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Journal articles on the topic 'Automorphic Lie Algebras'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Knibbeler, Vincent, Sara Lombardo, and Jan A. Sanders. "Hereditary automorphic Lie algebras." Communications in Contemporary Mathematics 22, no. 08 (December 20, 2019): 1950076. http://dx.doi.org/10.1142/s0219199719500767.

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We show that automorphic Lie algebras which contain a Cartan subalgebra with a constant-spectrum, called hereditary, are completely described by 2-cocycles on a classical root system taking only two different values. This observation suggests a novel approach to their classification. By determining the values of the cocycles on opposite roots, we obtain the Killing form and the abelianization of the automorphic Lie algebra. The results are obtained by studying equivariant vectors on the projective line. As a byproduct, we describe a method to reduce the computation of the infinite-dimensional space of said equivariant vectors to a finite-dimensional linear computation and the determination of the ring of automorphic functions on the projective line.

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2

Knibbeler, Vincent, Sara Lombardo, and Jan A. Sanders. "Higher-Dimensional Automorphic Lie Algebras." Foundations of Computational Mathematics 17, no. 4 (April 11, 2016): 987–1035. http://dx.doi.org/10.1007/s10208-016-9312-1.

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3

Borcherds, Richard. "Automorphic forms and Lie Algebras." Current Developments in Mathematics 1996, no. 1 (1996): 1–36. http://dx.doi.org/10.4310/cdm.1996.v1996.n1.a1.

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4

Karabanov, A. "Automorphic algebras of dynamical systems and generalised In¨on¨u-Wigner contractions." Proceedings of the Komi Science Centre of the Ural Division of the Russian Academy of Sciences, no. 5 (December 20, 2022): 5–14. http://dx.doi.org/10.19110/1994-5655-2022-5-5-14.

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Lie algebras a with a complex underlying vector space V are studied that are automorphic with respect to a given linear dynamical system on V , i.e., a 1-parameter subgroup Gt ⊂ Aut(a) ⊂ GL(V ). Each automorphic algebra imparts a Lie algebraic structure to the vector space of trajectories of the group Gt. The basics of the general structure of automorphic algebras a are described in terms of the eigenspace decomposition of the operatorM ∈ der(a) that determines the dynamics. Symmetries encoded by the presence of nonabelian automorphic algebras are pointed out connected to conservation laws, spectral relations and root systems. It is shown that, for a given dynamics Gt, automorphic algebras can be found via a limit transition in the space of Lie algebras on V along the trajectories of the group Gt itself. This procedure generalises the well-known Inönü-Wigner contraction and links adjoint representations of automorphic algebras to isospectral Lax representations on gl(V ). These results can be applied to physically important symmetry groups and their representations, including classical and relativistic mechanics, open quantum dynamics and nonlinear evolution equations. Simple examples are given.

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5

Mikhalev, Alexander A., and Jie-Tai Yu. "Test Elements, Retracts and Automorphic Orbits of Free Algebras." International Journal of Algebra and Computation 08, no. 03 (June 1998): 295–310. http://dx.doi.org/10.1142/s0218196798000144.

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A nonzero element a of an algebra A is called a test element if for any endomorphism φ of A it follows from φ(a)=a that φ is an automorphism of the algebra A. A subalgebra B of A is a retract if there is an ideal I of A such that A=B ⊕ I. We consider the main types of free algebras with the Nielsen–Schreier property: free nonassociative algebras, free commutative and anti-commutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie p-algebras and p-superalgebras. For any free algebra F of finite rank of such type we prove that an element u is a test element if and only if it does not belong to any proper retract of F. Test elements for monomorphisms of F are exactly elements that are not contained in proper free factors of F. These results give analogs of Turner's results on test elements of free groups. We also characterize retracts of the algebra F. We prove that if some endomorphism φ preserve the automorphic orbit of some nonzero element of F, then φ is a monomorphism. For free Lie algebras and superalgebras over a field of characteristic zero and for free Lie p-(super)algebras over a field of prime characteristic p we show that in this situation φ is an automorphism. We discuss some related topics and formulate open problems.

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6

Knibbeler, V., S. Lombardo, and J. A Sanders. "Automorphic Lie algebras with dihedral symmetry." Journal of Physics A: Mathematical and Theoretical 47, no. 36 (August 21, 2014): 365201. http://dx.doi.org/10.1088/1751-8113/47/36/365201.

