184 results on '"Restricted Lie algebra"'
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2. Naturality and innerness for morphisms of compact groups and (restricted) Lie algebras.
- Author
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Chirvasitu, Alexandru
- Subjects
- *
LIE algebras , *COMPACT groups , *ENDOMORPHISMS , *LIE groups , *BIJECTIONS , *AUTOMORPHISMS - Abstract
An extended derivation (endomorphism) of a (restricted) Lie algebra L is an assignment of a derivation (respectively) of L' for any (restricted) Lie morphism f:L\to L', functorial in f in the obvious sense. We show that (a) the only extended endomorphisms of a restricted Lie algebra are the two obvious ones, assigning either the identity or the zero map of L' to every f; and (b) if L is a Lie algebra in characteristic zero or a restricted Lie algebra in positive characteristic, then L is in canonical bijection with its space of extended derivations (so the latter are all, in a sense, inner). These results answer a number of questions of G. Bergman. In a similar vein, we show that the individual components of an extended endomorphism of a compact connected group are either all trivial or all inner automorphisms. [ABSTRACT FROM AUTHOR] more...
- Published
- 2024
- Full Text
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Catalog
3. Five-dimensional p-nilpotent restricted Lie algebras over algebraically closed fields of characteristic p > 3.
- Author
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Maletesta, Nicola and Siciliano, Salvatore
- Subjects
- *
LIE algebras , *CLASSIFICATION - Abstract
A classification of p -nilpotent 5-dimensional restricted Lie algebras over algebraically closed fields of characteristic p > 3 is provided. This is achieved by employing a natural restricted analogue of the known method by Skjelbred and Sund for classifying ordinary nilpotent Lie algebras as central extensions of Lie algebras of smaller dimension. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
- View/download PDF
4. Nilpotent varieties of some finite dimensional restricted Lie algebras
- Author
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Chen, Cong, Premet, Alexander, and Bazlov, Yuri
- Subjects
510 ,Restricted Lie algebra ,Semisimple Lie algebra ,The Zassenhaus algebra ,The minimal p-envelope ,Nilpotent variety ,Nilpotent element - Abstract
In the late 1980s, A. Premet conjectured that the variety of nilpotent elements of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic p > 0 is irreducible. This conjecture remains open, but it is known to hold for a large class of simple restricted Lie algebras, e.g. for Lie algebras of connected algebraic groups, and for Cartan series W, S and H. In this thesis we start by proving that Premet's conjecture can be reduced to the semisimple case. The proof is straightforward. However, the reduction of the semisimple case to the simple case is very non-trivial in prime characteristic as semisimple Lie algebras are not always direct sums of simple ideals. Then we consider some semisimple restricted Lie algebras. Under the assumption that p > 2, we prove that Premet's conjecture holds for the semisimple restricted Lie algebra whose socle involves the special linear Lie algebra sl_2 tensored by the truncated polynomial ring k[X]/(X^p). Then we extend this example to the semisimple restricted Lie algebra whose socle involves S ⊗ O(m;1), where S is any simple restricted Lie algebra such that ad S=Der S and its nilpotent variety N(S) is irreducible, and O(m;1)=k[X₁,...,X_m]/(X₁^p,...,X_m^p) is the truncated polynomial ring in m ≥ 2 variables. In the final chapter we assume that p > 3. We confirm Premet's conjecture for the minimal p-envelope W(1;n)_p of the Zassenhaus algebra W(1;n) for all n ⊂N≥₂. This is the main result of the author's research paper which was published in the Journal of Algebra and its Applications. more...
- Published
- 2019
5. On the subalgebra lattice of a restricted Lie algebra.
- Author
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Páez-Guillán, Pilar, Siciliano, Salvatore, and Towers, David A.
- Subjects
- *
ALGEBRA - Abstract
In this paper we study the lattice of restricted subalgebras of a restricted Lie algebra. In particular, we consider those algebras in which this lattice is dually atomistic, lower or upper semimodular, or in which every restricted subalgebra is a quasi-ideal. The fact that there are one-dimensional subalgebras which are not restricted results in some of these conditions being weaker than for the corresponding conditions in the non-restricted case. [ABSTRACT FROM AUTHOR] more...
- Published
- 2023
- Full Text
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6. Nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2.
- Author
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Krutov, Andrey, Lebedev, Alexei, Leites, Dimitry, and Shchepochkina, Irina
- Subjects
- *
LIE superalgebras , *BILINEAR forms , *LIE algebras - Abstract
As is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2. We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or 0 | 1 , or 1 | 1 ; it is equal to 1 | 1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic ≠2, see arXiv:1806.05505). We give examples of NISes on deformations (with both even and odd parameters) of several simple finite-dimensional Lie superalgebras in characteristic 2. We also recall examples of multiple NISes on simple Lie algebras over non-closed fields. [ABSTRACT FROM AUTHOR] more...
- Published
- 2022
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7. Restricted Lie algebras having a distributive lattice of restricted subalgebras.
- Author
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Maletesta, Nicola, Páez-Guillán, Pilar, and Siciliano, Salvatore
- Subjects
- *
DISTRIBUTIVE lattices , *POLYNOMIAL rings , *LIE algebras - Abstract
Let L be a restricted Lie algebra over a field of characteristic p>0. We investigate the structure of L when its lattice S (L) of restricted subalgebras satisfies some prescribed properties. In particular, we establish when S (L) is distributive. The special form that this result takes when F is either the field with p elements or an algebraically closed field is also discussed. Furthermore, we establish when S (L) is Boolean or the two-elements lattice. [ABSTRACT FROM AUTHOR] more...
- Published
- 2021
- Full Text
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8. Endotrivial modules for nilpotent restricted Lie algebras.
- Author
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Benson, David J. and Carlson, Jon F.
- Abstract
Let g be a finite dimensional nilpotent p-restricted Lie algebra over a field k of characteristic p. For p ⩾ 5 , we show that every endotrivial g -module is a direct sum of a syzygy of the trivial module and a projective module. The proof includes a theorem that the intersection of the maximal linear subspaces of the null cone of a nilpotent restricted p-Lie algebra for p ⩾ 5 has dimension at least two. We give an example to show that the statement about endotrivial modules is false in characteristic two. In characteristic three, another example shows that our proof fails, and we do not know a characterization of the endotrivial modules in this case. [ABSTRACT FROM AUTHOR] more...
