Linear Algebra and Its Applications, ISSN 0024-3795, 10/2012, Volume 437, Issue 7, pp. 1821 - 1834

We study the compatible left-symmetric algebra structures on the W-algebra W(2,2) with some natural grading conditions. The results of earlier work on...

W-algebra W(2, 2) | Virasoro algebra | Left-symmetric algebra | VIRASORO ALGEBRAS | MATHEMATICS, APPLIED | LIE-GROUPS | W-algebra W(2,2) | GEOMETRY | Algebra

W-algebra W(2, 2) | Virasoro algebra | Left-symmetric algebra | VIRASORO ALGEBRAS | MATHEMATICS, APPLIED | LIE-GROUPS | W-algebra W(2,2) | GEOMETRY | Algebra

Journal Article

Journal of Algebra, ISSN 0021-8693, 02/2013, Volume 376, pp. 174 - 195

In this paper, we introduce the notions of hom-Lie 2-algebras, which is the categorification of hom-Lie algebras, HL∞-algebras, which is the hom-analogue of...

Hom-left-symmetric algebras | [formula omitted]-algebras | Hom-Lie 2-algebras | Symplectic hom-Lie algebras | Crossed module of hom-Lie algebras | Hom-Lie algebras | Quadratic hom-Lie algebras | algebras | Ham-Lie 2-algebras | MATHEMATICS | ALGEBRAS | COHOMOLOGY | DEFORMATIONS | Quadratic ham-Lie algebras | HL infinity-algebras | Crossed module of ham-Lie algebras | Algebra

Hom-left-symmetric algebras | [formula omitted]-algebras | Hom-Lie 2-algebras | Symplectic hom-Lie algebras | Crossed module of hom-Lie algebras | Hom-Lie algebras | Quadratic hom-Lie algebras | algebras | Ham-Lie 2-algebras | MATHEMATICS | ALGEBRAS | COHOMOLOGY | DEFORMATIONS | Quadratic ham-Lie algebras | HL infinity-algebras | Crossed module of ham-Lie algebras | Algebra

Journal Article

Communications in Algebra, ISSN 0092-7872, 06/2018, Volume 46, Issue 6, pp. 2346 - 2355

In this paper, the biderivations without the skew-symmetric condition of the twisted Heisenberg-Virasoro algebra are presented. We find some non-inner and...

post-Lie algebra | twisted Heisenberg-Virasoro algebra | Biderivation | linear commuting maps | left-symmetric algebra | twisted Heisenberg–Virasoro algebra | MATHEMATICS | W(A | LIE-ALGEBRA | SUPER-BIDERIVATIONS | Quantum theory | Algebra | Lie groups

post-Lie algebra | twisted Heisenberg-Virasoro algebra | Biderivation | linear commuting maps | left-symmetric algebra | twisted Heisenberg–Virasoro algebra | MATHEMATICS | W(A | LIE-ALGEBRA | SUPER-BIDERIVATIONS | Quantum theory | Algebra | Lie groups

Journal Article

Journal of Algebra, ISSN 0021-8693, 08/2015, Volume 436, pp. 61 - 101

In this study, we consider Lie algebras that admit para-Kähler and hyper-para-Kähler structures. We provide new characterizations of these Lie algebras and...

Left symmetric algebra | S-matrix | Para-Kähler Lie algebra | Hyper-para-Kähler Lie algebra | Symplectic Lie algebra | MATHEMATICS | Hyper-para-Kahler Lie algebra | Para-Kahler Lie algebra | Algebra

Left symmetric algebra | S-matrix | Para-Kähler Lie algebra | Hyper-para-Kähler Lie algebra | Symplectic Lie algebra | MATHEMATICS | Hyper-para-Kahler Lie algebra | Para-Kahler Lie algebra | Algebra

Journal Article

Communications in Algebra, ISSN 0092-7872, 12/2018, Volume 46, Issue 12, pp. 5381 - 5398

In this article, under some natural condition, a complete classification of compatible left-symmetric conformal algebraic structures on the Lie conformal...

left-symmetric conformal algebra | Lie conformal algebra | left-symmetric algebra | Coefficient algebra | Compatibility | Algebra | Classification

left-symmetric conformal algebra | Lie conformal algebra | left-symmetric algebra | Coefficient algebra | Compatibility | Algebra | Classification

Journal Article

Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8121, 01/2008, Volume 41, Issue 1, p. 015201

Rota-Baxter algebras were introduced to solve some analytic and combinatorial problems and have appeared in many fields in mathematics and mathematical...

