Contractions of lie algebras with 2dimensional generic. Graded contractions of representations of lie algebras p. These are expanded lecture notes for the authors minicourse at msri in june 2012. Degenerations and contractions of algebraic groups we want to transfer the notions to algebraic groups. The first approach uses lie algebraic analysis alone, but already yields a whole range of alternative kinematics that are all in a quantifiable. If we consider deformations with base spec a, where a is a local algebra, this setup is adequate to study the problem of. S v is hence the set of common zeros of a certain system of second degree polynomials on the space. A definition for contraction of lie algebra representations based on this. Jump deformations and contractions of lie algebras are inverse concepts, but the approaches to their computations are quite different.
Gerstenhaber papers the series called on the deformation of rings and algebras is extremely. Contractions of lie algebras and algebraic groups universitat wien. Here we focus on contractions of lie algebras and algebraic groups. Group deformations and contractions physics notes by jakob. As a consequence it is shown that the only groups which can be contracted in. A maurercartan elements homotopy p algebra structure on a twisted dg lie. From the miniversal deformation, one can determine all jump deformations of a lie algebra. Algebraic deformation theory was introduced for associative algebras by gerstenhaber 14, and was extended to lie algebras by nijenhuis and richardson 23, 24. The theory of lie algebra deformations1,2,3 provides us with a systematic procedure which is an inverse to the more common lie algebra contractions4,5.
International journal of pure and applied mathematics. Lie algebras appear almost everywhere in mathematics. Contraction of a lie algebra encyclopedia of mathematics. Novotny faculty of nuclear sciences and physical engineering czech technical university in prague august 2012 star a lesn a p. Deformation and contraction of symmetries in special relativity. Byagradedvectorspaceweintendazgradedvector spaceoverk. Our main interest lies in a special case of this notion, a contraction. Deformations and contractions of lie algebras iopscience. Construction of miniversal deformations of lie algebras. An important result states that for any contraction of lie algebras g g0 there is a deformation of g0 that reverses it 3. Contractions and deformations of lie algebras request pdf. A jump deformation is precisely the inverse of a contraction of a lie algebra, so one can say that the miniversal deformations contain all the information about contractions as well as other interesting information about the moduli space. They did not study the versal property of deformations. However, the definition seems too general to be useful in our context.
When understood in a general and rather vague sense, deformations of lie algebras are continuous modi. Graded contractions of lie algebras and central extensions 4041 2. Furthermore, the theory of lie algebras is build on very simple arguments of linear algebra and the classification of semisimple lie algebras over an algebraically closed field of characteristic zero is very simple. Deformation theory is a natural generalization of lie theory, from lie groups and their linearization, lie algebras, to di. As a consequence it is shown that the only groups which can be contracted in the poincare group are so 4, 1 and so 3, 2. Then, after a brief introduction to operads, a strongly homotopy lie algebra governing deformations of diagrams of algebras of a given type is described, followed by examples and generalizations. In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution p of a problem to slightly different solutions p. For example, if, then, so for this contraction the limit algebra is always. Graded contractions of lie algebras and central extensions. This a generalization of the classical theory of deformations, which is only concerned with the deformation of the structure constants of finitedimensional lie algebras. Deformation and contraction of lie algebras aip publishing. On deformation theory and quantization semantic scholar. Sv is hence the set of common zeros of a certain system of second degree polynomials on the space a. Lie algebras, commutative algebras, poisson algebras, gerstenhaber algebras.
The parallel with lie theory suggests that deformation theory is a higher dimensional. The point of view of deformation theory is a bit di. There exist two main categories of such modifications of the structure con stants. This coursebased primer provides an introduction to lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. Also we note that some lie algebras belonging to parameterized families are singled out by the irreversibility of deformations and contractions. In 9, 14, the moduli space of lie algebras of dimension 3 was carefully analyzed, and in 10, a construction of the moduli space of 4dimensional lie algebras was given. Geometry and topology of coadjoint orbits of semisimple lie groups bernatska, julia and holod, petro, 2008. Deformation theory and differential graded lie algebras. Lie algebras and applications lecture notes in physics. Deformation theory requires solving maurercartan equation mce associated to an dgla linfinity algebra. Read contractions of lie algebras with 2dimensional generic coadjoint orbits, linear algebra and its applications on deepdyve, the largest online rental service for scholarly research with thousands of academic publications available at your fingertips. I stumbled across the following definition of the formal deformation of a lie algebra, and it looks like a group object in the category of formal schemes not necessarily commutative or 1dimensional. We first discuss lowdimensional lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation, the converse is. Pdf on deformations and contractions of lie algebras.
In this contributed presentation, we discuss and compare the mutually opposite procedures of deformations and contractions of lie algebras. This dissertation gives an account of the fundamental principles underlying two conceptionally different ways of embedding special relativity into a wider context. We first discuss lowdimensional lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation, the converse is not true in general. This approach is motivated by the belief in the existence of a quantum theory of gravity, and the assumption that such manifests itself in nonlinear modications of the laws of. Nonsolvable contractions of simple lie algebras in low. The article focuses on two basic constructions of deformation theory.
Now we move to the theory of deformations of lie algebras. Then the deformation theory of the inclusion morphism is the study of deformations. Formal deformations, contractions and moduli spaces of lie. Morrison2 1 department of advanced energy, university of tokyo, kashiwa, chiba 2778561, japan 2 department of physics and institute for fusion studies, university of texas at austin. Let a be an associative algebra over a eld kof characteristic zero. In this section, we compare deformations and contractions of complex and real threedimensional lie algebras. On contraction of lie algebra representations springerlink. In this paper, we contrast the two approaches, showing how to compute the jump deformations from the miniversal deformation of a lie algebra, and thus arrive at the contractions. On deformations and contractions of lie algebras 7 now, let us obtain other lie algebras by using contraction pro cedures similar to what we have done for the witt algebra. Novotny graded contractions of representations of lie alg. Deformation and contraction of symmetries in special. Group contraction and deformation the physics travel guide.
