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41 
Twisted derivations, quasihomLie algebras and their quasideformationsBergander, Philip January 2017 (has links)
QuasihomLie algebras (qhlalgebras) were introduced by Larsson and Silvestrov (2004) as a generalisation of homLie algebras, which are a deformation of Lie algebras. Lie algebras are defined by an operation called bracket, [·,·], and a threeterm Jacobi identity. By the theorem from Hartwig, Larsson, and Silvestrov (2003), this bracket and the threeterm Jacobi identity are deformed into a new bracket operation, <·,·>, and a sixterm Jacobi identity, making it a quasihomLie algebra. Throughout this thesis we deform the Lie algebra sl2(F), where F is a field of characteristic 0. We examine the quasideformed relations and sixterm Jacobi identities of the following polynomial algebras: F[t], F[t]/(t2), F[t]/(t3), F[t]/(t4), F[t]/(t5), F[t]/(tn), where n is a positive integer ≥2, and F[t]/((tt0)3). Larsson and Silvestrov (2005) and Larsson, Sigurdsson, and Silvestrov (2008) have already examined some of these cases, which we repeat for the reader's convenience. We further investigate the following σtwisted derivations, and how they act in the different cases of mentioned polynomial algebras: the ordinary differential operator, the shifted difference operator, the Jackson qderivation operator, the continuous qdifference operator, the Eulerian operator, the divided difference operator, and the nilpotent imaginary derivative operator. We also introduce a new, general, σtwisted derivation operator, which is σ(t) as a polynomial of degree k.

42 
An Introduction to Lie Theory and ApplicationsDickson, Anthony J. 06 May 2021 (has links)
No description available.

43 
Classification of Isometry Algebras of Solutions of Einstein's Field EquationsHwang, Eugene 01 August 2019 (has links)
Since Schwarzschild found the first solution of the Einstein’s equations, more than 800 solutions were found. Solutions of Einstein’s equations are classified according to their Lie algebras of isometries and their isotropy subalgebras. Solutions were taken from the USU electronic library of solutions of Einstein’s field equations and the classification used Maple code developed at USU. This classification adds to the data contained in the library of solutions and provides additional tools for addressing the equivalence problem for solutions to the Einstein field equations. In this thesis, homogeneous spacetimes, hypersurfacehomogeneous spacetimes, RobinsonTrautman solutions, and some famous black hole solutions have been classified.

44 
Equivalence Classes of Subquotients of Pseudodifferential Operator Modules on the LineLarsen, Jeannette M. 08 1900 (has links)
Certain subquotients of Vec(R)modules of pseudodifferential operators from one tensor density module to another are categorized, giving necessary and sufficient conditions under which two such subquotients are equivalent as Vec(R)representations. These subquotients split under the projective subalgebra, a copy of ????2, when the members of their composition series have distinct Casimir eigenvalues. Results were obtained using the explicit description of the action of Vec(R) with respect to this splitting. In the length five case, the equivalence classes of the subquotients are determined by two invariants. In an appropriate coordinate system, the level curves of one of these invariants are a pencil of conics, and those of the other are a pencil of cubics.

45 
Lowdimensional cohomology of current Lie algebrasZusmanovich, Pasha January 2010 (has links)
We deal with lowdimensional homology and cohomology of current Lie algebras, i.e., Lie algebras which are tensor products of a Lie algebra L and an associative commutative algebra A. We derive, in two different ways, a general formula expressing the second cohomology of current Lie algebra with coefficients in the trivial module through cohomology of L, cyclic cohomology of A, and other invariants of L and A. The first proof is achieved by using the Hopf formula expressing the second homology of a Lie algebra in terms of its presentation. The second proof employs a certain linearalgebraic technique, ideologically similar to “separation of variables” of differential equations. We also obtain formulas for the first and, in some particular cases, for the second cohomology of the current Lie algebra with coefficients in the “current” module, and the second cohomology with coefficients in the adjoint module in the case where L is the modular Zassenhaus algebra. Applications of these results include: description of modular semisimple Lie algebras with a solvable maximal subalgebra; computations of structure functions for manifolds of loops in compact Hermitian symmetric spaces; a unified treatment of periodizations of semisimple Lie algebras, derivation algebras (with prescribed semisimple part) of nilpotent Lie algebras, and presentations of affine KacMoody algebras.

