scott"https://www.zbmath.org/authors/?q=ai:carnahan.scottSummary: Given a collection of modules of a vertex algebra parametrized by an abelian group, together with one dimensional spaces of composable intertwining operators, we assign a canonical element of the cohomology of an Eilenberg-Mac Lane space. This element describes the obstruction to locality, as the vanishing of this element is equivalent to the existence of a vertex algebra structure with multiplication given by our intertwining operators, and given existence, the structure is unique up to isomorphism. The homological obstruction reduces to an ``evenness'' problem that naturally vanishes for 2-divisible groups, so simple currents organized into odd order abelian groups always produce vertex algebras. Furthermore, in cases most relevant to conformal field theory (i.e., when we have well-behaved contragradients and tensor products), we obtain our spaces of intertwining operators naturally, and the evenness obstruction reduces to the question of whether the contragradient bilinear form on certain order two currents is symmetric or skew-symmetric. We show that if we are given a simple regular VOA and integral-weight modules parametrized by a group of even units in the fusion ring, then the direct sum admits the structure of a simple regular VOA, called the simple current extension, and this structure is unique up to isomorphism.Derived representation theory of Lie algebras and stable homotopy categorification of \(sl_{k}\).https://www.zbmath.org/1453.570132021-02-27T13:50:00+00:00"Hu, Po"https://www.zbmath.org/authors/?q=ai:hu.po"Kriz, Igor"https://www.zbmath.org/authors/?q=ai:kriz.igor"Somberg, Petr"https://www.zbmath.org/authors/?q=ai:somberg.petrKhovanov homology [\textit{M. Khovanov}, Duke Math. J. 101, No. 3, 359--426 (2000; Zbl 0960.57005)] is now a well established categorification of the Jones polynomial which can be defined with fairly elementary means. But it can also be viewed as a categorification of representations of \(sl_2\). This less elementary point of view has led to the generalization of \(sl_k\)-Khovanov homology.
A different kind of generalization was the introduction of a stable homotopy type by \textit{R. Lipshitz} and \textit{S. Sarkar} [J. Am. Math. Soc. 27, No. 4, 983--1042 (2014; Zbl 1345.57014)] whose homology reduces to Khovanov homology. The paper here is now a step in obtaining a stable homotopy type for \(sl_k\)-Khovanov homology for \(k>2\). The approach is given by a generalization of a construction of \textit{J. Sussan} [``Category \(\mathcal O\) and \(\mathfrak{sl}_k\) link invariants'', Preprint, \url{arXiv:math/0701045}] for \(sl_k\)-homology which requires a representation theory over the sphere spectrum \(S\). This creates various technical difficulties that the authors circumvent by working over a large prime (linearly larger than \(k\)). While this is a notable restriction, one can still expect extra information for complicated links relative to \(sl_k\)-homology.
Reviewer: Dirk Schütz (Durham)Gradings on semisimple algebras.https://www.zbmath.org/1453.170182021-02-27T13:50:00+00:00"Córdova-Martínez, Alejandra S."https://www.zbmath.org/authors/?q=ai:cordova-martinez.alejandra-s"Elduque, Alberto"https://www.zbmath.org/authors/?q=ai:elduque.albertoSummary: The classification of gradings by abelian groups on finite direct sums of simple finite-dimensional nonassociative algebras over an algebraically closed field is reduced, by means of the use of loop algebras, to the corresponding problem for simple algebras. This requires a good definition of (free) products of group-gradings.Functorial PBW theorems for post-Lie algebras.https://www.zbmath.org/1453.170082021-02-27T13:50:00+00:00"Dotsenko, Vladimir"https://www.zbmath.org/authors/?q=ai:dotsenko.vladimir-vIn [``Universal enveloping Lie Rota-Baxter algebras of pre-Lie and post-Lie algebras'', Preprint, \url{arXiv:1708.06747}; ``Poincaré-Birkhoff-Witt theorem for pre-Lie and post-Lie algebras'', Preprint, \url{arXiv:1903.09260}] \textit{V. Gubarev} has proved appropriate PBW type theorems for post-Lie algebras. In the present paper the author relying on the categorical framework gives an alternative approach for PBW theorems in the considered case.
