Articles de revues : « Differential graded Lie algebras »

1

Piontkovskii, D. I. « On differential graded Lie algebras ». Russian Mathematical Surveys 58, n^o 1 (28 février 2003) : 189–90. http://dx.doi.org/10.1070/rm2003v058n01abeh000604.

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Bonezzi, Roberto, et Olaf Hohm. « Duality Hierarchies and Differential Graded Lie Algebras ». Communications in Mathematical Physics 382, n^o 1 (février 2021) : 277–315. http://dx.doi.org/10.1007/s00220-021-03973-8.

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AbstractThe gauge theories underlying gauged supergravity and exceptional field theory are based on tensor hierarchies: generalizations of Yang-Mills theory utilizing algebraic structures that generalize Lie algebras and, as a consequence, require higher-form gauge fields. Recently, we proposed that the algebraic structure allowing for consistent tensor hierarchies is axiomatized by ‘infinity-enhanced Leibniz algebras’ defined on graded vector spaces generalizing Leibniz algebras. It was subsequently shown that, upon appending additional vector spaces, this structure can be reinterpreted as a differential graded Lie algebra. We use this observation to streamline the construction of general tensor hierarchies, and we formulate dynamics in terms of a hierarchy of first-order duality relations, including scalar fields with a potential.

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Kaneyuki, Soji, et Hiroshi Asano. « Graded Lie algebras and generalized Jordan triple systems ». Nagoya Mathematical Journal 112 (décembre 1988) : 81–115. http://dx.doi.org/10.1017/s002776300000115x.

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One frequently encounters (real) semisimple graded Lie algebras in various branches of differential geometry (e.g. [16], [9], [14], [18]). It is therefore desirable to study semisimple graded Lie algebras, including those which have been studied individually, in a unified way. One of our concerns is to classify (finite-dimensional) semisimple graded Lie algebras in a way that enables us to construct them.

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Wulkenhaar, Raimar. « Noncommutative geometry with graded differential Lie algebras ». Journal of Mathematical Physics 38, n^o 6 (juin 1997) : 3358–90. http://dx.doi.org/10.1063/1.532048.

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Wulkenhaar, Raimar. « Gauge theories with graded differential Lie algebras ». Journal of Mathematical Physics 40, n^o 2 (février 1999) : 787–94. http://dx.doi.org/10.1063/1.532685.

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Wulkenhaar, Raimar. « Graded differential lie algebras and model building ». Journal of Geometry and Physics 25, n^o 3-4 (mai 1998) : 305–25. http://dx.doi.org/10.1016/s0393-0440(97)00029-6.

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Pei, Yufeng, et Jinwei Yang. « Strongly graded vertex algebras generated by vertex Lie algebras ». Communications in Contemporary Mathematics 21, n^o 08 (20 octobre 2019) : 1850069. http://dx.doi.org/10.1142/s0219199718500694.

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We construct three families of vertex algebras along with their modules from appropriate vertex Lie algebras, using the constructions in [Vertex Lie algebra, vertex Poisson algebras and vertex algebras, in Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory[Formula: see text] Proceedings of an International Conference at University of Virginia[Formula: see text] May 2000, in Contemporary Mathematics, Vol. 297 (American Mathematical Society, 2002), pp. 69–96] by Dong, Li and Mason. These vertex algebras are strongly graded vertex algebras introduced in [Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, I: Introduction and strongly graded algebras and their generalized modules, in Conformal Field Theories and Tensor Categories[Formula: see text] Proceedings of a Workshop Held at Beijing International Center for Mathematics Research, eds. C. Bai, J. Fuchs, Y.-Z. Huang, L. Kong, I. Runkel and C. Schweigert, Mathematical Lectures from Beijing University, Vol. 2 (Springer, New York, 2014), pp. 169–248] by Huang, Lepowsky and Zhang in their logarithmic tensor category theory and can also be realized as vertex algebras associated to certain well-known infinite dimensional Lie algebras. We classify irreducible [Formula: see text]-gradable weak modules for these vertex algebras by determining their Zhu’s algebras. We find examples of strongly graded generalized modules for these vertex algebras that satisfy the [Formula: see text]-cofiniteness condition introduced in [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009] by the second author. In particular, by a result of the second author [Differential equations and logarithmic intertwining operators for strongly graded vertex algebra, Comm. Contemp. Math. 19(2) (2017) 1650009, 26 pp.], the convergence and extension property for products and iterates of logarithmic intertwining operators in [Y.-Z. Huang, J. Lepowsky and L. Zhang, Logarithmic tensor category theory for generalized modules for a conformal vertex algebra, VII: Convergence and extension properties and applications to expansion for intertwining maps, preprint (2011); arXiv:1110.1929 ] among such strongly graded generalized modules is verified.