46 
SchurLike Forms for Matrix Lie Groups, Lie Algebras and Jordan AlgebrasAmmar, Gregory, Mehl, Christian, Mehrmann, Volker 09 September 2005 (has links)
We describe canonical forms for elements of a classical Lie group of matrices under similarity transformations in the group. Matrices in the associated Lie algebra and Jordan algebra of matrices inherit related forms under these similarity transformations. In general, one cannot achieve diagonal or Schur form, but the form that can be achieved displays the eigenvalues of the matrix. We also discuss matrices in intersections of these classes and their Schurlike forms. Such multistructered matrices arise in applications from quantum physics and quantum chemistry.

47 
Topological uniqueness results for the special linear and other classical Lie Algebras.Rees, Michael K. 12 1900 (has links)
Suppose L is a complete separable metric topological group (ring, field, etc.). L is topologically unique if the Polish topology on L is uniquely determined by its underlying algebraic structure. More specifically, L is topologically unique if an algebraic isomorphism of L with any other complete separable metric topological group (ring, field, etc.) induces a topological isomorphism. A local field is a locally compact topological field with nondiscrete topology. The only local fields (up to isomorphism) are the real, complex, and padic numbers, finite extensions of the padic numbers, and fields of formal power series over finite fields. We establish the topological uniqueness of the special linear Lie algebras over local fields other than the complex numbers (for which this result is not true) in the context of complete separable metric Lie rings. Along the way the topological uniqueness of all local fields other than the field of complex numbers is established, which is derived as a corollary to more general principles which can be applied to a larger class of topological fields. Lastly, also in the context of complete separable metric Lie rings, the topological uniqueness of the special linear Lie algebra over the real division algebra of quaternions, the special orthogonal Lie algebras, and the special unitary Lie algebras is proved.