Reviewer: Dmitry Artamonov (Moskva)Universal two-parameter even spin \(\mathcal{W}_\infty\)-algebra.https://www.zbmath.org/1453.170162021-02-27T13:50:00+00:00"Kanade, Shashank"https://www.zbmath.org/authors/?q=ai:kanade.shashank"Linshaw, Andrew R."https://www.zbmath.org/authors/?q=ai:linshaw.andrew-rSummary: We construct the unique two-parameter vertex algebra which is freely generated of type \(\mathcal{W}(2, 4, 6, \ldots)\), and generated by the weights 2 and 4 fields. Subject to some mild constraints, all vertex algebras of type \(\mathcal{W}(2, 4, \ldots, 2 N)\) for some \(N\), can be obtained as quotients of this universal algebra. This includes the \(B\) and \(C\) type principal \(\mathcal{W}\)-algebras, the \(\mathbb{Z}_2\)-orbifolds of the \(D\) type principal \(\mathcal{W}\)-algebras, and many others which arise as cosets of affine vertex algebras inside larger structures. As an application, we classify all coincidences among the simple quotients of the \(B\) and \(C\) type principal \(\mathcal{W}\)-algebras, as well as the \(\mathbb{Z}_2\)-orbifolds of the \(D\) type principal \(\mathcal{W}\)-algebras. Finally, we use our classification to give new examples of principal \(\mathcal{W}\)-algebras of \(B, C\), and \(D\) types, which are lisse and rational.Erratum to: ``A framework of Rogers-Ramanujan identities and their arithmetic properties''.https://www.zbmath.org/1453.111312021-02-27T13:50:00+00:00"Griffin, Michael J."https://www.zbmath.org/authors/?q=ai:griffin.michael-j"Ono, Ken"https://www.zbmath.org/authors/?q=ai:ono.ken"Warnaar, S. Ole"https://www.zbmath.org/authors/?q=ai:warnaar.s-oleErratum to the authors' paper [ibid. 165, No. 8, 1475--1527 (2016; Zbl 1405.11140)].Centers of generalized reflection equation algebras.https://www.zbmath.org/1453.810392021-02-27T13:50:00+00:00"Gurevich, D. I."https://www.zbmath.org/authors/?q=ai:gurevich.dmitrii-i"Saponov, P. A."https://www.zbmath.org/authors/?q=ai:saponov.pavel-aSummary: As is known, in the reflection equation (RE) algebra associated with an involutive or Hecke \(R\)-matrix, the elements \(\operatorname{Tr}_RL^k\) (called quantum power sums) are central. Here, \(L\) is the generating matrix of this algebra, and \(\operatorname{Tr}_R\) is the operation of taking the \(R\)-trace associated with a given \(R\)-matrix. We consider the problem of whether this is true in certain RE-like algebras depending on a spectral parameter. We mainly study algebras similar to those introduced by Reshetikhin and Semenov-Tian-Shansky (we call them algebras of RS type). These algebras are defined using some current \(R\)-matrices (i.e., depending on parameters) arising from involutive and Hecke \(R\)-matrices by so-called Baxterization. In algebras of RS type. we define quantum power sums and show that the lowest quantum power sum is central iff the value of the ``charge'' \(c\) in its definition takes a critical value. This critical value depends on the bi-rank \((m|n)\) of the initial \(R\)-matrix. Moreover, if the bi-rank is equal to \((m|m)\) and the charge \(c\) has a critical value, then all quantum power sums are central.Combinatorial Auslander-Reiten quivers and reduced expressions.https://www.zbmath.org/1453.160142021-02-27T13:50:00+00:00"Oh, Se-Jin"https://www.zbmath.org/authors/?q=ai:oh.se-jin"Suh, Uhi Rinn"https://www.zbmath.org/authors/?q=ai:suh.uhi-rinnSummary: In this paper, we introduce the notion of combinatorial Auslander-Reiten (AR) quivers for commutation classes \([\widetilde{w}]\) of \(w\) in a finite Weyl group. This combinatorial object is the Hasse diagram of the convex partial order \(\prec_{[\widetilde{w}]}\) on the subset \(\Phi(w)\) of positive roots. By analyzing properties of the combinatorial AR-quivers with labelings and reflection functors, we can apply their properties to the representation theory of KLR algebras and dual PBW-basis associated to any commutation class \([\widetilde{w}_0]\) of the longest element \(w_0\) of any finite type.Statement of retraction: Eight-dimensional octonion-like but associative normed division algebra.https://www.zbmath.org/1453.160162021-02-27T13:50:00+00:00The editors and publisher have retracted the article [\textit{J. Christian}, Commun. Algebra 49, No. 2, 905--914 (2021; Zbl 1453.16017)]. The error is obvious both from the title and abstract, which claims to construct an 8-dimensional normed division algebra over \(\mathbb R\), which doesn't exist due to Hurwitz's theorem [\textit{A. Hurwitz}, Math. Ann. 88, 1--25 (1922; JFM 48.1164.03)].Deducing the symmetry of the standard model from the automorphism and structure groups of the exceptional Jordan algebra.https://www.zbmath.org/1453.810362021-02-27T13:50:00+00:00"Todorov, Ivan"https://www.zbmath.org/authors/?q=ai:todorov.ivan-t"Dubois-Violette, Michel"https://www.zbmath.org/authors/?q=ai:dubois-violette.michelOn the tensor structure of modules for compact orbifold vertex operator algebras.https://www.zbmath.org/1453.170172021-02-27T13:50:00+00:00"McRae, Robert"https://www.zbmath.org/authors/?q=ai:mcrae.robertSuppose \(V\) is a vertex operator algebra and \(G\) is a compact Lie group acting faithfully and continuously (and by vertex operator algebra automorphisms) on \(V\); write \(V^G\) for the fixed subalgebra. \textit{C. Dong} et al. provided in [Int. Math. Res. Not. 1996, No. 18, 913--921 (1996; Zbl 0873.17028)] a Schur-Weyl type duality statement: \(V\) is semisimple as a \((V^G \times G)\)-module, and decomposes as \(V = \bigoplus_I V_I \otimes I\), where \(I\) ranges over the irreducible finite-dimensional \(G\)-modules, and \(V_I\) are nonzero distinct irreducible \(V^G\) modules. In particular, this theorem provides an identification of linear semisimple categories between \(\mathrm{Rep}(G)\) and the subcategory \(\mathcal{C}_V \subset \mathrm{Rep}(V^G)\) consisting of direct sums of the \(V_I\)'s.
The main result of the present paper improves this to an identification of braided monoidal categories. (The theorem is proved for arbitrary abelian intertwining algebras, which are a mild generalization of vertex operator algebras.) The only assumption needed is \(V^G\) indeed has a category of modules, containing \(\mathcal{C}_V\), with a braided tensor structure. This assumption is fairly deep in general: except in very special cases (unitary, strongly rational, etc.), the construction of braided tensor category structures on modules for vertex operator algebras requires the sophisticated theory of logarithmic vertex tensor categories developed by \textit{Y.-Z. Huang} and \textit{J. Lepowsky} [J. Phys. A, Math. Theor. 46, No. 49, Article ID 494009, 21 p. (2013; Zbl 1280.81125)], which in turn depends on subtle convergence properties of 4-point functions. Nevertheless, the assumption is known to hold for a vast assortment of examples. The proof uses a nice mixture of vertex-algebraic and tensor-categorical techniques.
