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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
Journal Article
Journal of Algebra, ISSN 0021-8693, 02/2013, Volume 376, pp. 174 - 195
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
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
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
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
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
Journal Article
中国科学:数学英文版, ISSN 1674-7283, 2014, Volume 57, Issue 3, pp. 469 - 476
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
Journal Article
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
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
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
Journal Article
Annals of Global Analysis and Geometry, ISSN 0232-704X, 4/2018, Volume 53, Issue 3, pp. 405 - 443
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
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
Journal Article
Journal of Lie Theory, ISSN 0949-5932, 2014, Volume 24, Issue 3, pp. 849 - 864
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
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
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
Journal Article
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