Recent zbMATH articles in MSC 17https://zbmath.org/atom/cc/172021-01-08T12:24:00+00:00WerkzeugRestricted Hom-Lie superalgebras.https://zbmath.org/1449.170082021-01-08T12:24:00+00:00"Shaqaqha, Shadi"https://zbmath.org/authors/?q=ai:shaqaqha.shadiSummary: The aim of this paper is to introduce the notion of restricted Hom-Lie superalgebras. This class of algebras is a generalization of both restricted Hom-Lie algebras and restricted Lie superalgebras. In this paper, we present a way to obtain restricted Hom-Lie superalgebras from the classical restricted Lie superalgebras along with algebra endomorphisms. Homomorphisms relations between restricted Hom-Lie superalgebras are defined and studied. Also, we obtain some properties of
\(p\)-maps and restrictable Hom-Lie superalgebras.Quasi-automorphism on general linear Lie algebras.https://zbmath.org/1449.170272021-01-08T12:24:00+00:00"Liao, Yang"https://zbmath.org/authors/?q=ai:liao.yang"Chen, Qinghua"https://zbmath.org/authors/?q=ai:chen.qinghuaSummary: Let \(\boldsymbol{F}\) be an algebraically closed field, \(n \ge 3\), let \(gl (n, \boldsymbol{F})\) be the general linear Lie algebra of all \(n \times n\) matrices on \(\boldsymbol{F}\), and let \(\varphi\) be a quasi-automorphism of \(gl (n, \boldsymbol{F})\). It is shown that a linear map over \(gl (n, \boldsymbol{F})\) is a quasi-automorphism if and only if there exist a non-zero element \(r\) in \(\boldsymbol{F}\), an invertible matrix \(\boldsymbol{T}\) in \(gl (n, \boldsymbol{F})\), and a linear function \(h:gl (n, \boldsymbol{F}) \to \boldsymbol{F}\) satisfying that for any \(\boldsymbol{A}\) in \(gl (n, \boldsymbol{F})\), \(\varphi (\boldsymbol{A}) = r\boldsymbol{TAT}^{-1} + h (\boldsymbol{A})\boldsymbol{I}\) or \(\varphi (\boldsymbol{A}) = r\boldsymbol{TA}^t\boldsymbol{T}^{-1} + h (\boldsymbol{A})\boldsymbol{I}\).On the nilpotent Leibniz-Poisson algebras.https://zbmath.org/1449.170042021-01-08T12:24:00+00:00"Ratseev, Sergeĭ Mikhaĭlovich"https://zbmath.org/authors/?q=ai:ratseev.sergej-mikhaijlovich"Cherevatenko, Ol'ga Ivanovna"https://zbmath.org/authors/?q=ai:cherevatenko.olga-ivanovnaSummary: In this article Leibniz and Leibniz-Poisson algebras in terms of correctness of different identities are investigated. We also examine varieties of these algebras. Let \(K\) be a base field of characteristics zero. It is well known that in this case all information about varieties of linear algebras \(V\) contains in its polylinear components \(P_n(V), n \in \mathbb{N} \), where \(P_n(V)\) is a linear span of polylinear words of \(n\) different letters in a free algebra \(K(X,V)\). In this article we give algebra constructions that generate class of nilpotent varieties of Leibniz algebras and also algebra constructions that generate class of nilpotent by Leibniz varieties of Leibniz-Poisson algebras with the identity \(\{ x_1, x_2 \} \cdot \{x_3, x_4 \} = 0\).Strongly ad-nilpotent elements of the Lie algebra of upper triangular matrices.https://zbmath.org/1449.170202021-01-08T12:24:00+00:00"Jin, Mengdan"https://zbmath.org/authors/?q=ai:jin.mengdan"Hu, Zhiguang"https://zbmath.org/authors/?q=ai:hu.zhiguangSummary: The Lie algebra consisting of upper triangular matrices over the field \(F\) of characteristics 0 is considered. The set of strongly ad-nilpotent elements of the Lie algebra is obtained by using derivation series and matrix eigenvalues, and their orbits under the automorphisms are also given.Description of regular Bihom-Lie algebras by coboundary operators.https://zbmath.org/1449.170212021-01-08T12:24:00+00:00"Xiong, Zhen"https://zbmath.org/authors/?q=ai:xiong.zhenSummary: We study trivial representation of regular Bihom-Lie algebra \( (L, [\cdot, \cdot], \alpha, \beta)\), and give coboundary operator \(d\) with respect to trivial representation. Then, we have some properties of coboundary operator \(d\). Lastly, we draw a conclusion that there is an one-to-one correspondence between regular Bihom-Lie algebra \( (L, [\cdot, \cdot], \alpha, \beta)\) and coboundary operator \(d\) on \( \wedge {L^*}\).A kind of infinite-dimensional Novikov algebras and its sub-adjacent Lie