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Budur, Nero, et Botong Wang. « Cohomology jump loci of differential graded Lie algebras ». Compositio Mathematica 151, n^o 8 (6 mars 2015) : 1499–528. http://dx.doi.org/10.1112/s0010437x14007970.

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To study infinitesimal deformation problems with cohomology constraints, we introduce and study cohomology jump functors for differential graded Lie algebra (DGLA) pairs. We apply this to local systems, vector bundles, Higgs bundles, and representations of fundamental groups. The results obtained describe the analytic germs of the cohomology jump loci inside the corresponding moduli space, extending previous results of Goldman–Millson, Green–Lazarsfeld, Nadel, Simpson, Dimca–Papadima, and of the second author.

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BENKHALIFA, MAHMOUD. « WHITEHEAD EXACT SEQUENCE AND DIFFERENTIAL GRADED FREE LIE ALGEBRA ». International Journal of Mathematics 15, n^o 10 (décembre 2004) : 987–1005. http://dx.doi.org/10.1142/s0129167x04002673.

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Let R be a principal and integral domain. We say that two differential graded free Lie algebras over R (free dgl for short) are weakly equivalent if and only if the homologies of their corresponding enveloping universal algebras are isomophic. This paper is devoted to the problem of how we can characterize the weakly equivalent class of a free dgl. Our tool to address this question is the Whitehead exact sequence. We show, under a certain condition, that two R-free dgls are weakly equivalent if and only if their Whitehead sequences are isomorphic.

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Yang, Jinwei. « Vertex algebras associated to the affine Lie algebras of abelian polynomial current algebras ». International Journal of Mathematics 27, n^o 05 (mai 2016) : 1650046. http://dx.doi.org/10.1142/s0129167x16500464.

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We construct a family of vertex algebras associated to the affine Lie algebra of polynomial current algebras of finite-dimensional abelian Lie algebras, along with their modules and logarithmic modules. These vertex algebras and their (logarithmic) modules are strongly [Formula: see text]-graded and quasi-conformal. We then show that matrix elements of products and iterates of logarithmic intertwining operators among these logarithmic modules satisfy certain systems of differential equations. Using these systems of differential equations, we verify the convergence and extension property needed in the logarithmic tensor category theory developed by Huang, Lepowsky and Zhang.

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Jurčo, Branislav. « From simplicial Lie algebras and hypercrossed complexes to differential graded Lie algebras via 1-jets ». Journal of Geometry and Physics 62, n^o 12 (décembre 2012) : 2389–400. http://dx.doi.org/10.1016/j.geomphys.2012.09.002.

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Iserles, Arieh, et Antonella Zanna. « On the Dimension of Certain Graded Lie Algebras Arising in Geometric Integration of Differential Equations ». LMS Journal of Computation and Mathematics 3 (2000) : 44–75. http://dx.doi.org/10.1112/s1461157000000206.

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AbstractMany discretization methods for differential equations that evolve in Lie groups and homogeneous spaces advance the solution in the underlying Lie algebra. The main expense of computation is the calculation of commutators, a task that can be made significantly cheaper by the introduction of appropriate bases of function values and by the exploitation of redundancies inherent in a Lie-algebraic structure by means of graded spaces. In many Lie groups of practical interest a convenient alternative to the exponential map is a Cayley transformation, and the subject of this paper is the investigation of graded algebras that occur in this context. To this end we introduce a new concept, a hierarchical algebra, a Lie algebra equipped with a countable number of m-nary multilinear operations which display alternating symmetry and a ‘hierarchy condition’. We present explicit formulae for the dimension of graded subspaces of free hierarchical algebras and an algorithm for the construction of their basis. The paper is concluded by reviewing a number of applications of our results to numerical methods in a Lie-algebraic setting.