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7

Lombardo, S., and A. V. Mikhailov. "Reduction Groups and Automorphic Lie Algebras." Communications in Mathematical Physics 258, no. 1 (March 30, 2005): 179–202. http://dx.doi.org/10.1007/s00220-005-1334-5.

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8

GRITSENKO, VALERI A., and VIACHESLAV V. NIKULIN. "AUTOMORPHIC FORMS AND LORENTZIAN KAC–MOODY ALGEBRAS PART I." International Journal of Mathematics 09, no. 02 (March 1998): 153–99. http://dx.doi.org/10.1142/s0129167x98000105.

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Using the general method which was applied to prove finiteness of the set of hyperbolic generalized Cartan matrices of elliptic and parabolic type, we classify all symmetric (and twisted to symmetric) hyperbolic generalized Cartan matrices of elliptic type and of rank 3 with a lattice Weyl vector. We develop the general theory of reflective lattices T with 2 negative squares and reflective automorphic forms on homogeneous domains of type IV defined by T. We consider this theory as mirror symmetric to the theory of elliptic and parabolic hyperbolic reflection groups and corresponding hyperbolic root systems. We formulate Arithmetic Mirror Symmetry Conjecture relating both these theories and prove some statements to support this Conjecture. This subject is connected with automorphic correction of Lorentzian Kac–Moody algebras. We define Lie reflective automorphic forms which are the most beautiful automorphic forms defining automorphic Lorentzian Kac–Moody algebras and formulate finiteness Conjecture for these forms. Detailed study of automorphic correction and Lie reflective automorphic forms for generalized Cartan matrices mentioned above will be given in Part II.

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9

Lombardo, Sara, and Jan A. Sanders. "On the Classification of Automorphic Lie Algebras." Communications in Mathematical Physics 299, no. 3 (July 24, 2010): 793–824. http://dx.doi.org/10.1007/s00220-010-1092-x.

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10

Bury, Rhys T., and Alexander V. Mikhailov. "Automorphic Lie algebras and corresponding integrable systems." Differential Geometry and its Applications 74 (February 2021): 101710. http://dx.doi.org/10.1016/j.difgeo.2020.101710.

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11

Tsurkov, A. "Automorphic equivalence of linear algebras." Journal of Algebra and Its Applications 13, no. 07 (May 2, 2014): 1450026. http://dx.doi.org/10.1142/s0219498814500261.

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This research is motivated by universal algebraic geometry. We consider in universal algebraic geometry the some variety of universal algebras Θ and algebras H ∈ Θ from this variety. One of the central question of the theory is the following: When do two algebras have the same geometry? What does it mean that the two algebras have the same geometry? The notion of geometric equivalence of algebras gives a sort of answer to this question. Algebras H1 and H2 are called geometrically equivalent if and only if the H1-closed sets coincide with the H2-closed sets. The notion of automorphic equivalence is a generalization of the first notion. Algebras H1 and H2 are called automorphically equivalent if and only if the H1-closed sets coincide with the H2-closed sets after some "changing of coordinates". We can detect the difference between geometric and automorphic equivalence of algebras of the variety Θ by researching of the automorphisms of the category Θ0 of the finitely generated free algebras of the variety Θ. By [5] the automorphic equivalence of algebras provided by inner automorphism coincide with the geometric equivalence. So the various differences between geometric and automorphic equivalence of algebras can be found in the variety Θ if the factor group 𝔄/𝔜 is big. Here 𝔄 is the group of all automorphisms of the category Θ0, 𝔜 is a normal subgroup of all inner automorphisms of the category Θ0. In [6] the variety of all Lie algebras and the variety of all associative algebras over the infinite field k were studied. If the field k has not nontrivial automorphisms then group 𝔄/𝔜 in the first case is trivial and in the second case has order 2. We consider in this paper the variety of all linear algebras over the infinite field k. We prove that group 𝔄/𝔜 is isomorphic to the group (U(kS2)/U(k{e}))λ Aut k, where S2 is the symmetric group of the set which has 2 elements, U(kS2) is the group of all invertible elements of the group algebra kS2, e ∈ S2, U(k{e}) is a group of all invertible elements of the subalgebra k{e}, Aut k is the group of all automorphisms of the field k. So even the field k has not nontrivial automorphisms the group 𝔄/𝔜 is infinite. This kind of result is obtained for the first time. The example of two linear algebras which are automorphically equivalent but not geometrically equivalent is presented in the last section of this paper. This kind of example is also obtained for the first time.

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12

Lange, Julia, and Javier de Lucas. "Geometric Models for Lie–Hamilton Systems on ℝ2." Mathematics 7, no. 11 (November 4, 2019): 1053. http://dx.doi.org/10.3390/math7111053.