algebras.https://zbmath.org/1449.170422021-01-08T12:24:00+00:00"Zhou, Xin"https://zbmath.org/authors/?q=ai:zhou.xin|zhou.xin.2|zhou.xin.3|zhou.xin.1|zhou.xin.4|zhou.xin.5Summary: In this paper, we construct a kind of infinite-dimensional Novikov algebras, which is specifically realized by exponential functions. Finally, we discuss the structure and property of its corresponding sub-adjacent Lie algebras.Atiyah classes of three-dimensional Lie bialgebras over \(\mathbb Z_3\).https://zbmath.org/1449.170392021-01-08T12:24:00+00:00"Shen, Dandan"https://zbmath.org/authors/?q=ai:shen.dandanSummary: Based on the theory of Lie algebra cohomology and the definition of the Atiyah class of a Lie bialgebra, Atiyah classes of all three-dimensional Lie bialgebras over \(\mathbb Z_3\) are calculated.Local derivations and 2-local derivations on the Lie algebra of antisymmetric matrices over a commutative ring.https://zbmath.org/1449.170282021-01-08T12:24:00+00:00"Wang, Di"https://zbmath.org/authors/?q=ai:wang.di"Wang, Yin"https://zbmath.org/authors/?q=ai:wang.yinSummary: Let \(R\) be a 2-torsion free commutative ring with identity 1 and \({L_n} (R)\) a Lie algebra consisting of all \(n \times n\) antisymmetric matrices over \(R\). The aim of this paper is to study the characters of the local derivations and 2-local derivations of \({L_n} (R)\). By using that \({L_n} (R)\) is a complete Lie algebra and the skill of matrix computation, it is proved that every local derivation and every 2-local derivation of \({L_n} (R)\) are derivations, which extends the main result of derivations of \({L_n} (R)\).Derivations of generalized extended Schrödinger-Virasoro algebras.https://zbmath.org/1449.170292021-01-08T12:24:00+00:00"Wang, Song"https://zbmath.org/authors/?q=ai:wang.song.2|wang.song|wang.song.1|wang.song.3"Wang, Xiaoming"https://zbmath.org/authors/?q=ai:wang.xiaoming.2Summary: Let \(\mathbb{F}\) be a field of characteristic 0, \(\Gamma\) be an additive subgroup of \(\mathbb{F}\), \(s \in \mathbb{F}\) satisfying \(s \notin \Gamma\) and \(2s \in \Gamma\). We define a class of infinite-dimensional Lie algebras which are called generalized extended Schrödinger-Virasoro algebras \(\mathcal{W}[\Gamma, s]\). In this paper, derivation algebras of \(\mathcal{W}[\Gamma, s]\) are completely determined.Second cohomology groups of the extended loop Schödinger-Virasoro algebras.https://zbmath.org/1449.170362021-01-08T12:24:00+00:00"Wang, Song"https://zbmath.org/authors/?q=ai:wang.song|wang.song.3|wang.song.1|wang.song.2"Wang, Xiaoming"https://zbmath.org/authors/?q=ai:wang.xiaoming.2Summary: In this paper, we study the extended loop Schrödinger-Virasoro algebras and give the second cohomology groups of the extended loop Schrödinger-Virasoro algebras. Moreover, we obtain the universal central extensions of the extended loop Schrödinger-Virasoro algebras.A characterization of the twisted Heisenberg-Virasoro vertex operator algebra.https://zbmath.org/1449.170432021-01-08T12:24:00+00:00"Cheng, Junfang"https://zbmath.org/authors/?q=ai:cheng.junfang"Chu, Yanjun"https://zbmath.org/authors/?q=ai:chu.yanjunSummary: The twisted Heisenberg-Virasoro algebra is the universal central extension of the Lie algebra of differential operators on a circle of order at most one. In this paper, we first study the variety of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra, which is a finite set consisting of two nontrivial elements. Based on this property, we also show that the twisted Heisenberg-Virasoro vertex operator algebra is a tensor product of two vertex operator algebras. Moreover, associating to properties of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra, we characterized twisted Heisenberg-Virasoro vertex operator algebras. This will be used to understand the classification problems of vertex operator algebras whose varieties of semi-conformal vectors are finite sets.Lie triple derivations of filiform Lie algebra \({R_n}\).https://zbmath.org/1449.170332021-01-08T12:24:00+00:00"Zhu, Kaixiao"https://zbmath.org/authors/?q=ai:zhu.kaixiao"Wu, Mingzhong"https://zbmath.org/authors/?q=ai:wu.mingzhongSummary: This paper studied the triple derivatives of the filiform Lie algebras \({R_n}\). Using the definition of the triple derivatives, we obtained