- Published
- 2020
- Full Text
- View/download PDF
9. Restricted one-dimensional central extensions of the restricted filiform Lie algebras [formula omitted].
- Author
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Evans, Tyler J. and Fialowski, Alice
- Subjects
- *
LIE algebras , *COHOMOLOGY theory , *LINEAR algebra , *MATRICES (Mathematics) , *MATHEMATICAL models - Abstract
Abstract Consider the filiform Lie algebra m 0 with nonzero Lie brackets [ e 1 , e i ] = e i + 1 for 1 < i < p , where the characteristic of the field F is p > 0. We show that there is a family m 0 λ (p) of restricted Lie algebra structures parameterized by elements λ ∈ F p. We explicitly describe both the ordinary and restricted 1-cohomology spaces and show that for p ≥ 3 these spaces are equal. We also describe the ordinary and restricted 2-cohomology spaces and interpret our results in the context of one-dimensional central extensions. [ABSTRACT FROM AUTHOR] more...
- Published
- 2019
- Full Text
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10. Varieties of a class of elementary subalgebras
- Author
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Yang Pan and Yanyong Hong
- Subjects
Physics ,General Mathematics ,Dimension (graph theory) ,Subalgebra ,elementary subalgebras ,commuting roots ,Type (model theory) ,Combinatorics ,Restricted Lie algebra ,Algebraic group ,Lie algebra ,QA1-939 ,Variety (universal algebra) ,Algebraically closed field ,Mathematics::Representation Theory ,irreducible components ,Mathematics - Abstract
Let $ G $ be a connected standard simple algebraic group of type $ C $ or $ D $ over an algebraically closed field $ \Bbbk $ of positive characteristic $ p > 0 $, and $ \mathfrak{g}: = \mathrm{Lie}(G) $ be the Lie algebra of $ G $. Motivated by the variety of $ \mathbb{E}(r, \mathfrak{g}) $ of $ r $-dimensional elementary subalgebras of a restricted Lie algebra $ \mathfrak{g} $, in this paper we characterize the irreducible components of $ \mathbb{E}(\mathrm{rk}_{p}(\mathfrak{g})-1, \mathfrak{g}) $ where the $ p $-rank $ \mathrm{rk}_{p}(\mathfrak{g}) $ is defined to be the maximal dimension of an elementary subalgebra of $ \mathfrak{g} $. more...
- Published
- 2022
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11. A Note on the Rank of a Restricted Lie Algebra.
- Author
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Chang, Hao
- Subjects
- *
LIE algebras , *DIMENSIONS , *ABSTRACT algebra , *MATHEMATICS , *GROUP theory - Abstract
In this short note, we study the rank of a restricted Lie algebra (𝔤, [p]), and give some applications which concern the dimensions of non-trivial irreducible modules. We also compute the rank of the restricted contact algebra. [ABSTRACT FROM AUTHOR] more...
- Published
- 2018
- Full Text
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12. Restricted Lie (Super)Algebras in Characteristic 3.
- Author
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Bouarroudj, S., Krutov, A. O., Lebedev, A. V., Leites, D. A., and Shchepochkina, I. M.
- Subjects
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LIE algebras , *SUPERALGEBRAS , *DEFORMATIONS of singularities , *MODULES (Algebra) , *DIVERGENCE theorem - Abstract
We give explicit formulas proving that the following Lie (super)algebras are restricted: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, simple subquotients over an algebraically closed field of characteristic 3 of these (super)algebras (under certain conditions), and deformed divergence-free Lie superalgebras of a certain type with any finite number of indeterminates in any characteristic. [ABSTRACT FROM AUTHOR] more...
- Published
- 2018
- Full Text
- View/download PDF
13. Stable invariance of the restricted Lie algebra structure of Hochschild cohomology
- Author
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Briggs, Benjamin, Rubio y Degrassi, Lleonard, Briggs, Benjamin, and Rubio y Degrassi, Lleonard
- Abstract
We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby, we obtain a number of positive characteristic stable invariants, such as the p-toral rank of HH1(A, A). We also prove a more general result concerning Iwanaga-Gorenstein algebras, using a generaliza-tion of stable equivalences of Morita type. Several applications are given to commutative algebra and modular representation theory.These results are proven by first establishing the stable invariance of the B∞-structure of the Hochschild cochain complex. In the appendix, we explain how the p-power operation on Hochschild cohomology can be seen as an artifact of this B∞-structure. In particular, we establish well-definedness of the p-power operation, following some-originally topological-methods due to May, Cohen and Turchin, using the language of operads. more...