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This paper provides a geometric description for Lie–Hamilton systems on R 2 with locally transitive Vessiot–Guldberg Lie algebras through two types of geometric models. The first one is the restriction of a class of Lie–Hamilton systems on the dual of a Lie algebra to even-dimensional symplectic leaves relative to the Kirillov-Kostant-Souriau bracket. The second is a projection onto a quotient space of an automorphic Lie–Hamilton system relative to a naturally defined Poisson structure or, more generally, an automorphic Lie system with a compatible bivector field. These models give a natural framework for the analysis of Lie–Hamilton systems on R 2 while retrieving known results in a natural manner. Our methods may be extended to study Lie–Hamilton systems on higher-dimensional manifolds and provide new approaches to Lie systems admitting compatible geometric structures.

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13

Tsurkov, A. "Automorphic equivalence in the varieties of representations of Lie algebras." Communications in Algebra 48, no. 1 (August 19, 2019): 397–409. http://dx.doi.org/10.1080/00927872.2019.1646270.

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14

Nikulin, Viacheslav V., and Valeri A. Gritsenko. "Siegel automorphic form corrections of some Lorentzian Kac-Moody Lie algebras." American Journal of Mathematics 119, no. 1 (1997): 181–224. http://dx.doi.org/10.1353/ajm.1997.0002.

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15

DRENSKY, VESSELIN, and JIE-TAI YU. "PRIMITIVE ELEMENTS OF FREE METABELIAN ALGEBRAS OF RANK TWO." International Journal of Algebra and Computation 13, no. 01 (February 2003): 17–33. http://dx.doi.org/10.1142/s021819670300133x.

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Let F(x,y) be a relatively free algebra of rank 2 in some variety of algebras over a field K of characteristic 0. In this paper we consider the problem whether p(x,y) ∈ F(x,y) is a primitive element (i.e. an automorphic image of x): (i) If F(x,y)/(p(x,y)) ≅ F(z), the relatively free algebra of rank 1 (ii) If p(f,g) is primitive for some injective endomorphism (f,g) of F(x,y) (iii) If p(x,y) is primitive in a relatively free algebra of larger rank. These problems have positive solutions for polynomial algebras in two variables. We give the complete answer for the free metabelian associative and Lie algebras and some partial results for free associative algebras.

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16

Merlini Giuliani, Maria De Lourdes, and Giliard Souza Dos Anjos. "Lie automorphic loops under half-automorphisms." Journal of Algebra and Its Applications 19, no. 11 (November 18, 2019): 2050221. http://dx.doi.org/10.1142/s0219498820502217.

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Automorphic loops or [Formula: see text]-loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. Given a Lie ring [Formula: see text] we can define an operation [Formula: see text] such that [Formula: see text] is an [Formula: see text]-loop. We call it Lie automorphic loop. A half-isomorphism [Formula: see text] between multiplicative systems [Formula: see text] and [Formula: see text] is a bijection from [Formula: see text] onto [Formula: see text] such that [Formula: see text] for any [Formula: see text]. It was shown by [W. R. Scott, Half-homomorphisms of groups, Proc. Amer. Math. Soc. 8 (1957) 1141–1144] that if [Formula: see text] is a group then [Formula: see text] is either an isomorphism or an anti-isomorphism. This was used to prove that a finite group is determined by its group determinant. Here, we show that every half-automorphism of a Lie automorphic loop of odd order is either an automorphism or an anti-automorphism.

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17

Cox, Ben, Xiangqian Guo, Rencai Lu, and Kaiming Zhao. "Simple superelliptic Lie algebras." Communications in Contemporary Mathematics 19, no. 03 (April 5, 2017): 1650032. http://dx.doi.org/10.1142/s0219199716500322.

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Let [Formula: see text], [Formula: see text]. Then we have the algebraic curve [Formula: see text], and its coordinate algebras (the Riemann surfaces) [Formula: see text] and [Formula: see text] The Lie algebras [Formula: see text] and [Formula: see text] are called the [Formula: see text]th superelliptic Lie algebras associated to [Formula: see text]. In this paper, we determine the necessary and sufficient conditions for such Lie algebras to be simple, and determine their universal central extensions and their derivation algebras. We also study the isomorphism and automorphism problem for these Lie algebras, which will help to understand the birational equivalence of some algebraic curves of the form [Formula: see text].