the matrix forms of the triple derivations by calculating the effect of the linear transformation on a particular basis, and found that the triple derivatives algebra is a \( (2n-1)\)-dimensional solvable Lie algebra.The semidirect sum of Lie algebras and its applications to C-KdV hierarchy.https://zbmath.org/1449.370482021-01-08T12:24:00+00:00"Dong, Xia"https://zbmath.org/authors/?q=ai:dong.xia"Xia, Tiecheng"https://zbmath.org/authors/?q=ai:xia.tie-cheng"Li, Desheng"https://zbmath.org/authors/?q=ai:li.deshengSummary: By use of the loop algebra \(\overset\simeq G\), integrable coupling of C-KdV hierarchy and its bi-Hamiltonian structures are obtained by \textit{G. Tu} scheme [J. Math. Phys. 30, No. 2, 330--338 (1989; Zbl 0678.70015)]
and the quadratic-form identity. The method can be used to produce the integrable coupling and its Hamiltonian structures to the other integrable systems.Property of a new infinite dimensional Block type Lie algebra with parameter \(q\).https://zbmath.org/1449.170412021-01-08T12:24:00+00:00"Yu, Demin"https://zbmath.org/authors/?q=ai:yu.de-ming"Fang, Chunhua"https://zbmath.org/authors/?q=ai:fang.chunhuaSummary: In this paper, Block type Lie algebra with parameter \(q\) is constructed. This kind of Lie algebras is a generalization of Virasoro-like Lie algebras. Virasoro-like Lie algebra is an important infinite dimensional Lie algebra. Subalgebra, isomorphisms and homomorphism of the infinite dimensional Lie algebra are studied.Fock representation of the type \({\mathrm{D}}_l\) Lie algebras.https://zbmath.org/1449.170232021-01-08T12:24:00+00:00"Zeng, Ziting"https://zbmath.org/authors/?q=ai:zeng.ziting"Zhou, Yingmei"https://zbmath.org/authors/?q=ai:zhou.yingmeiSummary: There have been some Wakimoto free field realizations of Lie algebra, Lie super algebra, and even type A quantum group, but little is known about other types. In this paper, root structure of type D Lie algebra is analyzed, a Fock representation of type \({\mathrm{D}}_l\) Lie algebras is constructed, the condition of irreducibility is given. Further, the realization for the untwisted affine Kac-Moody Lie algebra was done, the irreducibility is proved. Different from semi-simple algebra, most representations of affine Lie algebra are found irreducible.A matrix representation of outer derivations from \(\mathfrak{gl}_{0|2}\) to the generalized Witt Lie superalgebra.https://zbmath.org/1449.170242021-01-08T12:24:00+00:00"Zheng, Keli"https://zbmath.org/authors/?q=ai:zheng.keliSummary: Let \(\mathfrak{gl}_{0|2}\) be a subalgebra of the general linear Lie superalgebra. In this paper, outer derivations from \(\mathfrak{gl}_{0|2}\) to the generalized Witt Lie superalgebra are completely determined by matrices.Automorphisms preserving the diagonal matrices on general linear Lie algebras.https://zbmath.org/1449.170342021-01-08T12:24:00+00:00"Zhu, Xiangyi"https://zbmath.org/authors/?q=ai:zhu.xiangyi"Chen, Zhengxin"https://zbmath.org/authors/?q=ai:chen.zhengxinSummary: Let \(\boldsymbol{F}\) be a field, let \(\boldsymbol{L}\) be the general linear Lie algebra of all \(n \times n\) matrices on \(\boldsymbol{F}\). Let \(\mathcal{D}\) be a subalgebra consisting of all diagonal matrices, and let \(\varphi\) be an automorphism of \(\boldsymbol{L}\). It is shown that \(\varphi (\mathcal{D}) = \mathcal{D}\) if and only if \(\varphi\) is a composition of a graph automorphism, a central automorphism, a diagonal automorphism, and a special inner automorphism.Topological loops with six-dimensional solvable multiplication groups having five-dimensional nilradical.https://zbmath.org/1449.220032021-01-08T12:24:00+00:00"Figula, Ágota"https://zbmath.org/authors/?q=ai:figula.agota"Ficzere, Kornélia"https://zbmath.org/authors/?q=ai:ficzere.kornelia"Al-Abayechi, Ameer"https://zbmath.org/authors/?q=ai:al-abayechi.ameerSummary: Using connected transversals we determine the six-dimensional indecomposable solvable Lie groups with five-dimensional nilradical and their subgroups which are the multiplication groups and the inner mapping groups of three-dimensional connected simply connected topological loops. Together with this result we obtain that every six-dimensional indecomposable