HOPF-ALGEBRAS | PHYSICS, MULTIDISCIPLINARY | COMBINATORIAL IDENTITIES | NOVIKOV ALGEBRAS | LEFT-SYMMETRIC ALGEBRAS | EQUATIONS | QUANTUM-FIELD THEORY | REALIZATION | PHYSICS, MATHEMATICAL | CLASSICAL R-MATRIX | OPERATORS | RENORMALIZATION

HOPF-ALGEBRAS | PHYSICS, MULTIDISCIPLINARY | COMBINATORIAL IDENTITIES | NOVIKOV ALGEBRAS | LEFT-SYMMETRIC ALGEBRAS | EQUATIONS | QUANTUM-FIELD THEORY | REALIZATION | PHYSICS, MATHEMATICAL | CLASSICAL R-MATRIX | OPERATORS | RENORMALIZATION

Journal Article

Communications in Algebra, ISSN 0092-7872, 07/2017, Volume 45, Issue 7, pp. 2809 - 2820

A structure of a left-symmetric algebra on the set of all derivations of a free algebra is introduced such that its commutator algebra becomes the usual Lie...

Jacobian matrices | Secondary 17A36, 17A50 | Primary 17D25, 17A42, 14R15 | free algebras | Derivations | left-symmetric algebras | LIE-ALGEBRAS | MATHEMATICS | Derivation | Algebra | Lie groups | Commutators | Mathematical analysis | Symmetry

Jacobian matrices | Secondary 17A36, 17A50 | Primary 17D25, 17A42, 14R15 | free algebras | Derivations | left-symmetric algebras | LIE-ALGEBRAS | MATHEMATICS | Derivation | Algebra | Lie groups | Commutators | Mathematical analysis | Symmetry

Journal Article

中国科学：数学英文版, ISSN 1674-7283, 2014, Volume 57, Issue 3, pp. 469 - 476

The compatible left-symmetric algebra structures on the twisted Heisenberg-Virasoro algebra with some natural grading conditions are completely determined. The...

Virasoro代数 | 无限维 | 海森堡 | 子代数 | 代数结构 | 左对称代数 | 17B68 | left-symmetric algebra | Virasoro algebra | 17B65 | twisted Heisenberg-Virasoro algebra | Mathematics | Applications of Mathematics | 17D25 | Twisted Heisenberg-Virasoro algebra | Left-symmetric algebra | MATHEMATICS | MATHEMATICS, APPLIED | LIE-GROUPS | SPACES | CLASSIFICATION | GEOMETRY | Algebra | Grading | Compatibility | Mathematical analysis | Modules | China

Virasoro代数 | 无限维 | 海森堡 | 子代数 | 代数结构 | 左对称代数 | 17B68 | left-symmetric algebra | Virasoro algebra | 17B65 | twisted Heisenberg-Virasoro algebra | Mathematics | Applications of Mathematics | 17D25 | Twisted Heisenberg-Virasoro algebra | Left-symmetric algebra | MATHEMATICS | MATHEMATICS, APPLIED | LIE-GROUPS | SPACES | CLASSIFICATION | GEOMETRY | Algebra | Grading | Compatibility | Mathematical analysis | Modules | China

Journal Article

Linear Algebra and Its Applications, ISSN 0024-3795, 09/2012, Volume 437, Issue 5, pp. 1250 - 1263

We introduce post-Lie algebra structures on pairs of Lie algebras (g,n) defined on a fixed vector space V. Special cases are LR-structures and pre-Lie algebra...

Affine actions | Pre-Lie algebra | Post-Lie algebra | Left symmetric algebra | MATHEMATICS | MATHEMATICS, APPLIED | Algebra

Affine actions | Pre-Lie algebra | Post-Lie algebra | Left symmetric algebra | MATHEMATICS | MATHEMATICS, APPLIED | Algebra

Journal Article

Journal of Algebra and its Applications, ISSN 0219-4988, 04/2018

Journal Article

Communications in Algebra, ISSN 0092-7872, 07/2016, Volume 44, Issue 7, pp. 2919 - 2937

We discuss locally simply transitive affine actions of Lie groups G on finite-dimensional vector spaces such that the commutator subgroup [G, G] is acting by...