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18

Rezaei-Aghdam, A., and M. Sephid. "Classification of real low-dimensional Jacobi (generalized)–Lie bialgebras." International Journal of Geometric Methods in Modern Physics 14, no. 01 (December 20, 2016): 1750007. http://dx.doi.org/10.1142/s0219887817500074.

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We describe the definition of Jacobi (generalized)–Lie bialgebras [Formula: see text] in terms of structure constants of the Lie algebras [Formula: see text] and [Formula: see text] and components of their 1-cocycles [Formula: see text] and [Formula: see text] in the basis of the Lie algebras. Then, using adjoint representations and automorphism Lie groups of Lie algebras, we give a method for classification of real low-dimensional Jacobi–Lie bialgebras. In this way, we obtain and classify real two- and three-dimensional Jacobi–Lie bialgebras.

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19

Bolgar, J. R. "Uniqueness theorems for left-symmetric enveloping algebras." Mathematical Proceedings of the Cambridge Philosophical Society 120, no. 2 (August 1996): 193–206. http://dx.doi.org/10.1017/s030500410007479x.

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AbstractLet L be a Lie algebra over a field of characteristic zero. We study the uni versai left-symmetric enveloping algebra U(L) introduced Dan Segal in [9]. We prove some uniqueness results for these algebras and determine their automorphism groups, both as left-symmetric algebras and as Lie algebras.

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20

Arzikulov, Farhodjon, Furqatjon Urinboyev, and Shahlo Ergasheva. "A CHARACTERIZATION OF DERIVATIONS AND AUTOMORPHISMS ON SOME SIMPLE ALGEBRAS." Ural Mathematical Journal 8, no. 2 (December 29, 2022): 46. http://dx.doi.org/10.15826/umj.2022.2.004.

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In the present paper, we study simple algebras, which do not belong to the well-known classes of algebras (associative algebras, alternative algebras, Lie algebras, Jordan algebras, etc.). The simple finite-dimensional algebras over a field of characteristic 0 without finite basis of identities, constructed by Kislitsin, are such algebras. In the present paper, we consider two such algebras: the simple seven-dimensional anticommutative algebra \(\mathcal{D}\) and the seven-dimensional central simple commutative algebra \(\mathcal{C}\). We prove that every local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is a derivation, and every 2-local derivation of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also a derivation. We also prove that every local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is an automorphism, and every 2-local automorphism of these algebras \(\mathcal{D}\) and \(\mathcal{C}\) is also an automorphism.

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21

Kofinas, C. E., and A. I. Papistas. "Automorphisms of free metabelian Lie algebras." International Journal of Algebra and Computation 26, no. 04 (June 2016): 751–62. http://dx.doi.org/10.1142/s0218196716500326.

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We give a sharpening of a result of Bryant and Drensky [R. M. Bryant and V. Drensky, Dense subgroups of the automorphism groups of free algebras, Canad. J. Math. 45(6) (1993) 1135–1154] for the automorphism group [Formula: see text] of a free metabelian Lie algebra [Formula: see text], with [Formula: see text]. In particular, we prove that the subgroup of [Formula: see text] generated by [Formula: see text] and two more IA-automorphisms is dense in [Formula: see text] and, for [Formula: see text], the subgroup generated by [Formula: see text] and one more IA-automorphism is dense in [Formula: see text].

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22

Segal, Dan. "On the automorphism groups of certain Lie algebras." Mathematical Proceedings of the Cambridge Philosophical Society 106, no. 1 (July 1989): 67–76. http://dx.doi.org/10.1017/s0305004100067980.

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We fix a ground field k and a finite separable extension K of k. To a Lie algebra L over k is associated the Lie algebra KL = K ⊗kL over K. If we forget the action of K, we can think of KL as a larger Lie algebra over k; in particular we can ask what is the automorphism group Autk KL of KL as a k-algebra. There does not seem to be any simple answer to this question in general; the purpose of this note is to give a simple condition on L which makes Autk KL quite easy to determine. Examples of algebras which satisfy this condition include the free nilpotent Lie algebras and the algebras of all n × n triangular nilpotent matrices.

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23

Öğüşlü, N. Ş. "Normal automorphisms of the metabelian product of free abelian Lie algebras." Algebra and Discrete Mathematics 30, no. 2 (2020): 230–34. http://dx.doi.org/10.12958/adm1258.

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24

Salemkar, Ali Reza, Behrouz Edalatzadeh, and Hamid Mohammadzadeh. "On Covers of Perfect Lie Algebras." Algebra Colloquium 18, no. 03 (September 2011): 419–27. http://dx.doi.org/10.1142/s1005386711000307.