48 
Bases de monômes dans les algèbres préLie libres et applications / Monomial bases for free preLie algebras and applicationsAlKaabi, Mahdi Jasim Hasan 28 September 2015 (has links)
Dans cette thèse, nous étudions le concept d’algèbre préLie libre engendrée par un ensemble (nonvide). Nous rappelons la construction par A. Agrachev et R. Gamkrelidze des bases de monômes dans les algèbres préLie libres. Nous décrivons la matrice des vecteurs d’une base de monômes en termes de la base d’arbres enracinés exposée par F. Chapoton et M. Livernet. Nous montrons que cette matrice est unipotente et trouvons une expression explicite pour les coefficients de cette matrice, en adaptant une procédure suggérée par K. EbrahimiFard et D. Manchon pour l’algèbre magmatique libre. Nous construisons une structure d’algèbre préLie sur l’algèbre de Lie libre $\mathcal{L}$(E) engendrée par un ensemble E, donnant une présentation explicite de $\mathcal{L}$(E) comme quotient de l’algèbre préLie libre $\mathcal{T}$^E, engendrée par les arbres enracinés (nonplanaires) Edécorés, par un certain idéal I. Nous étudions les bases de Gröbner pour les algèbres de Lie libres dans une présentation à l’aide d’arbres. Nous décomposons la base d’arbres enracinés planaires Edécorés en deux parties O(J) et $\mathcal{T}$(J), où J est l’idéal définissant $\mathcal{L}$(E) comme quotient de l’algèbre magmatique libre engendrée par E. Ici, $\mathcal{T}$(J) est l’ensemble des termes maximaux des éléments de J, et son complément O(J) définit alors une base de $\mathcal{L}$(E). Nous obtenons un des résultats importants de cette thèse (Théorème 3.12) sur la description de l’ensemble O(J) en termes d’arbres. Nous décrivons des bases de monômes pour l’algèbre préLie (respectivement l’algèbre de Lie libre) $\mathcal{L}$(E), en utilisant les procédures de bases de Gröbner et la base de monômes pour l’algèbre préLie libre obtenue dans le Chapitre 2. Enfin, nous étudions les développements de Magnus classique et préLie, discutant comment nous pouvons trouver une formule de récurrence pour le cas préLie qui intègre déjà l’identité préLie. Nous donnons une vision combinatoire d’une méthode numérique proposée par S. Blanes, F. Casas, et J. Ros, sur une écriture du développement de Magnus classique, utilisant la structure préLie de $\mathcal{L}$(E). / In this thesis, we study the concept of free preLie algebra generated by a (nonempty) set. We review the construction by A. Agrachev and R. Gamkrelidze of monomial bases in free preLie algebras. We describe the matrix of the monomial basis vectors in terms of the rooted trees basis exhibited by F. Chapoton and M. Livernet. Also, we show that this matrix is unipotent and we find an explicit expression for its coefficients, adapting a procedure implemented for the free magmatic algebra by K. EbrahimiFard and D. Manchon. We construct a preLie structure on the free Lie algebra $\mathcal{L}$(E) generated by a set E, giving an explicit presentation of $\mathcal{L}$(E) as the quotient of the free preLie algebra $\mathcal{T}$^E, generated by the (nonplanar) Edecorated rooted trees, by some ideal I. We study the Gröbner bases for free Lie algebras in tree version. We split the basis of E decorated planar rooted trees into two parts O(J) and $\mathcal{T}$(J), where J is the ideal defining $\mathcal{L}$(E) as a quotient of the free magmatic algebra generated by E. Here $\mathcal{T}$(J) is the set of maximal terms of elements of J, and its complement O(J) then defines a basis of $\mathcal{L}$(E). We get one of the important results in this thesis (Theorem 3.12), on the description of the set O(J) in terms of trees. We describe monomial bases for the preLie (respectively free Lie) algebra $\mathcal{L}$(E), using the procedure of Gröbner bases and the monomial basis for the free preLie algebra obtained in Chapter 2. Finally, we study the socalled classical and preLie Magnus expansions, discussing how we can find a recursion for the preLie case which already incorporates the preLie identity. We give a combinatorial vision of a numerical method proposed by S. Blanes, F. Casas, and J. Ros, on a writing of the classical Magnus expansion in $\mathcal{L}$(E), using the preLie structure.

49 
Goldman Bracket : Center, Geometric Intersection Number & Length Equivalent CurvesKabiraj, Arpan January 2016 (has links) (PDF)
Goldman [Gol86] introduced a Lie algebra structure on the free vector space generated by the free homotopy classes of oriented closed curves in any orientable surface F . This Lie bracket is known as the Goldman bracket and the Lie algebra is known as the Goldman Lie algebra. In this dissertation, we compute the center of the Goldman Lie algebra for any hyperbolic surface of finite type. We use hyperbolic geometry and geometric group theory to prove our theorems. We show that for any hyperbolic surface of finite type, the center of the Goldman Lie algebra is generated by closed curves which are either homotopically trivial or homotopic to boundary components or punctures.
We use these results to identify the quotient of the Goldman Lie algebra of a nonclosed surface by its center as a subalgebra of the first Hochschild cohomology of the fundamental group.
Using hyperbolic geometry, we prove a special case of a theorem of Chas [Cha10], namely, the geometric intersection number between two simple closed geodesics is the same as the number of terms (counted with multiplicity) in the Goldman bracket between them.
We also construct infinitely many pairs of length equivalent curves in any hyperbolic surface F of finite type. Our construction shows that given a self intersecting geodesic x of F and any selfintersection point P of x, we get a sequence of such pairs.

50 
VERTEX ALGEBRAS AND STRONGLY HOMOTOPY LIE ALGEBRASPinzon, Daniel F. 01 January 2006 (has links)
Vertex algebras and strongly homotopy Lie algebras (SHLA) are extensively used in qunatum field theory and string theory. Recently, it was shown that a Courant algebroid can be naturally lifted to a SHLA. The 0product in the de Rham chiral algebra has an identical formula to the Courant bracket of vector fields and 1forms. We show that in general, a vertex algebra has an SHLA structure and that the de Rham chiral algebra has a nonzero l4 homotopy.

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