Left-invariant affine connections | Left-symmetric algebras | Derivation algebras | Novikov algebras | MATHEMATICS | AFFINE STRUCTURES

Left-invariant affine connections | Left-symmetric algebras | Derivation algebras | Novikov algebras | MATHEMATICS | AFFINE STRUCTURES

Journal Article

Journal of Geometry and Physics, ISSN 0393-0440, 01/2016, Volume 99, pp. 263 - 266

The purpose of this note is to correct the classification list of Novikov algebras admitting an invariant Lorentzian symmetric bilinear form in our paper...

Invariant Lorentzian forms | Novikov algebras | Extensions of left-symmetric algebras | Algebra

Invariant Lorentzian forms | Novikov algebras | Extensions of left-symmetric algebras | Algebra

Journal Article

International Journal of Algebra and Computation, ISSN 0218-1967, 03/2016, Volume 26, Issue 2, pp. 435 - 450

Let P n = k [ x 1 , x 2 , … , x n ] be the polynomial algebra over a field k of characteristic zero in the variables x 1 , x 2 , … , x n and ℒ n be the...

Algebras of derivations | multiplication algebras | free algebras | identities | left-symmetric algebras | MATHEMATICS | Algebra

Algebras of derivations | multiplication algebras | free algebras | identities | left-symmetric algebras | MATHEMATICS | Algebra

Journal Article

Annals of Global Analysis and Geometry, ISSN 0232-704X, 4/2018, Volume 53, Issue 3, pp. 405 - 443

The nonzero level sets in n-dimensional flat affine space of a translationally homogeneous function are improper affine spheres if and only if the Hessian...

Geometry | Cayley hypersurface | Statistics for Business/Economics/Mathematical Finance/Insurance | Analysis | Theoretical, Mathematical and Computational Physics | Affine spheres | Mathematics | Group Theory and Generalizations | Left-symmetric algebra | MATHEMATICS | LIE-GROUPS | Algebra | Functions (mathematics) | Operators (mathematics) | Multiplication | Mathematical analysis | Polynomials | Multiplication & division | Mathematics - Differential Geometry

Geometry | Cayley hypersurface | Statistics for Business/Economics/Mathematical Finance/Insurance | Analysis | Theoretical, Mathematical and Computational Physics | Affine spheres | Mathematics | Group Theory and Generalizations | Left-symmetric algebra | MATHEMATICS | LIE-GROUPS | Algebra | Functions (mathematics) | Operators (mathematics) | Multiplication | Mathematical analysis | Polynomials | Multiplication & division | Mathematics - Differential Geometry

Journal Article

Annals of Global Analysis and Geometry, ISSN 0232-704X, 6/2018, Volume 53, Issue 4, pp. 543 - 559

Hyper-para-Kähler structures on Lie algebras where the complex structure is abelian are studied. We show that there is a one-to-one correspondence between such...

Geometry | Hyper-para-Kähler manifold | Symplectic left-symmetric algebra | Mathematical Physics | 17B30 | Analysis | 53C55 | Global Analysis and Analysis on Manifolds | 53D05 | Abelian (para)complex structure | Mathematics | Differential Geometry | MATHEMATICS | Hyper-para-Kahler manifold | Algebra | Isomorphism | Classification | Lie groups

Geometry | Hyper-para-Kähler manifold | Symplectic left-symmetric algebra | Mathematical Physics | 17B30 | Analysis | 53C55 | Global Analysis and Analysis on Manifolds | 53D05 | Abelian (para)complex structure | Mathematics | Differential Geometry | MATHEMATICS | Hyper-para-Kahler manifold | Algebra | Isomorphism | Classification | Lie groups

Journal Article

Mathematische Nachrichten, ISSN 0025-584X, 05/2012, Volume 285, Issue 7, pp. 922 - 935

We classify the compatible left‐symmetric algebraic structures on the Witt algebra satisfying certain non‐graded conditions. It is unexpected that they are...

left‐symmetric algebra | Witt algebra | 17A30 | Novikov algebra | module MSC 17A32 | Left-symmetric algebra | Module | MATHEMATICS | left-symmetric algebra | NOVIKOV ALGEBRAS | SPACES | VIRASORO ALGEBRA

left‐symmetric algebra | Witt algebra | 17A30 | Novikov algebra | module MSC 17A32 | Left-symmetric algebra | Module | MATHEMATICS | left-symmetric algebra | NOVIKOV ALGEBRAS | SPACES | VIRASORO ALGEBRA

Journal Article

Journal of Lie Theory, ISSN 0949-5932, 2014, Volume 24, Issue 3, pp. 849 - 864

We investigate the properties of principal elements of Frobenius Lie algebras. We prove that any Lie algebra with a left symmetric algebra structure can be...