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A Lie algebra is said to be perfect when it coincides with its derived subalgebra. The paper is devoted to give a complete structure of covers of perfect Lie algebras. Also, similar to a result of Alperin and Gorenstein (1966) in group theory, it is shown that every automorphism of a finite dimensional perfect Lie algebra may be lifted to an automorphism of its cover. Moreover, we present the concepts of irreducible and primitive extensions of an arbitrary Lie algebra and give some equivalent conditions for a central extension to be irreducible or primitive. Finally, we study the connection between the primitive extensions and the stem covers of finite dimensional perfect Lie algebras.

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25

Ren, Bin, and Linsheng Zhu. "Quasi Qn-Filiform Lie Algebras." Algebra Colloquium 18, no. 01 (March 2011): 139–54. http://dx.doi.org/10.1142/s1005386711000083.

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In this paper, we explicitly determine the derivation algebra, automorphism group of quasi Qn-filiform Lie algebras, and by applying some properties of the root space decomposition, we obtain their isomorphism theorem.

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26

Nam, Ki-Bong, and Seul Hee Choi. "Degree Stable Lie Algebras I." Algebra Colloquium 13, no. 03 (September 2006): 487–94. http://dx.doi.org/10.1142/s1005386706000435.

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We define a degree stable Lie algebra. Since the special type Lie algebra S+(2) is degree stable, we find the automorphism group Aut Lie (S+(2)) of the Lie algebra S+(2) and prove the Jacobian conjecture of the Lie algebra S+(2).

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27

Xia, Chunguang, Xiu Han, and Wei Wang. "Structure of a class of Lie algebras of Block type, II." Journal of Algebra and Its Applications 15, no. 02 (October 6, 2015): 1650038. http://dx.doi.org/10.1142/s0219498816500389.

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28

BAUDISCH, ANDREAS. "FREE AMALGAMATION AND AUTOMORPHISM GROUPS." Journal of Symbolic Logic 81, no. 3 (August 12, 2016): 936–47. http://dx.doi.org/10.1017/jsl.2015.57.

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AbstractWe show that the class of graded c-nilpotent Lie algebras over a fixed field K is closed under free amalgamation. In [1] this result was applied, but its proof was incorrect. In case of a finite field K we obtain a Fraïssé limit of all finite graded c-nilpotent Lie algebras over K. This gives an example for the following more general considerations. The existence of free amalgamation for the age of a Fraïssé limit implies the universality of its automorphism group for all automorphism groups of substructures of that Fraïssé limit. We use [6] and [5].

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29

Bardakov, Valeriy G., and Mahender Singh. "Extensions and automorphisms of Lie algebras." Journal of Algebra and Its Applications 16, no. 09 (September 9, 2016): 1750162. http://dx.doi.org/10.1142/s0219498817501626.

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Let [Formula: see text] be a short exact sequence of Lie algebras over a field [Formula: see text], where [Formula: see text] is abelian. We show that the obstruction for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text] lies in the Lie algebra cohomology [Formula: see text]. As a consequence, we obtain a four term exact sequence relating automorphisms, derivations and cohomology of Lie algebras. We also obtain a more explicit necessary and sufficient condition for a pair of automorphisms in [Formula: see text] to be induced by an automorphism in [Formula: see text], where [Formula: see text] is a free nilpotent Lie algebra of rank [Formula: see text] and step [Formula: see text].

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30

Bryant, Roger M. "Automorphism Groups of Nilpotent Lie Algebras." Journal of the London Mathematical Society s2-36, no. 2 (October 1987): 257–74. http://dx.doi.org/10.1112/jlms/s2-36.2.257.

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31

Öztekin, Özge, and Naime Ekici. "Central automorphisms of free nilpotent Lie algebras." Journal of Algebra and Its Applications 16, no. 11 (October 4, 2017): 1750205. http://dx.doi.org/10.1142/s021949881750205x.

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Let [Formula: see text] be the free nilpotent Lie algebra of finite rank [Formula: see text] [Formula: see text] and nilpotency class [Formula: see text] over a field of characteristic zero. We give a characterization of central automorphisms of [Formula: see text] and we find sufficient conditions for an automorphism of [Formula: see text] to be a central automorphism.

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32

MASHEVITZKY, G., and B. I. PLOTKIN. "ON AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE UNIVERSAL ALGEBRA." International Journal of Algebra and Computation 17, no. 05n06 (August 2007): 1085–106. http://dx.doi.org/10.1142/s0218196707003974.