solvable Lie group which is the multiplication group of a three-dimensional topological loop has one-dimensional centre and two- or three-dimensional commutator subgroup.Second cohomology groups of the generalized map Schrödinger-Virasoro algebras.https://zbmath.org/1449.170352021-01-08T12:24:00+00:00"Wang, Song"https://zbmath.org/authors/?q=ai:wang.song.2|wang.song.1|wang.song.3|wang.song"Wang, Xiaoming"https://zbmath.org/authors/?q=ai:wang.xiaoming.2Summary: In this paper, the second cohomology groups of the generalized map Schrödinger-Virasoro algebras are determined. The universal central extensions of the generalized map Schrödinger-Virasoro algebras are given.The Schur multiplier and stem covers of Leibniz \(n\)-algebras.https://zbmath.org/1449.170072021-01-08T12:24:00+00:00"Casas, José Manuel"https://zbmath.org/authors/?q=ai:casas-miras.jose-manuel"Insua, Manuel Avelino"https://zbmath.org/authors/?q=ai:insua.manuel-avelino"rego, Natália Pacheco"https://zbmath.org/authors/?q=ai:rego.natalia-pachecoLeibniz \(n\)-algebras are the non-skewsymmetric version of \(n\)-Lie algebras. The authors study relations between the Schur multipliers and stem extensions of Leibniz \(n\)-algebras. In particular, they prove that every stem extension of a Leibniz \(n\)-algebra is an epimorphic image of a stem cover and that any two stem covers are isomorphic.
Reviewer: Norbert Knarr (Stuttgart)On the capability and tensor center of Lie algebras.https://zbmath.org/1449.170092021-01-08T12:24:00+00:00"Arabyani, Homayoon"https://zbmath.org/authors/?q=ai:arabyani.homayoonSummary: A Lie algebra \(L\) is called capable, if there exists a Lie algebra \(E\) such that \(L \cong E/Z (E)\). The concept of capability for Lie algebras was introduced by previous researchers. In this paper, the capability of the pair \( ({{L^2}, L})\) is determined, where \({L^2}\) is the derived subalgebra of \(L\). Moreover, we obtain some properties of the tensor center of a pair of Lie algebras.Torus and Cartan-demcomposition of restricted Hom-Lie superalgebras.https://zbmath.org/1449.170132021-01-08T12:24:00+00:00"Guan, Baoling"https://zbmath.org/authors/?q=ai:guan.baoling"Wang, Weihua"https://zbmath.org/authors/?q=ai:wang.weihua.1|wang.weihuaSummary: Some properties of restricted Hom-Lie algebras were extended to restricted Hom-Lie superalgebras. We gave definitions and some properties of semi-simple elements and torus elements of restricted Hom-Lie superalgebras, and definitions and some properties of Hom-Lie Cartan-subsuperalgebras. The necessary and sufficient condition for characterization of Hom-Lie Cartan-subsuperalgebras restricted by Hom-Lie superalgebras could be described by its maximal torus, and some properties of its maximal torus of the restricted Hom-Lie superalgebras were given.The centroid of a Jordan-Lie algebra.https://zbmath.org/1449.170182021-01-08T12:24:00+00:00"Zhou, Jia"https://zbmath.org/authors/?q=ai:zhou.jia"Cao, Yan"https://zbmath.org/authors/?q=ai:cao.yanSummary: Jordan-Lie algebra is a generation of Lie algebra. Centroids play an important role in studying algebraic structures. In this paper, we give definitions of derivation algebras \(Der (L)\), center derivation algebras \(ZDER (L)\) and centroids \(C (L)\) of a Jordan-Lie algebra \(L\). We prove analogs of results from the theory of Lie algebras to Jordan-Lie algebras and develop some properties on centroids of Jordan-Lie algebras. Furthermore, we have \(ZDER (L) = C (L) \cap Der (L)\).Generalized almost-Jordan algebras.https://zbmath.org/1449.170022021-01-08T12:24:00+00:00"Dembega, Abdoulaye"https://zbmath.org/authors/?q=ai:dembega.abdoulaye"Ouattara, Moussa"https://zbmath.org/authors/?q=ai:ouattara.moussaSummary: In this paper we deal with the variety of commutative algebras satisfying the identity \(\beta \{(yx^2)x-(yx\cdot x)x\}+\gamma \{y(x^2x)-(yx\cdot x)x\}=0\), where \(\beta \), \(\gamma\) are scalars. They are called generalized almost-Jordan algebras. We show under some conditions that generalized almost-Jordan algebras contain an idempotent. We revisit the study of these algebras. We show particularly that they contain an associative subalgebra with a unity. Thus, when the algebra