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COQUEREAUX, R., G. ESPOSITO FARÈSE et F. SCHECK. « NONCOMMUTATIVE GEOMETRY AND GRADED ALGEBRAS IN ELECTROWEAK INTERACTIONS ». International Journal of Modern Physics A 07, n^o 26 (20 octobre 1992) : 6555–93. http://dx.doi.org/10.1142/s0217751x9200301x.

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The Standard Model of Electroweak Interactions can be described by a generalized Yang-Mills field incorporating both the usual gauge bosons and the Higgs fields. The graded derivative by means of which the Yang-Mills field strength is constructed involves both a differential acting on space-time and a differential acting on an associative graded algebra of matrices. The square of the curvature for the corresponding covariant derivative yields the bosonic Lagrangian of the Standard Model. We show how to recover the whole fermionic part of the Standard Model in this framework. Quarks and leptons fit naturally into the smallest typical and nontypical irreducible representations of the graded algebra Lie SU(2|1) associated with the above associative ℤ2-graded algebra. The existence of reducible indecomposable representations leads naturally to flavor mixing in the quark sector, possibility of existence for a right neutrino and possible mixing in the leptonic sector. We therefore bridge the gap between noncommutative geometry and graded Lie algebras. The Z2 grading refers to left and right chiralities in the fermionic sector and to even and odd forms in the bosonic sector. Supergauge transformations could only be defined in an extension of the theory incorporating tensor fields of higher rank. The Standard Model contains only one-forms and zero-forms in the bosonic sector, therefore only the even part of the above graded Lie algebra — i.e. Lie[SU(2)×U(1)] — acts.

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Wagemann, Friedrich. « Differential Graded Cohomology and Lie Algebras¶of Holomorphic Vector Fields ». Communications in Mathematical Physics 208, n^o 2 (30 décembre 1999) : 521–40. http://dx.doi.org/10.1007/s002200050768.

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Retakh, Vladimir S. « Lie-Massey brackets and n-homotopically multiplicative maps of differential graded Lie algebras ». Journal of Pure and Applied Algebra 89, n^o 1-2 (octobre 1993) : 217–29. http://dx.doi.org/10.1016/0022-4049(93)90095-b.

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WULKENHAAR, RAIMAR. « GRADED DIFFERENTIAL LIE ALGEBRAS AND SU(5)×U(1)-GRAND UNIFICATION ». International Journal of Modern Physics A 13, n^o 15 (20 juin 1998) : 2627–92. http://dx.doi.org/10.1142/s0217751x98001359.

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We formulate the flipped SU(5)×U(1)-GUT within a Lie-algebraic approach to non-commutative geometry. It suffices to take the matrix Lie algebra su(5) as the input; the u(1)-part with its representation on the fermions is an algebraic consequence. The occurring Higgs multiplets (24, 5, 45, 50-representations of su(5)) are uniquely determined by the fermionic mass matrix and the spontaneous symmetry breaking pattern to SU(3)C×U(1)EM. We find the most general gauge invariant Higgs potential that is compatible with the given Higgs vacuum. Our formalism yields tree-level predictions for the masses of all gauge and Higgs bosons. It turns out that the low-energy sector is identical with the standard model. In particular, there exists precisely one light Higgs field, whose upper bound for the mass is 1.45 mt. All remaining 207 Higgs fields are extremely heavy.

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Popa, Mihail. « Problems of the theory of invariants and Lie algebras applied in the qualitative theory of differential systems ». Acta et commentationes : Ştiinţe Exacte şi ale Naturii 14, n^o 2 (janvier 2023) : 15–23. http://dx.doi.org/10.36120/2587-3644.v14i2.15-23.

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In this work there were formulated 18 problems from the theory of invariant processes, Lie algebras, commutative graded algebras, generating functions and Hilbert series, orbit theory and Lyapunov stability theory that are important to be solved. There was substantiated the necessity of using the solutions of these problems in the qualitative theory of differential systems.

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Goldman, William M., et John J. Millson. « Differential graded Lie algebras and singularities of level sets of momentum mappings ». Communications in Mathematical Physics 131, n^o 3 (août 1990) : 495–515. http://dx.doi.org/10.1007/bf02098273.