- Published
- 2022
- Full Text
- View/download PDF
14. Fractal nil graded Lie, associative, Poisson, and Jordan superalgebras
- Author
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Ivan P. Shestakov and V. M. Petrogradsky
- Subjects
Poisson superalgebra ,Pure mathematics ,Algebra and Number Theory ,Mathematics::Rings and Algebras ,010102 general mathematics ,Graded ring ,SUPERÁLGEBRAS DE LIE ,Lie superalgebra ,Grigorchuk group ,01 natural sciences ,Superalgebra ,Restricted Lie algebra ,0103 physical sciences ,Lie algebra ,010307 mathematical physics ,0101 mathematics ,Group theory ,Mathematics - Abstract
The Grigorchuk and Gupta-Sidki groups play fundamental role in modern group theory. They are natural examples of self-similar finitely generated periodic groups. The first author constructed their analogue in case of restricted Lie algebras of characteristic 2 [50] , Shestakov and Zelmanov extended this construction to an arbitrary positive characteristic [68] . Thus, we have examples of finitely generated restricted Lie algebras with a nil p-mapping. In characteristic zero, similar examples of Lie and Jordan algebras do not exist by results of Martinez and Zelmanov [43] and [78] . The first author constructed analogues of the Grigorchuk and Gupta-Sidki groups in the world of Lie superalgebras of arbitrary characteristic, the virtue of that construction is that Lie superalgebras have clear monomial bases [51] , they have slow polynomial growth. As an analogue of periodicity, Z 2 -homogeneous elements are ad-nilpotent. A recent example of a Lie superalgebra is of linear growth, of finite width 4, just infinite but not hereditary just infinite [13] . By that examples, an extension of the result of Martinez and Zelmanov [43] for Lie superalgebras of characteristic zero is not valid. Now, we construct a just infinite fractal 3-generated Lie superalgebra Q over arbitrary field, which gives rise to an associative hull A, a Poisson superalgebra P, and two Jordan superalgebras J and K, the latter being a factor algebra of J. In case char K ≠ 2 , A has a natural filtration, which associated graded algebra has a structure of a Poisson superalgebra such that gr A ≅ P , also P admits an algebraic quantization using a deformed superalgebra A ( t ) . The Lie superalgebra Q is finely Z 3 -graded by multidegree in the generators, A, P are also Z 3 -graded, while J and K are Z 4 -graded by multidegree in four generators. By virtue of our construction, these five superalgebras have clear monomial bases and slow polynomial growth. We describe multihomogeneous coordinates of bases of Q, A, P in space as bounded by “almost cubic paraboloids”. We determine a similar hypersurface in R 4 that bounds monomials of J and K. Constructions of the paper can be applied to Lie (super)algebras studied before to obtain Poisson and Jordan superalgebras as well. The algebras Q, A, and the algebras without unit P o , J o , K o are direct sums of two locally nilpotent subalgebras and there are continuum such decompositions. Also, Q = Q 0 ¯ ⊕ Q 1 ¯ is a nil graded Lie superalgebra, so, Q again shows that an extension of the result of Martinez and Zelmanov for Lie superalgebras of characteristic zero is not valid. In case char K = 2 , Q has a structure of a restricted Lie algebra with a nil p-mapping. The Jordan superalgebra K is nil finely Z 4 -graded, in contrast with non-existence of such examples (roughly speaking, analogues of the Grigorchuk group) of Jordan algebras in characteristic distinct from 2 [78] . Also, K is of slow polynomial growth, just infinite, but not hereditary just infinite. We call the superalgebras Q, A, P, J, K fractal because they contain infinitely many copies of themselves. more...
- Published
- 2021
- Full Text
- View/download PDF
15. Moduli of Lie p-algebras
- Author
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Bouillet, Alice, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro) more...
- Subjects
Mathematics - Algebraic Geometry ,FOS: Mathematics ,moduli space ,positive characteristic ,group scheme of height one ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Algebraic Geometry (math.AG) ,restricted Lie algebra - Abstract
In this paper, we study moduli spaces of finite-dimensional Lie algebras with flat center, proving that the forgetful map from Lie p-algebras to Lie algebras is an affine fibration, and we point out a new case of existence of a p-mapping. Then we illustrate these results for the special case of Lie algebras of rank 3, whose moduli space we build and study over Z. We extend the classical equivalence of categories between locally free Lie p-algebras of finite rank with finite locally free group schemes of height 1, showing that the centers of these objects correspond to each other. We finish by analysing the smoothness of the moduli of p-Lie algebras of rank 3, in particular identifying some smooth components., The introduction has been changed, in order to give more motivations more...
- Published
- 2022
16. Restricted one-dimensional central extensions of restricted simple Lie algebras.
- Author
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Evans, Tyler J. and Fialowski, Alice
- Subjects
- *
SPANNING trees , *ARBITRARY constants , *LIE algebras , *COCYCLES , *COHOMOLOGY theory , *GROUP extensions (Mathematics) - Abstract
We study the restricted one-dimensional central extensions of an arbitrary finite dimensional restricted simple Lie algebra for p ≥ 5 . For H 2 ( g ) = 0 , we explicitly describe the cocycles spanning H ⁎ 2 ( g ) , and in the case H 2 ( g ) ≠ 0 , we give a procedure to describe a basis for H ⁎ 2 ( g ) . [ABSTRACT FROM AUTHOR] more...
- Published
- 2017
- Full Text
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17. Twisted modules for affine vertex algebras over fields of prime characteristic
- Author
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Qiang Mu and Haisheng Li
- Subjects
Vertex (graph theory) ,Algebra and Number Theory ,Coprime integers ,010102 general mathematics ,Subalgebra ,Automorphism ,01 natural sciences ,Combinatorics ,Restricted Lie algebra ,0103 physical sciences ,Lie algebra ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Quotient ,Mathematics - Abstract
In this paper, twisted modules for modular affine vertex algebras V g ˆ ( l , 0 ) and for their quotient vertex algebras V g ˆ χ ( l , 0 ) with g a restricted Lie algebra are studied. Let σ be an automorphism of g and let T be a positive integer relatively prime with the characteristic p such that σ T = 1 . It is proved that 1 T N -graded irreducible σ-twisted V g ˆ 0 ( l , 0 ) -modules are in one-to-one correspondence with irreducible modules for the restricted enveloping algebra u ( g 0 ) , where g 0 is the subalgebra of σ-fixed points in g . It is also proved that when g = h is abelian, the twisted Heisenberg Lie algebra h ˆ [ σ ] is actually isomorphic to the untwisted Heisenberg Lie algebra h ˆ , unlike in the case of characteristic zero. Furthermore, for any nonzero level l, irreducible σ-twisted L h ˆ ( l , 0 ) -modules are explicitly classified and the complete reducibility of every σ-twisted L h ˆ ( l , 0 ) -module is obtained. more...
- Published
- 2020
- Full Text
- View/download PDF
18. RESTRICTED COMTRANS ALGEBRAS OVER SMALL ODD PRIMES.
- Author
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Daǧli, Mehmet, Garcia, Luis A., and Smith, Jonathan D. H.
- Subjects
PRIME numbers ,LIE algebras ,TANGENT bundles ,MANIFOLDS (Mathematics) ,REPRESENTATION theory - Abstract
Comtrans algebras, as analogues of Lie algebras, provide tangent bundle structure corresponding to web geometry in a manifold. In this article, restricted comtrans algebras over rings of small odd prime characteristic are introduced, as analogues of restricted Lie algebras. It is shown that their representations are equivalent to modules over a restricted universal enveloping algebra. [ABSTRACT FROM AUTHOR] more...