is simple, it is an associative field. The special cases \(\beta =0\), \(\gamma =0\), \(\beta +2\gamma =0\) and \(\beta +\gamma =0\) have been studied. In each case we give examples.A complement on representations of Hom-Lie algebras.https://zbmath.org/1449.170222021-01-08T12:24:00+00:00"Xiong, Zhen"https://zbmath.org/authors/?q=ai:xiong.zhenSummary: In this paper, we give a new series of coboundary operators of Hom-Lie algebras, and prove that cohomology groups with respect to coboundary operators are isomorphic. Then, we revisit representations of Hom-Lie algebras, and prove that there is a one-to-one correspondence between Hom-Lie algebraic structure on vector space \(g\) and these coboundary operators on \(\Lambda {g^*} \otimes V\).3-pre-Lie superalgebras.https://zbmath.org/1449.170372021-01-08T12:24:00+00:00"Hu, Mengru"https://zbmath.org/authors/?q=ai:hu.mengru"Wang, Bo"https://zbmath.org/authors/?q=ai:wang.bo.1|wang.bo|wang.bo.2"Zhang, Qingcheng"https://zbmath.org/authors/?q=ai:zhang.qingchengSummary: By giving the concept of 3-pre-Lie superalgebras and the definition of \(\mathcal{O}\)-super-operators, we discussed the basic properties of 3-pre-Lie superalgebras, and constructed the solution of the 3-Lie super classical Yang-Baxter equation by means of the sub-adjacent 3-Lie superalgebras derived from 3-pre-Lie superalgebras.On dimension of \(c\)-nilpotent multiplier of a pair of Lie algebras.https://zbmath.org/1449.170252021-01-08T12:24:00+00:00"Arabyani, H."https://zbmath.org/authors/?q=ai:arabyani.homayoonSummary: The notion of the Schur multiplier of a Lie algebra \(L\) was introduced and generalized to the \(c\)-nilpotent multiplier by previous researchers. Recently, the author introduced the notion of the \(c\)-nilpotent multiplier of a pair of Lie algebras and proved some equalities and inequalities for the dimension of the \(c\)-nilpotent multiplier of a pair of Lie algebras. In this paper, we extend these results to obtain several inequalities for the dimension of the \(c\)-nilpotent multiplier of a pair of Lie algebras.Properties of post Lie superalgebra structure.https://zbmath.org/1449.170142021-01-08T12:24:00+00:00"Wang, Xu"https://zbmath.org/authors/?q=ai:wang.xu.4|wang.xu.1|wang.xu.3|wang.xu.2|wang.xu|wang.xu.5"Wei, Zhu"https://zbmath.org/authors/?q=ai:wei.zhu"Zhang, Qingcheng"https://zbmath.org/authors/?q=ai:zhang.qingchengSummary: Through the definition of post Lie superalgebra, we discussed the basic properties of post Lie superalgebra structure, and gave some existence results of post Lie superalgebra. The LR superalgebra and left symmetric superalgebra could be constructed by post Lie superalgebra, and the related results of commutative post Lie superalgebra structure were given.Some properties of subalgebras of Lie-Rinehart algebras.https://zbmath.org/1449.170382021-01-08T12:24:00+00:00"Wang, Xuebing"https://zbmath.org/authors/?q=ai:wang.xuebing"Niu, Yanjun"https://zbmath.org/authors/?q=ai:niu.yanjun"Chen, Liangyun"https://zbmath.org/authors/?q=ai:chen.liangyunSummary: We develop \(c\)-supplemented subalgebras, \(E\)-algebras and Frattini theory of Lie algebras for Lie-Rinehart algebras, obtain some important properties and give a necessary condition for solvable Lie-Rinehart algebras. Moreover, we obtain a necessary and sufficient condition for \(E\)-Lie-Rinehart algebras and \(c\)-supplemented Lie-Rinehart algebras, respectively.An optimal system of one-dimensional subalgebras for the symmetry algebra of three-dimensional equations of the perfect plasticity.https://zbmath.org/1449.740512021-01-08T12:24:00+00:00"Kovalëv, Vladimir Aleksandrovich"https://zbmath.org/authors/?q=ai:kovalev.vladimir-aleksandrovich"Radaev, Yuriĭ Nikolaevich"https://zbmath.org/authors/?q=ai:radaev.yu-nSummary: The present paper is devoted to a study of a natural 12-dimensional symmetry algebra of the three-dimensional hyperbolic differential equations of the perfect plasticity, obtained by \textit{D. D. Ivlev} [Sov. Phys., Dokl. 4, 217--220 (1959; Zbl 0088.40801); translation from Dokl. Akad. Nauk SSSR 124, 546--549 (1959)] and formulated in isostatic coordinates. An optimal system of one-dimensional subalgebras constructing algorithm