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Let U be a universal algebra. An automorphism α of the endomorphism semigroup of U defined by α(φ) = sφs-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.

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33

Varea, V. R., and J. J. Varea. "On Automorphisms and Derivations of a Lie Algebra." Algebra Colloquium 13, no. 01 (March 2006): 119–32. http://dx.doi.org/10.1142/s1005386706000149.

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We study automorphisms and derivations of a Lie algebra L of finite dimension satisfying certain centrality conditions. As a consequence, we obtain that every nilpotent normal subgroup of the automorphism group of L is unipotent for a very large class of Lie algebras. This result extends one of Leger and Luks. We show that the automorphism group of a nilpotent Lie algebra can have trivial center and have yet a unipotent identity component.

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34

Choi, Seul Hee, and Ki-Bong Nam. "Automorphism Group of a Special Type Lie Algebra I." Algebra Colloquium 17, spec01 (December 2010): 815–28. http://dx.doi.org/10.1142/s1005386710000763.

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In an earlier paper, we defined a degree stable Lie algebra, and determined the Lie algebra automorphism group AutLie(S+(2)) of the Lie algebra S+(2). In this paper, we determine the Lie algebra automorphism group AutLie(S(1,0,2)) of the Lie algebra S(1,0,2).

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35

Gao, Shoulan, Cuipo Jiang, and Yufeng Pei. "Structure of the Extended Schrödinger–Virasoro Lie Algebra $\widetilde{\mathfrak{sv}}$." Algebra Colloquium 16, no. 04 (December 2009): 549–66. http://dx.doi.org/10.1142/s1005386709000522.

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We study the derivations, the central extensions and the automorphism group of the extended Schrödinger–Virasoro Lie algebra [Formula: see text], introduced by Unterberger in the context of two-dimensional conformal field theory and statistical physics. Moreover, we show that [Formula: see text] is an infinite-dimensional complete Lie algebra, and the universal central extension of [Formula: see text] in the category of Leibniz algebras is the same as that in the category of Lie algebras.

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36

Belov, A. I. "Lie algebras admitting a hypercentrally regular automorphism." Mathematical Notes 52, no. 4 (October 1992): 1003–5. http://dx.doi.org/10.1007/bf01210431.

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37

Kofinas, C. E. "On certain subgroups of the McCool group." International Journal of Algebra and Computation 30, no. 05 (March 23, 2020): 1081–96. http://dx.doi.org/10.1142/s0218196720500320.

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For a positive integer [Formula: see text], with [Formula: see text], let [Formula: see text] be a free group of rank [Formula: see text] and let [Formula: see text] be the subgroup of the automorphism group of [Formula: see text] consisting of all automorphisms which induce the identity on the abelianization of [Formula: see text]. We write [Formula: see text] and [Formula: see text] for the upper McCool group and the partial inner automorphism group, respectively. We show that [Formula: see text] is isomorphic to the quotient of [Formula: see text] by its center and we prove similar results for their associated graded Lie algebras and their Andreadakis–Johnson Lie algebras. Furthermore, we give a presentation of the associated graded Lie algebra over the integers of [Formula: see text] and we prove that it admits a natural embedding into the Andreadakis–Johnson Lie algebra of [Formula: see text]. Although the latter results are known, we present proofs based on different methods.

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38

Azam, Saeid, and Valiollah Khalili. "Lie Tori and Their Fixed Point Subalgebras." Algebra Colloquium 16, no. 03 (September 2009): 381–96. http://dx.doi.org/10.1142/s1005386709000376.

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We study the fixed point subalgebra of a centerless irreducible Lie torus under a certain finite order automorphism. We investigate which axioms of a Lie torus hold for the fixed points and which do not. We relate our study to some recent results about the fixed points of extended affine Lie algebras under a class of finite order automorphisms.

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39

Wu, Yongping, Ying Xu, and Lamei Yuan. "Derivations and Automorphism Group of Completed Witt Lie Algebra." Algebra Colloquium 19, no. 03 (July 5, 2012): 581–90. http://dx.doi.org/10.1142/s1005386712000454.

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In this paper, a simple Lie algebra, referred to as the completed Witt Lie algebra, is introduced. Its derivation algebra and automorphism group are completely described. As a by-product, it is obtained that the first cohomology group of this Lie algebra with coefficients in its adjoint module is trivial. Furthermore, we completely determine the conjugate classes of this Lie algebra under its automorphism group, and also obtain that this Lie algebra does not contain any nonzero ad -locally finite element.