Reviewer: Theo Johnson-Freyd (Waterloo)Relation between categories of representations of the super-Yangian of a special linear Lie superalgebra and quantum loop superalgebra.https://www.zbmath.org/1453.810402021-02-27T13:50:00+00:00"Stukopin, V. A."https://www.zbmath.org/authors/?q=ai:stukopin.vladimir-alekseevichSummary: Using the approach developed by Gautam and Toledano Laredo, we introduce analogues of the category \(\mathfrak{O}\) for representations of the Yangian \(Y_{\hbar}(A(m,n))\) of a special linear Lie superalgebra and the quantum loop superalgebra \(U_q(LA(m,n))\). We investigate the relation between them and conjecture that these categories are equivalent.Monomial bases and pre-Lie structure for free Lie algebras.https://www.zbmath.org/1453.170192021-02-27T13:50:00+00:00"Al-Kaabi, Mahdi J. Hasan"https://www.zbmath.org/authors/?q=ai:al-kaabi.mahdi-j-hasan"Manchon, Dominique"https://www.zbmath.org/authors/?q=ai:manchon.dominique"Patras, Frédéric"https://www.zbmath.org/authors/?q=ai:patras.fredericA tree is a undirected connected finite graph, without cycles. A rooted tree is defined as a tree with one designated vertex called the root. The other remaining vertices are partitioned into \(k \geq 0\) disjoint subsets such that each of them in turn represents a rooted tree, and a subtree of the whole tree. A rooted tree is said to be planar, if it is endowed with an embedding in the plane. Otherwise, its called a non-planar rooted tree. Let \(E\) be a non-empty set. An \(E\)-decorated rooted tree is a pair \((t, d)\) of a rooted tree \(t\) together with a map \(d: V (t) \rightarrow E\), which decorates each vertex \(v\) of \(t\) by an element \(a\) of \(E\), i.e. \(d(v) = a\), where \(V (t)\) is the set of all vertices of \(t\).
Recall that a pre-Lie algebra is a vector space \(\mathcal{A}\) over a field \(K\), together with a bilinear operation \(\triangleright\) that satisfies the identity: \((x \triangleright y) \triangleright z - x \triangleright (y \triangleright z) = (y \triangleright x) \triangleright z - y \triangleright (x \triangleright z)\). The authors construct a pre-Lie structure on the free Lie algebra \(\mathcal{L}(E)\) generated by a set \(E\), giving an explicit presentation of \(\mathcal{L}(E)\) as the quotient of the free pre-Lie algebra \(\mathcal{T}^E\), generated by the (non-planar) \(E\)-decorated rooted trees, by some ideal \(I\). They describe a monomial bases in tree version for the free Lie (respectively pre-Lie) algebras using the procedures of Gr\(\ddot{o}\)bner bases, comparing with the one (i.e., the monomial basis) obtained for the free pre-Lie algebra in the work [\textit{M. J. H. Al-Kaabi}, Sémin. Lothar. Comb. 71, B71b, 19 p. (2014; Zbl 1303.17023)].
Reviewer: Leonid M. Martynov (Omsk)The algebraic approach to duality: an introduction.https://www.zbmath.org/1453.820592021-02-27T13:50:00+00:00"Sturm, Anja"https://www.zbmath.org/authors/?q=ai:sturm.anja-k"Swart, Jan M."https://www.zbmath.org/authors/?q=ai:swart.jan-m"Völlering, Florian"https://www.zbmath.org/authors/?q=ai:vollering.florianHidden symmetry algebra and construction of quadratic algebras of superintegrable systems.https://www.zbmath.org/1453.810372021-02-27T13:50:00+00:00"Campoamor-Stursberg, Rutwig"https://www.zbmath.org/authors/?q=ai:campoamor-stursberg.rutwig"Marquette, Ian"https://www.zbmath.org/authors/?q=ai:marquette.ianSummary: The notion of hidden symmetry algebra used in the context of exactly solvable systems (typically a non semisimple Lie algebra) is re-examined from the purely algebraic way, analyzing subspaces of commuting polynomials that generate finite-dimensional quadratic algebras. By construction, these algebras do not depend on the choice of realizations by vector fields of the