for the Lie algebra is proposed. The optimal system (total 187 elements) is shown consisting of of a 3-parametrical element, twelve 2-parametrical elements, sixty six 1-parametrical elements and one hundred and eight individual elements.The mixed Lie triple \(\xi\)-derivation on prime \(*\)-algebras.https://zbmath.org/1449.170322021-01-08T12:24:00+00:00"Zhou, You"https://zbmath.org/authors/?q=ai:zhou.you"Yang, Zhujun"https://zbmath.org/authors/?q=ai:yang.zhujun"Zhang, Jianhua"https://zbmath.org/authors/?q=ai:zhang.jianhuaSummary: The aim of this paper is to characterize the nonlinear mixed Lie triple \(\xi \)-derivation \( (\xi \ne 1)\) of a prime \(*\)-algebra. By using Peirce decomposition and the main proposition of mixed Lie triple \(\xi\)-derivation, it is proved that the nonlinear mixed Lie triple \(\xi\)-derivation \( (\xi \ne 1)\) of a prime \(*\)-algebra with unit and non-trivial projection is an additive \(*\)-derivation and linear about \(\xi\).The structure of the Heisenberg Lie algebras automorphism group.https://zbmath.org/1449.170172021-01-08T12:24:00+00:00"Zhang, Yan"https://zbmath.org/authors/?q=ai:zhang.yan.4|zhang.yan.2|zhang.yan.3"Ren, Bin"https://zbmath.org/authors/?q=ai:ren.binSummary: This paper mainly discussed the automorphism of Heisenberg Lie algebra \(N\). A sufficient and necessary condition for automorphism of the \( (2n + 1)\) dimensional Heisenberg Lie algebras was obtained by way of matrix representation. We also gained the decomposition structure of the five-dimensional Heisenberg Lie algebra automorphism group.The singular integrals on the closed piecewise smooth manifolds of octonions.https://zbmath.org/1449.320082021-01-08T12:24:00+00:00"Gong, Dingdong"https://zbmath.org/authors/?q=ai:gong.dingdongSummary: The solid-angle coefficient method is used to study the principal value on the closed piecewise smooth manifolds in octonions, and a corresponding Sokhotski-Plemel formula is obtained. These results are proved to be useful in the further study of the singular integral theory in octonions.Some properties of restricted Hom-Lie superalgebras.https://zbmath.org/1449.170122021-01-08T12:24:00+00:00"Guan, Baoling"https://zbmath.org/authors/?q=ai:guan.baoling"Wang, Chunyan"https://zbmath.org/authors/?q=ai:wang.chunyan"Wu, Xianfeng"https://zbmath.org/authors/?q=ai:wu.xianfengSummary: The definitions and some properties of Hom-Lie \(p\)-subsuperalgebras for restricted Hom-Lie superalgebras are given, and definitions and some properties of its semisimple elements and torus elements are proposed. The decomposition theorem of elements for restricted Hom-Lie superalgebras is determined.On the structures of split \(\delta\)-Jordan Lie superalgebras.https://zbmath.org/1449.170112021-01-08T12:24:00+00:00"Cao, Yan"https://zbmath.org/authors/?q=ai:cao.yan"Liu, Yanli"https://zbmath.org/authors/?q=ai:liu.yanli"Liu, Junting"https://zbmath.org/authors/?q=ai:liu.juntingSummary: The structures of arbitrary split \(\delta\)-Jordan Lie superalgebras with symmetric root systems are studied. The concept of split \(\delta\)-Jordan Lie superalgebras is introduced. The techniques of connections of roots for this kind of algebras are developed. By the techniques of connections of roots, the algebras \(L\) of the form \(L = U + \sum\limits_{[j] \in \Lambda / \sim} {I_{[j]}}\) is characterized.The structure of a Lie algebra attached to a unit form.https://zbmath.org/1449.170162021-01-08T12:24:00+00:00"Yu, Yalong"https://zbmath.org/authors/?q=ai:yu.yalong"Chen, Zhengxin"https://zbmath.org/authors/?q=ai:chen.zhengxinSummary: Let \(n \ge 4\). The complex Lie algebra, which is attached to the unit form \(q ({x_1}, {x_2} \ldots, {x_n}) = \sum_{i = 1}^n {x_i^2}- (\sum_{i=1}^{n-1}{x_i}{x_{i+1}}) + {x_1}{x_n}\) and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type \({\mathbb{D}_n}\), and realized by the Ringel-Hall Lie algebra of a Nakayama algebra. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.On split \(\delta\)-Jordan Lie triple systems.https://zbmath.org/1449.170052021-01-08T12:24:00+00:00"Cao, Yan"https://zbmath.org/authors/?q=ai:cao.yan"Chen, Liangyun"https://zbmath.org/authors/?q=ai:chen.liangyunSummary: The aim of this article is to study the structures of arbitrary split \(\delta\)-Jordan Lie triple systems, which are a generalization of split Lie triple systems. By developing techniques of connections of roots for this kind of triple systems, we show that any of such \(\delta\)-Jordan Lie triple systems \(T\) with a symmetric root system is of the form \(T = U+\Sigma_{[\alpha]\in \Lambda^1/\sim}I_{[\alpha]}\) with \(U\) a subspace of \({T_0}\) and any \(I_{[\alpha]}\) a well described ideal of \(T\), satisfying \(\{I_{[\alpha]}, T, I_{[\beta]}\} = \{I_{[\alpha]}, I_{[\beta]}, T\} = \{T, I_{[\alpha]}, I_{[\beta]}\} = 0\) if \([\alpha] \ne [\beta]\).On split regular Hom-Poisson color algebras.https://zbmath.org/1449.170032021-01-08T12:24:00+00:00"Guo, Shuangjian"https://zbmath.org/authors/?q=ai:guo.shuangjianSummary: We introduce the class of split regular Hom-Poisson color algebras as the natural generalization of split regular Hom-Poisson algebras and the one of split regular Hom-Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular Hom-Poisson color algebra \(A\) is of the form \(A = U + \sum_\alpha {I_\alpha}\) with \(U\) a subspace of a maximal abelian subalgebra \(H\) and any \({I_\alpha}\), a well described ideal of \(A\), satisfying \([{I_\alpha}, {I_\beta}] + {I_\alpha}{I_\beta} = 0\) if \([\alpha] \ne [\beta]\). Under certain conditions, in the case of \(A\) being of maximal length, the simplicity of the algebra is characterized.The structure of a Hom-Poisson color algebras.https://zbmath.org/1449.170152021-01-08T12:24:00+00:00"Xu, Jianguo"https://zbmath.org/authors/?q=ai:xu.jianguo"Wang, Song"https://zbmath.org/authors/?q=ai:wang.song.2|wang.song.1|wang.song.3|wang.song"Wang, Cong"https://zbmath.org/authors/?q=ai:wang.congSummary: The concepts of Hom-Poisson color algebra and Hom-admissible Poisson color algebra are given. It is proved that Hom-Poisson color algebra is closed under the tensor products operation. Moreover, Hom-Poisson color algebra is defined by Hom-admissible Poisson color algebra and its nonassociative binary operation.Atiyah classes of three-dimensional real Lie bialgebras.https://zbmath.org/1449.170402021-01-08T12:24:00+00:00"Shen, Dandan"https://zbmath.org/authors/?q=ai:shen.dandan"Qiao, Yu"https://zbmath.org/authors/?q=ai:qiao.yu.1|qiao.yu.2|qiao.yuSummary: Let \( (g, g^*, \gamma)\) be a Lie bialgebra, where the map \(\gamma: g \to g \otimes g\) is a 1-cocycle, that is, \([\gamma] \in {H^1} (g, g \otimes g)\). First, the definition of Atiyah class of a Lie bialgebra based on the map \(\gamma\) was given. Combining the classification results of three-dimensional real Lie bialgebras, Atiyah classes of these Lie bialgebras were computed and analyzed.Representation of the double of a quasi-bialgebra in its exterior algebra.https://zbmath.org/1449.170012021-01-08T12:24:00+00:00"Bangoura, Momo"https://zbmath.org/authors/?q=ai:bangoura.momoSummary: Lie quasi-bialgebras are natural generalisations of Lie bialgebras introduced by Drinfeld. To any Lie quasi-bialgebra structure of finite-dimensional \((\mathcal{G}, \mu, \gamma, \phi)\), correspond a unique Lie algebra structure on \(\mathcal{D}= \mathcal{G}\oplus \mathcal{G}^*\) which leaves invariant the canonical scalar product on \(\mathcal{D} \), called the double of the given Lie quasi-bialgebra. We show that there exist on \(\Lambda\mathcal{G} \), the exterior algebra of \(\mathcal{G} \), a \(\mathcal{D} \)-module structure.\(k\)-order generalized derivations of weight \(\lambda\) on \(\delta\)-Lie supertriple systems.https://zbmath.org/1449.170062021-01-08T12:24:00+00:00"Liu, Guilai"https://zbmath.org/authors/?q=ai:liu.guilai"Hao, Zhongjie"https://zbmath.org/authors/?q=ai:hao.zhongjie"Zhang, Qingcheng"https://zbmath.org/authors/?q=ai:zhang.qingchengSummary: The concepts of \(k\)-order (generalized) \( (\theta, \varphi)\)-derivations of weight \(\lambda\) and \(k\)-order (generalized) Jordan \( (\theta, \varphi)\)-derivations of weight \(\lambda\) on a \(\delta\)-Lie supertriple system are introduced. We conclude that \(k\)-order (generalized) Jordan \( (\theta, \varphi)\)-derivations of weight \(\lambda\) are \(k\)-order (generalized) \( (\theta, \varphi)\)-derivations of