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Abramov, Viktor. « Matrix 3-Lie superalgebras and BRST supersymmetry ». International Journal of Geometric Methods in Modern Physics 14, n^o 11 (23 octobre 2017) : 1750160. http://dx.doi.org/10.1142/s0219887817501602.

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Given a matrix Lie algebra one can construct the 3-Lie algebra by means of the trace of a matrix. In the present paper, we show that this approach can be extended to the infinite-dimensional Lie algebra of vector fields on a manifold if instead of the trace of a matrix we consider a differential 1-form which satisfies certain conditions. Then we show that the same approach can be extended to matrix Lie superalgebras [Formula: see text] if instead of the trace of a matrix we make use of the supertrace of a matrix. It is proved that a graded triple commutator of matrices constructed with the help of the graded commutator and the supertrace satisfies a graded ternary Filippov–Jacobi identity. In two particular cases of [Formula: see text] and [Formula: see text], we show that the Pauli and Dirac matrices generate the matrix 3-Lie superalgebras, and we find the non-trivial graded triple commutators of these algebras. We propose a Clifford algebra approach to 3-Lie superalgebras induced by Lie superalgebras. We also discuss an application of matrix 3-Lie superalgebras in BRST-formalism.

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Gualtieri, Marco, Mykola Matviichuk et Geoffrey Scott. « Deformation of Dirac Structures via L∞ Algebras ». International Mathematics Research Notices 2020, n^o 14 (22 juin 2018) : 4295–323. http://dx.doi.org/10.1093/imrn/rny134.

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Abstract The deformation theory of a Dirac structure is controlled by a differential graded Lie algebra that depends on the choice of an auxiliary transversal Dirac structure; if the transversal is not involutive, one obtains an $L_\infty $ algebra instead. We develop a simplified method for describing this $L_\infty $ algebra and use it to prove that the $L_\infty $ algebras corresponding to different transversals are canonically $L_\infty $–isomorphic. In some cases, this isomorphism provides a formality map, as we show in several examples including (quasi)-Poisson geometry, Dirac structures on Lie groups, and Lie bialgebras. Finally, we apply our result to a classical problem in the deformation theory of complex manifolds; we provide explicit formulas for the Kodaira–Spencer deformation complex of a fixed small deformation of a complex manifold, in terms of the deformation complex of the original manifold.

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NODA, Takahiro. « On a certain invariant of differential equations associated with nilpotent graded Lie algebras ». Hokkaido Mathematical Journal 47, n^o 3 (octobre 2018) : 445–64. http://dx.doi.org/10.14492/hokmj/1537948824.

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Bandiera, Ruggero, Marco Manetti et Francesco Meazzini. « Formality conjecture for minimal surfaces of Kodaira dimension 0 ». Compositio Mathematica 157, n^o 2 (février 2021) : 215–35. http://dx.doi.org/10.1112/s0010437x20007605.

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Let $\mathcal {F}$ be a polystable sheaf on a smooth minimal projective surface of Kodaira dimension 0. Then the differential graded (DG) Lie algebra $R\operatorname {Hom}(\mathcal {F},\mathcal {F})$ of derived endomorphisms of $\mathcal {F}$ is formal. The proof is based on the study of equivariant $L_{\infty }$ minimal models of DG Lie algebras equipped with a cyclic structure of degree 2 which is non-degenerate in cohomology, and does not rely (even for K3 surfaces) on previous results on the same subject.

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Buijs, Urtzi, José G. Carrasquel-Vera et Aniceto Murillo. « The gauge action, DG Lie algebras and identities for Bernoulli numbers ». Forum Mathematicum 29, n^o 2 (1 mars 2017) : 277–86. http://dx.doi.org/10.1515/forum-2015-0257.

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AbstractIn this paper we prove a family of identities for Bernoulli numbers parameterized by triples of integers ${(a,b,c)}$ with ${a+b+c=n-1}$, ${n\geq 4}$. These identities are deduced by translating into homotopical terms the gauge action on the Maurer–Cartan set of a differential graded Lie algebra. We show that Euler and Miki’s identities, well-known and apparently non-related formulas, are linear combinations of our family and they satisfy a particular symmetry relation.