- Published
- 2016
- Full Text
- View/download PDF
19. Restricted enveloping algebras whose skew and symmetric elements are Lie metabelian.
- Author
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Siciliano, Salvatore and Usefi, Hamid
- Subjects
- *
UNIVERSAL enveloping algebras , *MATHEMATICAL symmetry , *LIE algebras , *POLYNOMIALS , *FREE metabelian groups - Abstract
Let L be a restricted Lie algebra over a field of characteristic and denote by its restricted enveloping algebra. We establish when the symmetric or skew elements of under the principal involution are Lie metabelian. [ABSTRACT FROM AUTHOR] more...
- Published
- 2016
- Full Text
- View/download PDF
20. Restricted Lie (super)algebras, central extensions of non-associative algebras and some tapas
- Author
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Ladra González, Manuel, Kaygorodov, Ivan, Universidade de Santiago de Compostela. Escola de Doutoramento Internacional (EDIUS), Universidade de Santiago de Compostela. Programa de Doutoramento en Matemáticas, Páez Guillán, María Pilar, Ladra González, Manuel, Kaygorodov, Ivan, Universidade de Santiago de Compostela. Escola de Doutoramento Internacional (EDIUS), Universidade de Santiago de Compostela. Programa de Doutoramento en Matemáticas, and Páez Guillán, María Pilar more...
- Abstract
The general framework of this dissertation is the theory of non-associative algebras. We tackle diverse problems regarding restricted Lie algebras and superalgebras, central extensions of different classes of algebras and crossed modules of Lie superalgebras. Namely, we study the relations between the structural properties of a restricted Lie algebra and those of its lattice of restricted subalgebras; we define a non-abelian tensor product for restricted Lie superalgebras and for graded ideal crossed submodules of a crossed module of Lie superalgebras, and explore their properties from structural, categorical and homological points of view; we employ central extensions to classify nilpotent bicommutative algebras; and we compute central extensions of the associative null-filiform algebras and of axial algebras. Also, we include a final chapter devoted to compare the two main methods (Rabinowitsch's trick and saturation) to introduce negative conditions in the standard procedures of the theory of automated proving and discovery. more...
- Published
- 2021
21. Restricted one-dimensional central extensions of the restricted filiform Lie algebras m0λ(p)
- Author
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Alice Fialowski and Tyler J. Evans
- Subjects
Numerical Analysis ,Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Parameterized complexity ,Context (language use) ,Field (mathematics) ,010103 numerical & computational mathematics ,01 natural sciences ,Representation theory ,Restricted Lie algebra ,Lie algebra ,Discrete Mathematics and Combinatorics ,Geometry and Topology ,0101 mathematics ,Mathematics - Abstract
Consider the filiform Lie algebra m 0 with nonzero Lie brackets [ e 1 , e i ] = e i + 1 for 1 i p , where the characteristic of the field F is p > 0 . We show that there is a family m 0 λ ( p ) of restricted Lie algebra structures parameterized by elements λ ∈ F p . We explicitly describe both the ordinary and restricted 1-cohomology spaces and show that for p ≥ 3 these spaces are equal. We also describe the ordinary and restricted 2-cohomology spaces and interpret our results in the context of one-dimensional central extensions. more...
- Published
- 2019
- Full Text
- View/download PDF
22. Ideal growth in metabelian Lie P-algebras.
- Author
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Petrogradsky, V. and Subbotin, I.
- Subjects
- *
FREE metabelian groups , *FUNCTION algebras , *FINITE fields , *LAMBDA algebra , *POLYNOMIALS - Abstract
Consider a finitely generated restricted Lie algebra L over the finite field F and, given n ≥ 0, denote the number of restricted ideals H ⊂ L with $${\dim _{{F_q}}}$$ L/H = n by c( L). We show for the free metabelian restricted Lie algebra L of finite rank that the ideal growth sequence grows superpolynomially; namely, there exist positive constants λ and λ such that $${q^{{\lambda _1}{n^2}}} \leqslant {c_n}\left( L \right) \leqslant {q^{{\lambda _2}{n^2}}}$$ for n large enough. [ABSTRACT FROM AUTHOR] more...
- Published
- 2015
- Full Text
- View/download PDF
23. The variety of nilpotent elements and invariant polynomial functions on the special algebra Sn.
- Author
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Wei, Junyan, Chang, Hao, and Lu, Xin
- Subjects
- *
VARIETIES (Universal algebra) , *NILPOTENT groups , *INVARIANTS (Mathematics) , *POLYNOMIALS , *MATHEMATICAL functions , *LIE algebras - Abstract
In the study of the variety of nilpotent elements in a Lie algebra, Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. In this paper, with the assumption that the ground field is algebraically closed of characteristic p > 3, we confirm this conjecture for the Lie algebras of Cartan type S˜ n and Sn. Moreover, we show that the variety of nilpotent elements in Sn is a complete intersection. Motivated by the proof of the irreducibility, we describe explicitly the ring of invariant polynomial functions on Sn. [ABSTRACT FROM AUTHOR] more...
- Published
- 2015
- Full Text
- View/download PDF
24. The variety of nilpotent elements and invariant polynomial functions on the special algebra Sn.
- Author
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Wei, Junyan, Chang, Hao, and Lu, Xin
- Subjects
VARIETIES (Universal algebra) ,NILPOTENT groups ,MATHEMATICAL invariants ,POLYNOMIALS ,MATHEMATICAL functions ,LIE algebras - Abstract
In the study of the variety of nilpotent elements in a Lie algebra, Premet conjectured that this variety is irreducible for any finite dimensional restricted Lie algebra. In this paper, with the assumption that the ground field is algebraically closed of characteristic p > 3, we confirm this conjecture for the Lie algebras of Cartan type S˜
n and Sn . Moreover, we show that the variety of nilpotent elements in Sn is a complete intersection. Motivated by the proof of the irreducibility, we describe explicitly the ring of invariant polynomial functions on Sn . [ABSTRACT FROM AUTHOR] more...- Published
- 2015
- Full Text
- View/download PDF
25. c-Sections of Lie algebras.
- Author
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Towers, David A.