A construction by deformation of unitary irreducible. Also we note that some lie algebras belonging to parameterized families are singled. To achieve this a different kind of deformation than the usual one is needed, because one must consider nonlinear transformations of the generators. Some features of rank one real solvable cohomologically. Although, in general these algebras are not isomorphic. The second direction of departures from special relativity stresses and develops the algebraic view on spacetime by considering hopf instead of lie algebras as candidates for the description of kinematical transformations and hence spacetime symmetry. Roger, extensions and contractions of lie algebra of qpseudodifferential. On deformations and contractions of lie algebras nasaads. Deformations of algebras in noncommutative algebraic geometry travis schedler abstract. In the first part, it concisely presents the basic concepts of lie algebras, their representations and their invariants. In this sense, deformations add new possibilities that cannot appear in limiting processes.
Contractions, degenerations and deformations of lie algebras. Buy lie algebras and applications lecture notes in physics 891 on free shipping on qualified orders. Let o be the associative operad, let p be the operad for associative algebras with an action of a comm. Request pdf contractions and deformations of lie algebras this is a brief survey of contractions and deformations of lie algebras. Nonsolvable contractions of simple lie algebras in low dimension. A comment concerning cohomology and invariants of lie. We expect these new lie algebras to be of interest, particularly in conformal. Although we will be primarily interested in contracting semisimple lie algebras, our method can, in principle, be applied to nonsemisimple lie algebras, as well as lie superalgebras and in. If is the lie algebra of a lie group, then the lie group corresponding to is called a contraction of the group. We first discuss lowdimensional lie algebras and illustrate thereby that whereas for every contraction there exists a reverse deformation, the. Both of them root in the attempt to explore the full scope of the relativity postulate. Hereafter, the deformations of real threedimensional lie algebras are classified for the first time by using the recent deformation classification of the corresponding complex lie algebras 15.
This paper deals with the theory of deformation of lie algebras. As a consequence it is shown that the only groups which can be contracted in the poincare group are so4, 1 and so3, 2. Deformation of liepoisson algebras and chirality zensho yoshida1 and philip j. Deformations of algebras we wish to rst illustrate the theory with a technically simple case, namely deformations of algebras. These show that to understand the deformation of algebras one must consider not just that of single algebras but of diagrams even if at the start one was concerned with the deformation of but a single speci c. On deformation of associative algebras and graph homology.
An introduction to algebraic deformation theory sciencedirect. Degenerations, contractions and deformations of various algebraic structures play an important role in mathematics and physics. The main idea which we used in our analysis was the computation of the miniversal deformation, which allows one to determine all possible deformations of the lie algebra. Dec 21, 2004 this paper deals with the theory of deformation of lie algebras. A formal deformation of ais a ktbilinear multiplication law m t. Differential graded lie algebras and toy examples of deformation problems let k be a.
The infinitesimal conditions are therefore the result of applying the approach of differential calculus to solving a problem with constraints. Group deformations and contractions physics notes by. By the preceding results, contractions of lie algebras can be analyzed using cohomological tools. We suggest that with appropriate combinations of both procedures one may construct new lie algebras. The contraction is also a lie algebra, moreover can be obtained by means of a deformation of.
I stumbled across the following definition of the formal deformation of a lie algebra, and it looks like a group object in the category of formal schemes not necessarily commutative or 1dimension. To illustrate the method, we consider in detail two physical applications. Graded contractions of representations of lie algebras. Some features of rank one real solvable cohomologically rigid. We follow \on the deformation of rings and algebras by murray gerstenhaber annals 1964 closely. Ouraimis to show that the same scheme holds for a wide class of algebras and their diagrams, though instead of dglie one sometimes needs an l. Jul 28, 2007 jump deformations and contractions of lie algebras are inverse concepts, but the approaches to their computations are quite different. Lie algebras and applications francesco iachello springer. Deformation quantization in the teaching of lie group representations balsomo, alexander j. Deformations and contractions of lie algebras elte. For instance in synthetic differential geometry then a lie algebra of a lie group is just the firstorder infinitesimal neighbourhood of the unit element e. We begin with a naive definition of deformation of lie algebras.
A modern approach to deformations theory is via di. Among other properties, contractions of lie algebras imply various numerical relations among invariants of contracting and contracted lie algebras, such as dimensions of cohomology groups or number of generalized casimir invariants. This generalizes naturally the approach of roots systems of semisimple lie algebras 9,10. Lie algebras are equivalently groups in infinitesimal geometry. Let v be a linear space, and s v 2 the set of all linear maps. The role of the kuranishi functor in this construction is emphasized. Lie algebras dgla of graphs 4 which controls the deformation theory of associative algebras 1,2, etc. The universal solution of hs is obtained iteratively, as a fixed point of a contraction, analogous to the picard method. Inonuwigner contractions and deformations of finite. See also deformation and contraction of lie algebras by levynahas. A number of results about deformations have been proved, for example 21, 22. Contractions are important in physics because they explain in terms of lie algebras why some theories arise as a limit regime of more exact theories. Note that in the case of lie algebras the underlying space does not change under deformation, degeneration or contraction. Deformation theory of algebras and structures and applications il ciocco, 1986, 11264, nato adv.
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