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Post, Gerhard. « A class of graded Lie algebras of vector fields and first order differential operators ». Journal of Mathematical Physics 35, n^o 12 (décembre 1994) : 6838–56. http://dx.doi.org/10.1063/1.530645.

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Das, Apurba. « Cohomology and deformations of weighted Rota–Baxter operators ». Journal of Mathematical Physics 63, n^o 9 (1 septembre 2022) : 091703. http://dx.doi.org/10.1063/5.0093066.

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Weighted Rota–Baxter operators on associative algebras are closely related to modified Yang–Baxter equations, splitting of algebras, and weighted infinitesimal bialgebras and play an important role in mathematical physics. For any λ ∈ k, we construct a differential graded Lie algebra whose Maurer–Cartan elements are given by λ-weighted relative Rota–Baxter operators. Using such characterization, we define the cohomology of a λ-weighted relative Rota-Baxter operator T and interpret this as the Hochschild cohomology of a suitable algebra with coefficients in an appropriate bimodule. We study linear, formal, and finite order deformations of T from cohomological points of view. Among others, we introduce Nijenhuis elements that generate trivial linear deformations and define a second cohomology class to any finite order deformation, which is the obstruction to extend the deformation. In the end, we also consider the cohomology of λ-weighted relative Rota–Baxter operators in the Lie case and find a connection with the case of associative algebras.

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Scott, Jonathan A. « A factorization of the homology of a differential graded Lie algebra ». Journal of Pure and Applied Algebra 167, n^o 2-3 (février 2002) : 329–40. http://dx.doi.org/10.1016/s0022-4049(01)00037-8.

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Sharygin, G., et D. Talalaev. « On the Lie-formality of Poisson manifolds ». Journal of K-Theory 2, n^o 2 (4 mars 2008) : 361–84. http://dx.doi.org/10.1017/is008001011jkt030.

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MAGNOT, JEAN-PIERRE. « AMBROSE–SINGER THEOREM ON DIFFEOLOGICAL BUNDLES AND COMPLETE INTEGRABILITY OF THE KP EQUATION ». International Journal of Geometric Methods in Modern Physics 10, n^o 09 (30 août 2013) : 1350043. http://dx.doi.org/10.1142/s0219887813500436.

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In this paper, we start from an extension of the notion of holonomy on diffeological bundles, reformulate the notion of regular Lie group or Frölicher Lie groups, state an Ambrose–Singer theorem that enlarges the one stated in [J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math.128 (2004) 513–529], and conclude with a differential geometric treatment of KP hierarchy. The examples of Lie groups that are studied are principally those obtained by enlarging some graded Frölicher (Lie) algebras such as formal q-series of the quantum algebra of pseudo-differential operators. These deformations can be defined for classical pseudo-differential operators but they are used here on formal pseudo-differential operators in order to get a differential geometric framework to deal with the KP hierarchy that is known to be completely integrable with formal power series. Here, we get an integration of the Zakharov–Shabat connection form by means of smooth sections of a (differential geometric) bundle with structure group, some groups of q-deformed operators. The integration obtained by Mulase [Complete integrability of the Kadomtsev–Petviashvili equation Adv. Math.54 (1984) 57–66], and the key tools he developed, are totally recovered on the germs of the smooth maps of our construction. The tool coming from (classical) differential geometry used in this construction is the holonomy group, on which we have an Ambrose–Singer-like theorem: the Lie algebra is spanned by the curvature elements. This result is proved for any connection a diffeological principal bundle with structure group a regular Frölicher Lie group. The case of a (classical) Lie group modeled on a complete locally convex topological vector space is also recovered and the work developed in [J.-P. Magnot, Difféologie du fibré d'Holonomie en dimension infinie, Math. Rep. Canadian Roy. Math. Soc.28(4) (2006); J.-P. Magnot, Structure groups and holonomy in infinite dimensions, Bull. Sci. Math. 128 (2004) 513–529] is completed.

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Nagai, Yasunari, et Fumitoshi Sato. « Deformation of a smooth Deligne–Mumford stack via differential graded Lie algebra ». Journal of Algebra 320, n^o 9 (novembre 2008) : 3481–92. http://dx.doi.org/10.1016/j.jalgebra.2008.08.020.