- Subjects
- *
LIE algebras , *ISOMORPHISM (Mathematics) , *NILPOTENT groups , *ABSTRACT algebra , *LINEAR algebra , *MATHEMATICAL analysis - Abstract
Let M be a maximal subalgebra of a Lie algebra L and A / B a chief factor of L such that B ⊆ M and A ⊈ M . We call the factor algebra M ∩ A / B a c -section of M . All such c -sections are isomorphic, and this concept is related those of c -ideals and ideal index previously introduced by the author. Properties of c -sections are studied and some new characterizations of solvable Lie algebras are obtained. [ABSTRACT FROM AUTHOR] more...
- Published
- 2015
- Full Text
- View/download PDF
26. The p.i.m.s for the restricted Zassenhaus algebras in characteristic 2.
- Author
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Lancellotti, B. and Weigel, Th.
- Abstract
It is shown that for the restricted Zassenhaus algebra $${\mathfrak{W} = \mathfrak{W}(1; n)}$$ , n > 1, defined over an algebraically closed field $${\mathbb{F}}$$ of characteristic 2, any projective indecomposable restricted $${\mathfrak{W}}$$ -module has maximal possible dimension $${2^{2^n-1}}$$ , and thus is isomorphic to some induced module $${{\rm idn}^{\mathfrak{W}}_{\mathfrak{t}}(\mathbb{F}(\mu))}$$ for some torus of maximal dimension $${\mathfrak{t}}$$ . This phenomenon is in contrast to the behavior of finite-dimensional non-solvable restricted Lie algebras in characteristic p > 3 (cf. Feldvoss et al. Restricted Lie algebras with maximal 0-pim, , Theorem 6.3). [ABSTRACT FROM AUTHOR] more...
- Published
- 2015
- Full Text
- View/download PDF
27. Criteria for the existence of a Jordan–Chevalley–Seligman decomposition.
- Author
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Kim, Kyoung-Tark
- Subjects
- *
EXISTENCE theorems , *MATHEMATICAL decomposition , *CHARACTERISTIC functions , *NILPOTENT groups , *SEMISIMPLE Lie groups , *POLYNOMIALS - Abstract
Let ( L , [ p ] ) be a finite dimensional restricted Lie algebra over a field K of positive characteristic p . A Jordan–Chevalley–Seligman decomposition of x ∈ L is a unique expression of x as a sum of commuting semisimple and nilpotent elements in L . It is well-known that each x ∈ L has such a decomposition when K is perfect. When K is non-perfect, the present paper gives several criteria for the existence of a Jordan–Chevalley–Seligman decomposition for a given x ∈ L as well as for determining when an element in the restricted subalgebra generated by x is semisimple or nilpotent. [ABSTRACT FROM AUTHOR] more...
- Published
- 2015
- Full Text
- View/download PDF
28. TORSORS AND THE QUILLEN-BARR-BECK COHOMOLOGY FOR RESTRICTED LIE ALGEBRAS.
- Author
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DOKAS, IOANNIS
- Subjects
- *
COHOMOLOGY theory , *LIE algebras , *MATHEMATICAL sequences , *MODULES (Algebra) , *MEMORY - Abstract
In this paper we study Duskin-Glenn torsor cohomology in the context of restricted Lie algebras. In particular, we give an interpretation of the torsor cohomology groups which appear in Cegarra-Aznar’s eight-term exact sequence. Thus, we prove a classification theorem for the second Quillen-Barr-Beck cohomology in terms of 2-fold extensions of restricted Lie algebras. This paper is dedicated to the memory of Jean-Louis Loday. [ABSTRACT FROM AUTHOR] more...
- Published
- 2015
- Full Text
- View/download PDF
29. Restricted Lie algebras having a distributive lattice of restricted subalgebras
- Author
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Nicola Maletesta, Salvatore Siciliano, Pilar Páez-Guillán, Maletesta, Nicola, Pilar, Paez-Guillan, and Siciliano, Salvatore
- Subjects
Restricted Lie algebra ,Pure mathematics ,Algebra and Number Theory ,Structure (category theory) ,Field (mathematics) ,Distributive lattice ,010103 numerical & computational mathematics ,Lattice (discrete subgroup) ,distributive lattice ,01 natural sciences ,Lie algebra ,restricted subalgebra ,0101 mathematics ,Mathematics - Abstract
Let L be a restricted Lie algebra over a field of characteristic p>0. We investigate the structure of L when its lattice S(L) of restricted subalgebras satisfies some prescribed properties. In particular, we establish when S(L) is distributive. The special form that this result takes when F is either the field with p elements or an algebraically closed field is also discussed. Furthermore, we establish when S(L) is Boolean or the two-elements lattice. more...
- Published
- 2021
30. Restricted Lie (super)algebras, central extensions of non-associative algebras and some tapas
- Author
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Páez Guillán, María Pilar, Ladra González, Manuel, Kaygorodov, Ivan, Universidade de Santiago de Compostela. Escola de Doutoramento Internacional (EDIUS), and Universidade de Santiago de Compostela. Programa de Doutoramento en Matemáticas more...
- Subjects
Restricted Lie algebra ,non-abelian tensor product ,Investigación::12 Matemáticas::1201 Algebra::120107 Algebra homologica [Materias] ,automated proving and discovery ,Mathematics::Rings and Algebras ,restricted Lie superalgebra ,central extension ,Investigación::12 Matemáticas::1201 Algebra::120109 Algebra de Lie [Materias] ,Mathematics::Representation Theory ,lattice of restricted subalgebras ,Investigación::12 Matemáticas::1201 Algebra::120112 Algebras no asociativas [Materias] - Abstract
The general framework of this dissertation is the theory of non-associative algebras. We tackle diverse problems regarding restricted Lie algebras and superalgebras, central extensions of different classes of algebras and crossed modules of Lie superalgebras. Namely, we study the relations between the structural properties of a restricted Lie algebra and those of its lattice of restricted subalgebras; we define a non-abelian tensor product for restricted Lie superalgebras and for graded ideal crossed submodules of a crossed module of Lie superalgebras, and explore their properties from structural, categorical and homological points of view; we employ central extensions to classify nilpotent bicommutative algebras; and we compute central extensions of the associative null-filiform algebras and of axial algebras. Also, we include a final chapter devoted to compare the two main methods (Rabinowitsch's trick and saturation) to introduce negative conditions in the standard procedures of the theory of automated proving and discovery. more...