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MAGNIN, L. "COMPLEX STRUCTURES ON INDECOMPOSABLE 6-DIMENSIONAL NILPOTENT REAL LIE ALGEBRAS." International Journal of Algebra and Computation 17, no. 01 (February 2007): 77–113. http://dx.doi.org/10.1142/s0218196707003500.

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We show how resorting to dependable computer calculations makes it possible to compute all integrable complex structures on indecomposable 6-dimensional nilpotent real Lie algebras and their equivalence classes under the automorphism group of the Lie algebra. We also prove that the set comprised of all integrable complex structures on such a Lie algebra is a smooth submanifold of ℝ36.

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41

PAPISTAS, A. I. "AUTOMORPHISMS OF CERTAIN RELATIVELY FREE GROUPS AND LIE ALGEBRAS." International Journal of Algebra and Computation 14, no. 03 (June 2004): 311–23. http://dx.doi.org/10.1142/s0218196704001761.

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For positive integers n and c, with n≥2, let Gn,c be a relatively free group of rank n in the variety N2A∧AN2∧Nc. It is shown that there exists an explicitly described finite subset Ω of IA-automorphisms of Gn,c such that the cardinality of Ω is independent upon n and c and the subgroup of the automorphism group Aut (Gn,c) of Gn,c generated by the tame automorphisms and Ω has finite index in Aut (Gn,c). This is a simpler result than one given in [12, Theorem 1(I)]. Let L(Gn,c) be the associated Lie ring of Gn,c and K be a field of characteristic zero. The method developed in the proof of the aforementioned result is applied in order to find an explicitly described finite subset ΩL of the IA-automorphism group of K⊗L(Gn,c) such that the automorphism group of K⊗L(Gn,c) is generated by GL (n,K) and ΩL. In particular, for n≥3, the cardinality of ΩL is independent upon n and c.

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42

Drensky, Vesselin, and C. K. Gupta. "Automorphisms of Free Nilpotent Lie Algebras." Canadian Journal of Mathematics 42, no. 2 (April 1, 1990): 259–79. http://dx.doi.org/10.4153/cjm-1990-015-1.

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Let Fm be the free Lie algebra of rank m over a field K of characteristic 0 freely generated by the set ﹛x1,… ,xm﹜, m ≧ 2. Cohn [7] proved that the automorphism group Aut Fm of the K-algebra Fm is generated by the following automorphisms: (i) automorphisms which are induced by the action of the general linear group GLm (= GLm(K)) on the subspace of Fm spanned by ﹛x1, … ,xm﹜; (ii) automorphisms of the form x1 → x1 +f(x2,… ,xm),Xk → xk, k ≠ 1, where the polynomial f(x2,…,xm) does not depend on x1.

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43

Tikaradze, Akaki. "On Automorphisms of Enveloping Algebras." International Mathematics Research Notices 2020, no. 21 (February 27, 2019): 8183–96. http://dx.doi.org/10.1093/imrn/rnz039.

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Abstract Given an algebraic Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we canonically associate to it a Lie algebra $\mathcal{L}_{\infty }(\mathfrak{g})$ defined over $\mathbb{C}_{\infty }$, the reduction of $\mathbb{C}$ modulo the infinitely large prime, and show that for a class of Lie algebras, $\mathcal{L}_{\infty }(\mathfrak{g})$ is an invariant of the derived category of $\mathfrak{g}$-modules. We give two applications of this construction. First, we show that the bounded derived category of $\mathfrak{g}$-modules determines algebra $\mathfrak{g}$ for a class of Lie algebras. Second, given a semi-simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, we construct a canonical homomorphism from the group of automorphisms of the enveloping algebra $\mathfrak{U}\mathfrak{g}$ to the group of Lie algebra automorphisms of $\mathfrak{g}$, such that its kernel does not contain a non-trivial semi-simple automorphism. As a corollary, we obtain that any finite subgroup of automorphisms of $\mathfrak{U}\mathfrak{g}$ is isomorphic to a subgroup of Lie algebra automorphisms of $\mathfrak{g}.$

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MIKHALEV, ALEXANDER A., and ANDREJ A. ZOLOTYKH. "RANK AND PRIMITIVITY OF ELEMENTS OF FREE COLOUR LIE (p-)SUPERALGEBRAS." International Journal of Algebra and Computation 04, no. 04 (December 1994): 617–55. http://dx.doi.org/10.1142/s021819679400018x.