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CRANE, LOUIS. « RELATIONAL SPACETIME, MODEL CATEGORIES AND QUANTUM GRAVITY ». International Journal of Modern Physics A 24, n^o 15 (20 juin 2009) : 2753–75. http://dx.doi.org/10.1142/s0217751x0904614x.

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We propose a mathematically concrete way of modelling the suggestion that in quantum gravity the spacetime manifold disappears. We replace the underlying point set topological space with several apparently different models, which are actually related by pairs of adjoint functors from rational homotopy theory. One is a discrete approximation to the causal null path space derived from the multiple images in the spacetime theory of gravitational lensing, described as an object in the model category of differential graded Lie algebras. Another of our models appears as a thickening of spacetime, which we interpret as a formulation of relational geometry. This model is produced from the finite dimensional differential graded algebra of differential forms which can be transmitted out of a finite region consistent with the Bekenstein bound by another functor, called geometric realisation. The thickening of spacetime, which we propose as a version of relational spacetime, has a surprizingly rich structure. Information which would make up a spin bundle over spacetime is contained in it, making it possible to include fermionic fields in a geometric state sum over it. Avenues toward constructing an actual quantum theory of gravity on our models are given a preliminary exploration.

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Kotov, Alexei, et Thomas Strobl. « Characteristic classes associated to Q-bundles ». International Journal of Geometric Methods in Modern Physics 12, n^o 01 (28 décembre 2014) : 1550006. http://dx.doi.org/10.1142/s0219887815500061.

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A Q-manifold is a graded manifold endowed with a vector field of degree 1 squaring to zero. We consider the notion of a Q-bundle, that is, a fiber bundle in the category of Q-manifolds. To each homotopy class of "gauge fields" (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. As any principal bundle yields canonically a Q-bundle, this construction generalizes Chern–Weil classes. Novel examples include cohomology classes that are locally de Rham differential of the integrands of topological sigma models obtained by the AKSZ-formalism in arbitrary dimensions. For Hamiltonian Poisson fibrations one obtains a characteristic 3-class in this manner. We also relate the framework to equivariant cohomology and Lecomte's characteristic classes of exact sequences of Lie algebras.

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Filip, Matej. « A differential graded Lie algebra controlling the Poisson deformations of an affine Poisson variety ». Communications in Algebra 48, n^o 5 (11 janvier 2020) : 2183–95. http://dx.doi.org/10.1080/00927872.2019.1710520.

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Hidalgo, Rubén A., Irina Markina et Alexander Vasil'ev. « Finite Dimensional Grading of the Virasoro Algebra ». gmj 14, n^o 3 (septembre 2007) : 419–34. http://dx.doi.org/10.1515/gmj.2007.419.

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Abstract The Virasoro algebra is a central extension of the Witt algebra, the complexified Lie algebra of the sense preserving diffeomorphism group of the circle Diff 𝑆1. It appears in Quantum Field Theories as an infinite dimensional algebra generated by the coefficients of the Laurent expansion of the analytic component of the momentum-energy tensor, Virasoro generators. The background for the construction of the theory of unitary representations of Diff 𝑆1 is found in the study of Kirillov's manifold Diff 𝑆1=𝑆1. It possesses a natural Kählerian embedding into the universal Teichmüller space with the projection into the moduli space realized as an infinite-dimensional body of the coefficients of univalent quasiconformally extendable functions. The differential of this embedding leads to an analytic representation of the Virasoro algebra based on Kirillov's operators. In this paper we overview several interesting connections between the Virasoro algebra, Teichmüller theory, Löwner representation of univalent functions, and propose a finite-dimensional grading of the Virasoro algebra such that the grades form a hierarchy of finite dimensional algebras which, in their turn, are the first integrals of Liouville partially integrable systems for coefficients of univalent functions.

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Qingyun, Fei, et Shen Guangyu. « Universal graded Lie algebras ». Journal of Algebra 152, n^o 2 (novembre 1992) : 439–53. http://dx.doi.org/10.1016/0021-8693(92)90042-k.

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Larsson, Daniel, et Sergei D. Silvestrov. « Graded quasi-Lie algebras ». Czechoslovak Journal of Physics 55, n^o 11 (novembre 2005) : 1473–78. http://dx.doi.org/10.1007/s10582-006-0028-3.