- Published
- 2021
31. Stable invariance of the restricted Lie algebra structure of Hochschild cohomology
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Benjamin Briggs and Lleonard Rubio y Degrassi
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16E40, 16D90 (Primary) 17B50, 13D03, 18D50 (Secondary) ,stable equivalence of Morita type ,General Mathematics ,Mathematics - Rings and Algebras ,singularity category ,Mathematics::Algebraic Topology ,Hochschild cohomology ,B-infinity algebra ,Rings and Algebras (math.RA) ,Mathematics::K-Theory and Homology ,FOS: Mathematics ,Representation Theory (math.RT) ,Gerstenhaber bracket ,Mathematics - Representation Theory ,restricted Lie algebra - Abstract
We show that the restricted Lie algebra structure on Hochschild cohomology is invariant under stable equivalences of Morita type between self-injective algebras. Thereby we obtain a number of positive characteristic stable invariants, such as the $p$-toral rank of $\mathrm{HH}^1(A,A)$. We also prove a more general result concerning Iwanaga-Gorenstein algebras, using a more general notion of stable equivalences of Morita type. Several applications are given to commutative algebra and modular representation theory. These results are proven by first establishing the stable invariance of the $B_\infty$-structure of the Hochschild cochain complex. In the appendix we explain how the $p$-power operation on Hochschild cohomology can be seen as an artifact of this $B_\infty$-structure. In particular, we establish well-definedness of the $p$-power operation, following some -- originally topological -- methods due to May, Cohen and Turchin, using the language of operads., 20 pages, v3: small changes in the abstract and in the introduction more...
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- 2020
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32. A Note on the Rank of a Restricted Lie Algebra
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Hao Chang
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Algebra and Number Theory ,Applied Mathematics ,010102 general mathematics ,Mathematics - Rings and Algebras ,01 natural sciences ,Combinatorics ,Restricted Lie algebra ,Rings and Algebras (math.RA) ,0103 physical sciences ,FOS: Mathematics ,Rank (graph theory) ,010307 mathematical physics ,Representation Theory (math.RT) ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
In this short note, we study the rank of a restricted Lie algebra $(\mathfrak{g},[p])$ and give some applications, which concerns the dimensions of non-trivial irreducible modules. We also compute the rank of the restricted contact algebra $K(n)$. more...
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- 2018
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33. Ограниченные простые (супер)алгебры Ли в характеристике 3
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Sofian Bouarroudj, Aleksei Vital'evich Lebedev, Irina Shchepochkina, Andrey Krutov, and Dmitrii Aleksandrovich Leites
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Pure mathematics ,Restricted Lie algebra ,Mathematics - Published
- 2018
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34. OUTER RESTRICTED DERIVATIONS OF NILPOTENT RESTRICTED LIE ALGEBRAS.
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FELDVOSS, JÖRG, SICILIANO, SALVATORE, and WEIGEL, THOMAS
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- *
NILPOTENT groups , *LIE algebras , *IDEALS (Algebra) , *MODULES (Algebra) , *COHOMOLOGY theory , *AUTOMORPHISMS , *EXISTENCE theorems , *ISOMORPHISM (Mathematics) - Abstract
In this paper we prove that every finite-dimensional nilpotent restricted Lie algebra over a field of prime characteristic has an outer restricted derivation whose square is zero unless the restricted Lie algebra is a torus or it is one-dimensional or it is isomorphic to the three-dimensional Heisenberg algebra in characteristic two as an ordinary Lie algebra. This result is the restricted analogue of a result of Tôgô on the existence of nilpotent outer derivations of ordinary nilpotent Lie algebras in arbitrary characteristic and the Lie-theoretic analogue of a classical group-theoretic result of Gaschütz on the existence of p-power automorphisms of p-groups. As a consequence we obtain that every finite-dimensional non-toral nilpotent restricted Lie algebra has an outer restricted derivation. [ABSTRACT FROM AUTHOR] more...
- Published
- 2013
35. Inverse limits in representations of a restricted Lie algebra.
- Author
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Yao, Yu, Shu, Bin, and Li, Yi
- Subjects
- *
LIE algebras , *INFINITY (Mathematics) , *DIMENSIONAL analysis , *PROJECTIVE modules (Algebra) , *MATHEMATICAL forms , *GENERALIZATION - Abstract
Let ( g, [ p]) be a restricted Lie algebra over an algebraically closed field of characteristic p > 0. Then the inverse limits of 'higher' reduced enveloping algebras { u s ( g) | s ∈ ℕ} with χ running over g* make representations of g split into different 'blocks'. In this paper, we study such an infinite-dimensional algebra [Figure not available: see fulltext.] for a given χ ∈ g*. A module category equivalence is built between subcategories of U( g)- mod and [Figure not available: see fulltext.]. In the case of reductive Lie algebras, (quasi) generalized baby Verma modules and their properties are described. Furthermore, the dimensions of projective covers of simple modules with characters of standard Levi form in the generalized χ-reduced module category are precisely determined, and a higher reciprocity in the case of regular nilpotent is obtained, generalizing the ordinary reciprocity. [ABSTRACT FROM AUTHOR] more...
- Published
- 2012
- Full Text
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36. Cohomology of restricted Lie–Rinehart algebras and the Brauer group
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Dokas, Ioannis
- Subjects
- *
COHOMOLOGY theory , *LIE algebras , *BRAUER groups , *SEPARABLE algebras , *EXPONENTS , *COMMUTATIVE algebra , *MODULES (Algebra) - Abstract
Abstract: We give an interpretation of the Brauer group of a purely inseparable extension of exponent , in terms of restricted Lie–Rinehart cohomology. In particular, we define and study the category - of restricted Lie–Rinehart algebras over a commutative algebra . We define cotriple cohomology groups for - and a Beck -module. We classify restricted Lie–Rinehart extensions. Thus, we obtain a classification theorem for regular extensions considered by Hochschild. [Copyright &y& Elsevier] more...
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- 2012
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37. Differentiating the Weyl generic dimension formula with applications to support varieties
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Drupieski, Christopher M., Nakano, Daniel K., and Parshall, Brian J.