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Using Fox differential calculus we study characteristics of orbits of elements of free colour Lie (p-)superalgebras under action of the automorphism groups of these algebras. In particular, an effective criterion for an element to be primitive and an algorithm for finding the rank of an element are obtained.

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45

MIKHALEV, ALEXANDER A., VLADIMIR SHPILRAIN, and UALBAI U. UMIRBAEV. "ON ISOMORPHISM OF LIE ALGEBRAS WITH ONE DEFINING RELATION." International Journal of Algebra and Computation 14, no. 03 (June 2004): 389–93. http://dx.doi.org/10.1142/s0218196704001773.

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Let L be a finitely generated free Lie algebra. We construct an example of two elements u and v of L such that the factor algebras L/(u) and L/(v) are isomorphic, where (u) and (v) are ideals of L generated by u and v, respectively, but there is no automorphism φ of L such that φ(u)=v.

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46

Kofinas, C. E. "Automorphisms of the completion of relatively free Lie algebras." International Journal of Algebra and Computation 28, no. 06 (September 2018): 1091–100. http://dx.doi.org/10.1142/s0218196718500479.

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Let [Formula: see text] be a relatively free Lie algebra of finite rank [Formula: see text], with [Formula: see text], [Formula: see text] be the completion of [Formula: see text] with respect to the topology defined by the lower central series [Formula: see text] of [Formula: see text] and [Formula: see text], with [Formula: see text]. We prove that, with respect to the formal power series topology, the automorphism group [Formula: see text] of [Formula: see text] is dense in the automorphism group [Formula: see text] of [Formula: see text] if and only if [Formula: see text] is nilpotent. Furthermore, we show that there exists a dense subgroup of [Formula: see text] generated by [Formula: see text] and a finite set of IA-automorphisms if and only if [Formula: see text] is generated by [Formula: see text] and a finite set of IA-automorphisms independent upon [Formula: see text] for all [Formula: see text]. We apply our study to several varieties of Lie algebras.

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47

Cox, Ben, Xiangqian Guo, Rencai Lu, and Kaiming Zhao. "n-Point Virasoro algebras and their modules of densities." Communications in Contemporary Mathematics 16, no. 03 (May 26, 2014): 1350047. http://dx.doi.org/10.1142/s0219199713500478.

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In this paper we introduce and study n-point Virasoro algebras, [Formula: see text], which are natural generalizations of the classical Virasoro algebra and have as quotients multipoint genus zero Krichever–Novikov type algebras. We determine necessary and sufficient conditions for the latter two such Lie algebras to be isomorphic. Moreover we determine their automorphisms, their derivation algebras, their universal central extensions, and some other properties. The list of automorphism groups that occur is Cn, Dn, A4, S4 and A5. We also construct a large class of modules which we call modules of densities, and determine necessary and sufficient conditions for them to be irreducible.

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JORDAN, DAVID A., and NONGKHRAN SASOM. "REVERSIBLE SKEW LAURENT POLYNOMIAL RINGS AND DEFORMATIONS OF POISSON AUTOMORPHISMS." Journal of Algebra and Its Applications 08, no. 05 (October 2009): 733–57. http://dx.doi.org/10.1142/s0219498809003564.

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A skew Laurent polynomial ring S = R[x±1;α] is reversible if it has a reversing automorphism, that is, an automorphism θ of period 2 that transposes x and x-1 and restricts to an automorphism γ of R with γ = γ-1. We study invariants for reversing automorphisms and apply our methods to determine the rings of invariants of reversing automorphisms of the two most familiar examples of simple skew Laurent polynomial rings, namely a localization of the enveloping algebra of the two-dimensional non-abelian solvable Lie algebra and the coordinate ring of the quantum torus, both of which are deformations of Poisson algebras over the base field 𝔽. Their reversing automorphisms are deformations of Poisson automorphisms of those Poisson algebras. In each case, the ring of invariants of the Poisson automorphism is the coordinate ring B of a surface in 𝔽3 and the ring of invariants Sθ of the reversing automorphism is a deformation of B and is a factor of a deformation of 𝔽[x1, x2, x3] for a Poisson bracket determined by the appropriate surface.

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49

Papistas, A. I. "On Automorphism groups of 2-generator metabelian lie algebras." Communications in Algebra 20, no. 7 (January 1992): 1937–53. http://dx.doi.org/10.1080/00927879208824441.

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50

Makarenko, N. Yu. "Lie type algebras with an automorphism of finite order." Journal of Algebra 439 (October 2015): 33–66. http://dx.doi.org/10.1016/j.jalgebra.2015.04.033.

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