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Sánchez Ortega, Juana, et Mercedes Siles Molina. « Algebras of quotients of graded Lie algebras ». Journal of Algebra 323, n^o 7 (avril 2010) : 2002–15. http://dx.doi.org/10.1016/j.jalgebra.2010.01.005.

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Millionshchikov, D. « Narrow Positively Graded Lie Algebras ». Доклады академии наук 483, n^o 5 (décembre 2018) : 492–94. http://dx.doi.org/10.31857/s086956520003295-7.

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Millionshchikov, D. V. « Narrow Positively Graded Lie Algebras ». Doklady Mathematics 98, n^o 3 (novembre 2018) : 626–28. http://dx.doi.org/10.1134/s1064562418070244.

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Calderón Martín, Antonio J. « Graded extended Lie-type algebras ». Communications in Algebra 45, n^o 2 (7 octobre 2016) : 866–77. http://dx.doi.org/10.1080/00927872.2016.1175611.

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Echarte, F. J., M. C. Márquez et J. Núñez. « c-Graded filiform Lie algebras ». Bulletin of the Brazilian Mathematical Society, New Series 36, n^o 1 (avril 2005) : 59–77. http://dx.doi.org/10.1007/s00574-005-0028-0.

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Caranti, A., et S. Mattarei. « Some thin Lie algebras related to Albert-Frank algebras and algebras of maximal class ». Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 67, n^o 2 (octobre 1999) : 157–84. http://dx.doi.org/10.1017/s1446788700001142.

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AbstractWe investigate a class of infinite-dimensional, modular, graded Lie algebra in which the homogeneous components have dimension at most two. A subclass of these algebras can be obtained via a twisted loop algebra construction from certain finite-dimensional, simple Lie algebras of Albert-Frank type.Another subclass of these algebras is strictly related to certain graded Lie algebras of maximal class, and exhibits a wide range of behaviours.

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Mazorchuk, Volodymyr, et Kaiming Zhao. « Graded simple Lie algebras and graded simple representations ». manuscripta mathematica 156, n^o 1-2 (4 août 2017) : 215–40. http://dx.doi.org/10.1007/s00229-017-0960-5.

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Zhao, Kaiming. « Simple Lie color algebras from graded associative algebras ». Journal of Algebra 269, n^o 2 (novembre 2003) : 439–55. http://dx.doi.org/10.1016/s0021-8693(02)00564-1.

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YATSUI, Tomoaki. « On pseudo-product graded Lie algebras ». Hokkaido Mathematical Journal 17, n^o 3 (octobre 1988) : 333–43. http://dx.doi.org/10.14492/hokmj/1381517817.

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Caranti, A., S. Mattarei et M. F. Newman. « Graded Lie Algebras of Maximal Class ». Transactions of the American Mathematical Society 349, n^o 10 (1997) : 4021–51. http://dx.doi.org/10.1090/s0002-9947-97-02005-9.

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Gómez, J. R., et A. Jiménez-Merchán. « Naturally graded quasi-filiform Lie algebras ». Journal of Algebra 256, n^o 1 (octobre 2002) : 211–28. http://dx.doi.org/10.1016/s0021-8693(02)00130-8.

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Leznov, A. N., et M. V. Savel'ev. « Nonlinear equations and graded Lie algebras ». Journal of Soviet Mathematics 36, n^o 6 (mars 1987) : 699–721. http://dx.doi.org/10.1007/bf01085505.

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Bøgvad, Rikard, et Carl Jacobsson. « Graded lie algebras of depth one ». Manuscripta Mathematica 66, n^o 1 (décembre 1990) : 153–59. http://dx.doi.org/10.1007/bf02568488.

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Bierwirth, Hannes, et Mercedes Siles Molina. « Lie ideals of graded associative algebras ». Israel Journal of Mathematics 191, n^o 1 (5 décembre 2011) : 111–36. http://dx.doi.org/10.1007/s11856-011-0201-7.

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Weigel, Th. « Graded Lie algebras of type FP ». Israel Journal of Mathematics 205, n^o 1 (5 décembre 2014) : 185–209. http://dx.doi.org/10.1007/s11856-014-1131-y.

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