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- *
WEYL groups , *VARIETIES (Universal algebra) , *MODULES (Algebra) , *COXETER groups , *LIE algebras , *AFFINE algebraic groups , *MATHEMATICAL formulas - Abstract
Abstract: The authors compute the support varieties of all irreducible modules for the small quantum group , where is a finite-dimensional simple complex Lie algebra, and ζ is a primitive ℓ-th root of unity with ℓ larger than the Coxeter number of . The calculation employs the prior calculations and techniques of Ostrik and of Nakano, Parshall, and Vella, as well as deep results involving the validity of the Lusztig character formula for quantum groups and the positivity of parabolic Kazhdan–Lusztig polynomials for the affine Weyl group. Analogous support variety calculations are provided for the first Frobenius kernel of a reductive algebraic group scheme G defined over the prime field . [Copyright &y& Elsevier] more...
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- 2012
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38. Restricted Lie algebras all whose elements are semisimple.
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Chen, Liangyun, Xu, Xiaoning, and Zhang, Yongzheng
- Abstract
People studied the properties and structures of restricted Lie algebras all whose elements are semisimple. It is the main objective of this paper to continue the investigation in order to obtain deeper structure theorems. We obtain some sufficient conditions for the commutativity of restricted Lie algebras, generalize some results of R. Farnsteiner and characterize some properties of a finite-dimensional semisimple restricted Lie algebra all whose elements are semisimple. Moreover, we show that a centralsimple restricted Lie algebra all whose elements are semisimple over a field of characteristic p > 7 is a form of a classical Lie algebra. [ABSTRACT FROM AUTHOR] more...
- Published
- 2011
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39. Restricted Enveloping Algebras with Minimal Lie Derived Length.
- Author
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Catino, Francesco, Siciliano, Salvatore, and Spinelli, Ernesto
- Abstract
Let L be a non-abelian restricted Lie algebra over a field of characteristic p > 0 and let u( L) denote its restricted enveloping algebra. In Siciliano (Publ Math (Debr) 68:503-513, ) it was proved that if u( L) is Lie solvable then the Lie derived length of u( L) is at least ⌈log( p + 1)⌉. In the present paper we characterize the restricted enveloping algebras whose Lie derived length coincides with this lower bound. [ABSTRACT FROM AUTHOR] more...
- Published
- 2010
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40. Enumeration of maximal subalgebras in free restricted lie algebras.
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Petrogradskiĭ, V. M. and Smirnov, A. A.
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- *
LIE algebras , *LINEAR algebra , *FINITE fields , *MATHEMATICAL analysis , *RING theory - Abstract
Given a finitely generated restricted Lie algebra L over the finite field $$ \mathbb{F}_q $$ , and n ≥ 0, denote by a n ( L) the number of restricted subalgebras H ⊆ L with $$ \dim _{\mathbb{F} _q} $$ L/H = n. Denote by ã n ( L) the number of the subalgebras satisfying the maximality condition as well. Considering the free restricted Lie algebra L = F d of rank d ≥ 2, we find the asymptotics of ã n ( F d ) and show that it coincides with the asymptotics of a n ( F d ) which was found previously by the first author. Our approach is based on studying the actions of restricted algebras by derivations on the truncated polynomial rings. We establish that the maximal subalgebras correspond to the so-called primitive actions. This means that “almost all” restricted subalgebras H ⊂ F d of finite codimension are maximal, which is analogous to the corresponding results for free groups and free associative algebras. [ABSTRACT FROM AUTHOR] more...
- Published
- 2008
- Full Text
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41. Fine Hochschild invariants of derived categories for symmetric algebras
- Author
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Zimmermann, Alexander
- Subjects
- *
MATHEMATICS , *ALGEBRA , *MATHEMATICAL analysis , *HOMOLOGY (Biology) - Abstract
Abstract: Let A be a symmetric k-algebra over a perfect field k. Külshammer defined for any integer n a mapping on the degree 0 Hochschild cohomology and a mapping on the degree 0 Hochschild homology of A as adjoint mappings of the respective p-power mappings with respect to the symmetrising bilinear form. In an earlier paper it is shown that is invariant under derived equivalences. In the present paper we generalise the definition of to higher Hochschild homology and show the invariance of κ and its generalisation under derived equivalences. This provides fine invariants of derived categories. [Copyright &y& Elsevier] more...
- Published
- 2007
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42. On Restricted Leibniz Algebras.
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Dokas, Ioannis and Loday, Jean-Louis
- Subjects
ASSOCIATIVE algebras ,LIE algebras ,TENSOR products ,FROBENIUS algebras ,KOSZUL algebras ,FUNCTOR theory - Abstract
In this article we prove that in prime characteristic there is a functor - p-Leib from the category of diassociative algebras to the category of restricted Leibniz algebras, generalizing the functor from associative algebras to restricted Lie algebras. Moreover, we define the notion of restricted universal enveloping diassociative algebra Udp(𝔤) of a restricted Leibniz algebra 𝔤 and we show that Udp is left adjoint to the functor - p-Leib. We also construct the restricted enveloping algebra, which classifies the restricted Leibniz modules. In the last section we put a restricted pre-Lie structure on the tensor product of a Leibniz algebra by a Zinbiel algebra. [ABSTRACT FROM AUTHOR] more...
- Published
- 2006
- Full Text
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43. Restricted one-dimensional central extensions of restricted simple Lie algebras
- Author
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Alice Fialowski and Tyler J. Evans
- Subjects
Discrete mathematics ,Numerical Analysis ,Pure mathematics ,Algebra and Number Theory ,Simple Lie group ,010102 general mathematics ,Adjoint representation ,010103 numerical & computational mathematics ,(g,K)-module ,Killing form ,01 natural sciences ,Affine Lie algebra ,Graded Lie algebra ,Lie conformal algebra ,Restricted Lie algebra ,Discrete Mathematics and Combinatorics ,Geometry and Topology ,0101 mathematics ,Mathematics - Abstract
We study the restricted one-dimensional central extensions of an arbitrary finite dimensional restricted simple Lie algebra for p ≥ 5 . For H 2 ( g ) = 0 , we explicitly describe the cocycles spanning H ⁎ 2 ( g ) , and in the case H 2 ( g ) ≠ 0 , we give a procedure to describe a basis for H ⁎ 2 ( g ) . more...
- Published
- 2017
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44. On Some Restricted Lie Algebras.