weight \(\lambda\) on a \(\delta\)-Lie supertriple system under some conditions. In particular, \(k\)-order Jordan \(\theta\)-derivations of weight \(\lambda\) are \(k\)-order \(\theta\)-derivations of weight \(\lambda\).The construct of Rota-Baxter algebra on the Sweedler 4-dimensional Hopf algebra.https://zbmath.org/1449.160672021-01-08T12:24:00+00:00"Zhang, Qian"https://zbmath.org/authors/?q=ai:zhang.qian"Li, Xuan"https://zbmath.org/authors/?q=ai:li.xuan"Li, Xin"https://zbmath.org/authors/?q=ai:li.xin|li.xin.7|li.xin.12|li.xin.2|li.xin.3|li.xin.9|li.xin.10|li.xin.6|li.xin.11|li.xin.1|li.xin.13|li.xin.4|li.xin.14|li.xin.5"Zheng, Huihui"https://zbmath.org/authors/?q=ai:zheng.huihui"Li, Linhan"https://zbmath.org/authors/?q=ai:li.linhan"Zhang, Liangyun"https://zbmath.org/authors/?q=ai:zhang.liangyunSummary: The nontrivial Rota-Baxter operators with weight \(-1\) are constructed from the Sweedler four-dimensional Hopf algebra and its subalgebras.8-dimensional Manin Triple of 3-Lie algebras.https://zbmath.org/1449.170102021-01-08T12:24:00+00:00"Bai, Ruipu"https://zbmath.org/authors/?q=ai:bai.ruipu"Wu, Yingli"https://zbmath.org/authors/?q=ai:wu.yingli"Hou, Shuai"https://zbmath.org/authors/?q=ai:hou.shuaiSummary: Dual 3-Lie algebras are constructed by involutive derivations and dual modules. It is proved that there exist involutive derivations on 4-dimensional 3-Lie algebras over an algebraically closed field of characteristic zero. By the involutive derivations on 4-dimensional 3-Lie algebras, seven non-isomorphic 8-dimensional Manian Triples of 3-Lie algebras are constructed.Derivations of the extended loop Schrödinger-Virasoro algebras.https://zbmath.org/1449.170302021-01-08T12:24:00+00:00"Wang, Song"https://zbmath.org/authors/?q=ai:wang.song.3|wang.song.1|wang.song|wang.song.2"Wang, Xiaoming"https://zbmath.org/authors/?q=ai:wang.xiaoming.2Summary: The infinite dimensional Lie algebras related to the extended Schrödinger-Virasoro algebras are introduced in this paper, and their derivation algebras are completely determined.Jordan derivation of generalized stochastic Jordan algebra.https://zbmath.org/1449.170312021-01-08T12:24:00+00:00"Wen, Liuting"https://zbmath.org/authors/?q=ai:wen.liuting"Chen, Qinghua"https://zbmath.org/authors/?q=ai:chen.qinghua"Chen, Zhengxin"https://zbmath.org/authors/?q=ai:chen.zhengxinSummary: Let \(\mathbb{F}\) be a field with char \( (\mathbb{F}) \ne 2\), \(M (n, \mathbb{F})\) be the matrix algebra consisting of all \(n \times n\) matrices over \(\mathbb{F}\), and let \(\alpha\) be an \(n\) dimensional non-zero column vector over \(\mathbb{F}\). Denote \(L (\alpha) = \{A \in M (n, \mathbb{F})A\alpha = 0 \}\). This paper proves that \(L (\alpha)\) is a Jordan subalgebra over \(\mathbb{F}\) (called generalized stochastic Jordan algebra), and any Jordan derivation of \(L (\alpha)\) is an inner derivation.Anti-commuting maps of upper triangular matrix Lie algebras.https://zbmath.org/1449.170192021-01-08T12:24:00+00:00"Zhu, Chundan"https://zbmath.org/authors/?q=ai:zhu.chundan"Chen, Zhengxin"https://zbmath.org/authors/?q=ai:chen.zhengxinSummary: Let \(\mathbb{F}\) be a field with char \( (\mathbb{F}) \ne 2\), let \({T_n} (\mathbb{F})\) be the Lie algebra consisting of all upper triangular matrices over \(\mathbb{F}\), and let \(\varphi:{T_n} (\mathbb{F}) \to {T_n} (\mathbb{F})\) be a linear map. If \([\varphi (X), Y] = -[X,\varphi (Y)]\) for all \(X, Y\) in \({T_n} (\mathbb{F})\), \(\varphi\) is called an anti-commuting map of \({T_n} (\mathbb{F})\). In this paper, if \(n \ge 3\), it is proved that a linear map \(\varphi\) of \({T_n} (\mathbb{F})\) is an anti-commuting map if and only if \(\varphi\) is a sum of a central anti-commuting map and an extremal inner derivation.Quasi-derivations of Lie color triple systems.https://zbmath.org/1449.170262021-01-08T12:24:00+00:00"Cao, Yan"https://zbmath.org/authors/?q=ai:cao.yanSummary: The author proved that a derivation of a larger Lie color triple system \(\check{T}\) could be embedded in a quasi-derivation of Lie color triple system \(T\). In particular, when \(Z (T) = 0\), \({\mathrm{Der}} (\check{T}) = \varphi ({\mathrm{QDer}} (T))\bigoplus {\mathrm{ZDer}} (\check{T})\).