Affine motion | Seaweed Lie algebra | Frobenius Lie algebra | Invariant affine connection | Affine Lie algebra | Classical Yang Baxter equation | Left symmetric algebra | Symplectic Lie algebra | Kähler algebra | YANG-BAXTER EQUATION | invariant affine connection | symplectic Lie algebra | seaweed Lie algebra | Kahler algebra | MATHEMATICS | classical Yang Baxter equation | affine motion | DOUBLE EXTENSION | left symmetric algebra | MANIFOLDS | affine Lie algebra

Affine motion | Seaweed Lie algebra | Frobenius Lie algebra | Invariant affine connection | Affine Lie algebra | Classical Yang Baxter equation | Left symmetric algebra | Symplectic Lie algebra | Kähler algebra | YANG-BAXTER EQUATION | invariant affine connection | symplectic Lie algebra | seaweed Lie algebra | Kahler algebra | MATHEMATICS | classical Yang Baxter equation | affine motion | DOUBLE EXTENSION | left symmetric algebra | MANIFOLDS | affine Lie algebra

Journal Article

Journal of the Korean Mathematical Society, ISSN 0304-9914, 2017, Volume 54, Issue 5, pp. 1537 - 1556

In this paper, we introduce the notion of generating index I-1(A) (2-generating index I-2(A), resp.) of a left-symmetric algebra A, which is the maximum of the...

Non-associative algebra | Left-symmetric algebra | Linear function | Generating index | generating index | LIE-ALGEBRAS | MATHEMATICS | MATHEMATICS, APPLIED | left-symmetric algebra | linear function | EQUATIONS | non-associative algebra | NONASSOCIATIVE-ALGEBRAS

Non-associative algebra | Left-symmetric algebra | Linear function | Generating index | generating index | LIE-ALGEBRAS | MATHEMATICS | MATHEMATICS, APPLIED | left-symmetric algebra | linear function | EQUATIONS | non-associative algebra | NONASSOCIATIVE-ALGEBRAS

Journal Article

Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, 01/2016, Volume 23, Issue 1, pp. 47 - 73

Motivated by the work of Kupershmidt (J. Nonlin. Math. Phys. 6 (1998), 222 -245) we discuss the occurrence of left symmetry in a generalized Virasoro algebra....

Nonlinear systems of hydrodynamic type | Left-symmetric algebras | Quasi-associativity | Virasoro algebra | Coboundary operators | MATHEMATICS, APPLIED | INFINITE CONFORMAL SYMMETRY | RINGS | DEFORMATIONS | WITT | PHYSICS, MATHEMATICAL | Mathematics

Nonlinear systems of hydrodynamic type | Left-symmetric algebras | Quasi-associativity | Virasoro algebra | Coboundary operators | MATHEMATICS, APPLIED | INFINITE CONFORMAL SYMMETRY | RINGS | DEFORMATIONS | WITT | PHYSICS, MATHEMATICAL | Mathematics

Journal Article

Communications in Contemporary Mathematics, ISSN 0219-1997, 04/2008, Volume 10, Issue 2, pp. 221 - 260

We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent...

Left-symmetric algebra | Parakähler Lie algebra | S-equation | Left-symmetric bialgebra | MATHEMATICS, APPLIED | HOPF-ALGEBRAS | left-symmetric algebra | HOMOGENEOUS SPACES | parakahler Lie algebra | MATHEMATICS | LIE-GROUPS | left-symmetric bialgebra | DOUBLE EXTENSION | NOVIKOV ALGEBRAS | ABELIAN PHASE SPACES | REALIZATION | MATCHED PAIRS | GEOMETRY | RENORMALIZATION | Algebra

Left-symmetric algebra | Parakähler Lie algebra | S-equation | Left-symmetric bialgebra | MATHEMATICS, APPLIED | HOPF-ALGEBRAS | left-symmetric algebra | HOMOGENEOUS SPACES | parakahler Lie algebra | MATHEMATICS | LIE-GROUPS | left-symmetric bialgebra | DOUBLE EXTENSION | NOVIKOV ALGEBRAS | ABELIAN PHASE SPACES | REALIZATION | MATCHED PAIRS | GEOMETRY | RENORMALIZATION | Algebra

Journal Article

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