- Author
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Jong-Sook Lee
- Subjects
- *
LIE algebras , *LINEAR algebra , *MATHEMATICS , *LOGIC , *GENERALIZED spaces - Abstract
In this paper we define and characterize a simply restricted Lie algebra g over a field of characteristic p > 2. [ABSTRACT FROM AUTHOR]
- Published
- 2005
45. On the Restricted Lie Algebra Structure of the Witt Lie Algebra in Finite Characteristic.
- Author
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Evans, T. and Fuchs, D.
- Abstract
The article contains an explicit formula for the restricted Lie algebra structure in the Witt Lie algebra over a field of finite characteristic. Some combinatorial lemmas can be of independent interest. [ABSTRACT FROM AUTHOR] more...
- Published
- 2002
- Full Text
- View/download PDF
46. Cohomology of Restricted Filiform Lie Algebras m2λ(p)
- Author
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Alice Fialowski and Tyler J. Evans
- Subjects
Physics ,Combinatorics ,Restricted Lie algebra ,Lie algebra ,Parameterized complexity ,Field (mathematics) ,Prime characteristic ,Geometry and Topology ,Lambda ,Mathematical Physics ,Analysis ,Cohomology - Abstract
For the $p$-dimensional filiform Lie algebra ${\mathfrak m}_2(p)$ over a field ${\mathbb F}$ of prime characteristic $p\ge 5$ with nonzero Lie brackets $[e_1,e_i] = e_{i+1}$ for $1
- Published
- 2019
- Full Text
- View/download PDF
47. On the first Hochschild cohomology of cocommutative Hopf algebras of finite representation type
- Author
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Hao Chang
- Subjects
General Mathematics ,010102 general mathematics ,Block (permutation group theory) ,Group algebra ,Type (model theory) ,Hopf algebra ,01 natural sciences ,Cohomology ,010101 applied mathematics ,Combinatorics ,Restricted Lie algebra ,Group scheme ,FOS: Mathematics ,0101 mathematics ,Algebraically closed field ,Representation Theory (math.RT) ,Mathematics - Representation Theory ,Mathematics - Abstract
Let $\mathscr{B}_0({\mathcal{G}})\subseteq k\,{\mathcal{G}}$ be the principal block algebra of the group algebra $k\,{\mathcal{G}}$ of an infinitesimal group scheme ${\mathcal{G}}$ over an algebraically closed field $k$ of characteristic ${\operatorname{char}}(k)=:p\geq 3$. We calculate the restricted Lie algebra structure of the first Hochschild cohomology ${\mathcal{L}}:={\operatorname{H}}^1(\mathscr{B}_0({\mathcal{G}}),\mathscr{B}_0({\mathcal{G}}))$ whenever $\mathscr{B}_0({\mathcal{G}})$ has finite representation type. As a consequence, we prove that the complexity of the trivial ${\mathcal{G}}$-module $k$ coincides with the maximal toral rank of ${\mathcal{L}}$. more...
- Published
- 2019
48. The classification of 5-dimensional p-nilpotent restricted Lie algebras over perfect fields, I
- Author
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Hamid Usefi and Iren Darijani
- Subjects
Discrete mathematics ,Pure mathematics ,Algebra and Number Theory ,Simple Lie group ,Mathematics::Rings and Algebras ,010102 general mathematics ,010103 numerical & computational mathematics ,Killing form ,01 natural sciences ,Affine Lie algebra ,Lie conformal algebra ,Graded Lie algebra ,Mathematics::Group Theory ,Adjoint representation of a Lie algebra ,Representation of a Lie group ,Restricted Lie algebra ,0101 mathematics ,Mathematics::Representation Theory ,Mathematics - Abstract
We first adapt a method due to Skjelbred–Sund to classify p-nilpotent restricted Lie algebras. It turns out that any p-nilpotent restricted Lie algebra of dimension n can be constructed as a central extension of a p-nilpotent restricted Lie algebra of dimension n − 1 . We apply these techniques to classify all p-nilpotent restricted Lie algebras of dimension 5 over a perfect field of characteristic p ⩾ 5 . more...
- Published
- 2016
- Full Text
- View/download PDF
49. Restricted Comtrans Algebras over Small Odd Primes
- Author
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Luis A. Garcia, Jonathan D. H. Smith, and Mehmet Daǧli
- Subjects
Discrete mathematics ,Pure mathematics ,Algebra and Number Theory ,Quantum group ,010102 general mathematics ,Non-associative algebra ,Universal enveloping algebra ,010103 numerical & computational mathematics ,01 natural sciences ,Affine Lie algebra ,Lie conformal algebra ,Adjoint representation of a Lie algebra ,Restricted Lie algebra ,0101 mathematics ,Generalized Kac–Moody algebra ,Mathematics - Abstract
Comtrans algebras, as analogues of Lie algebras, provide tangent bundle structure corresponding to web geometry in a manifold. In this article, restricted comtrans algebras over rings of small odd prime characteristic are introduced, as analogues of restricted Lie algebras. It is shown that their representations are equivalent to modules over a restricted universal enveloping algebra. more...
- Published
- 2016
- Full Text
- View/download PDF
50. Varieties of Elementary Subalgebras of Maximal Dimension for Modular Lie Algebras
- Author
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Jim Stark and Julia Pevtsova
- Subjects
Modular representation theory ,Pure mathematics ,business.industry ,010102 general mathematics ,Dimension (graph theory) ,Subalgebra ,Modular design ,01 natural sciences ,Restricted Lie algebra ,0103 physical sciences ,Lie algebra ,010307 mathematical physics ,0101 mathematics ,Abelian group ,Mathematics::Representation Theory ,business ,Mathematics - Abstract
Motivated by questions in modular representation theory, Carlson, Friedlander, and the first author introduced the varieties \(\mathbb E(r, \mathfrak g)\) of r-dimensional abelian p-nilpotent subalgebras of a p-restricted Lie algebra \(\mathfrak g\) in [8]. In this paper, we identify the varieties \(\mathbb E(r, \mathfrak g)\) for a reductive restricted Lie algebra \(\mathfrak g\) and r the maximal dimension of an abelian p-nilpotent subalgebra of \(\mathfrak g\). more...
- Published
- 2018
- Full Text
- View/download PDF
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