underlying Lie algebra, allowing to propose a new approach to analyze polynomial algebras as those subspaces in an enveloping algebra that commute with a given algebraic Hamiltonian. These polynomial algebras play an important role in context of superintegrability, but are still poorly understood from an algebraic point of view. Among the main results, we present finitely generated quadratic algebras of dimensions 4, 5 and 6, as well as cubic algebras of dimensions 3 and 5, and various abelian algebras, all of dimension 3. Based on the observation how superintegrability is associated with exact solvability, we propose a procedure that connects the underlying Lie algebra with algebraic integrals of motion. As the integrals constructed in such way are now independent on the realization, alternative choices of realizations can provide new explicit models with the same symmetry algebra. In this paper, we consider examples of such equivalent Hamiltonians in terms of differential operators for the three cases and connected to the underlying Lie algebra \(\mathfrak{gl}(2,\mathbb{R})\ltimes \mathbb{R}^2\oplus T_1\) as well as to the maximal parabolic subalgebra of \(\mathfrak{gl}(3,\mathbb{R})\). We also point out differences between the enveloping algebra of Lie algebras and the enveloping algebra of the related differential operators realization.Some properties of the \(c\)-nilpotent multiplier of a pair of Lie algebras.https://www.zbmath.org/1453.170062021-02-27T13:50:00+00:00"Arabyani, Homayoon"https://www.zbmath.org/authors/?q=ai:arabyani.homayoonThe notion of the Schur multiplier of a group originated from works of \textit{I. Schur} [J. Reine Angew. Math. 127, 20--50 (1904; JFM 35.0155.01)]. Since then, it has appeared in many investigations related to group theory, Lie (super)algebras and Leibniz algebras. Nearly a century after Schur, \textit{G. Ellis} [Appl. Categ. Struct. 6, 355--371 (1998; Zbl 0948.20026)] extended the Schur multiplier to a pair of groups. This study relies on the concept of relative central extension discussed by L. Loday in 1978.
The Schur multiplier of a Lie algebra \(L\), denoted by \(\mathcal{M}(L)\), which has been introduced by \textit{P. Batten} et al. [Commun. Algebra 24, 4319--4330 (1996; Zbl 0893.17008)], is defined to be the factor Lie algebra \[\mathcal{M}(L)=\frac{R\cap F^2}{[R,F]}\] on a free presentation \(0\rightarrow R\rightarrow F\rightarrow L\rightarrow 0\) of \(L\), where \(F^2\) is the derived subalgebra of \(F\). \textit{F. Saeedi} et al. [J. Lie Theory 21, 491--498 (2011; Zbl 1216.17010)] generalized it to the Schur multiplier of a pair \((N,L)\) of Lie algebras as \(\mathcal{M}(N,L)=(R\cap [S,F])/[R,F]\), where \(N\cong S/R\) such that \(N\) and \(S\) are ideals in \(L\) and \(F\), respectively. Another generalization of this notion is due to \textit{A. R. Salemkar} et al. [J. Algebra 322, 1575--1585 (2009; Zbl 1229.17010)], which is the \(c\)-nilpotent multiplier \(\mathcal{M}^{(c)}(L)=(R\cap F^{c+1})/[R,_cF]\) of a Lie algebra \(L\), where \(F^{c+1}\) is the \((c+1)\)st term of the lower central series of \(F\) and \([R,_cF]=[R,F,\ldots,F]\) consisting \(c\) copies of \(F\).
Finally in [the reviewer and the author, Commun. Algebra 45, 4429--4434 (2017; Zbl 1427.17021)], the \(c\)-nilpotent multiplier of a pair \((N,L)\) of Lie algebras is similarly defined as
\[\mathcal{M}^{(c)}(N,L)=\frac{R\cap [S,_cF]}{[R,_cF]}.\]
In the paper under review, it is given some inequalities on the dimension of the \(c\)-nilpotent multiplier of a pair of Lie algebras and their quotients by an ideal. It is also provided a necessary and sufficient condition under which \(\mathcal{M}^{(c)}(N,L)\) can be embedded into \(\mathcal{M}^{(c)}(N/K,L/K)\).
Reviewer: Hesam Safa (Bojnord)