By Professor A. N. Leznov, Professor M. V. Saveliev (auth.)

ISBN-10: 3034886381

ISBN-13: 9783034886383

ISBN-10: 303489709X

ISBN-13: 9783034897099

The ebook studies a great number of 1- and 2-dimensional equations that describe nonlinear phenomena in numerous parts of recent theoretical and mathematical physics. it's intended, particularly, for physicists who focus on the sphere idea and physics of straightforward debris and plasma, for mathe­ maticians facing nonlinear differential equations, differential geometry, and algebra, and the idea of Lie algebras and teams and their representa­ tions, and for college kids and post-graduates in those fields. we are hoping that the ebook may be necessary additionally for specialists in hydrodynamics, solid-state physics, nonlinear optics electrophysics, biophysics and physics of the Earth. the 1st chapters of the e-book current a few effects from the repre­ sentation conception of Lie teams and Lie algebras and their counterpart on supermanifolds in a kind handy in what follows. they're addressed to those that have an interest in integrable platforms yet have a scanty vocabulary within the language of illustration conception. The specialists might confer with the 1st chapters simply sometimes. As we would have liked to offer the reader a chance not just to return to grips with the matter at the ideological point but in addition to combine her or his personal concrete nonlinear equations regardless of the literature, we needed to reveal in a self-contained method the ideal components of the illustration thought from a selected aspect of view.

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X U(l) , v q [~] ChT q ( eXf{T X ... X [~] ShT) ChT S U(l) , iShT) ChT ish T 0 ChT 0 ChT 0 ish T 0 ( ChT ShT 2 G=KAK L(n, G) (A) ChT -ishT J 28 Chapter 1: Background of the theory of Lie algebras and ... •. • X 71.. •. 1 Embeddings of sl(2) into Lie algebras. The description of all nonequivalent embeddings of semisimple algebras gl into a finite-dimensional simple Lie algebra 9 is completely solved in [32]-[34], where, in particular, there is a list of all sl(2)-subalgebras. Thanks to Gantmakher's theorem [16], [17] for any embedding gl C 9 we can select a Cartan sub algebra ~ c 9 so that the elements of the Cartan subalgebra ~I C gl would belong to ~.

Yi o E 90/go) vanish. 2) they also hold for Yv± = FJ. 2). 5) follow from the Jacobi identity for 9and distinguish nonzero elements y;o. e. 5) and hence the Y;0 belong to the center of the whole 9. One can recover the whole 9 by introducing the invariant symmetric bilinear form on 9and its subsequent continuation onto 9, cf. 3. 4) for the finite-dimensional Lie algebra 9 whose extension is 9. 5) are well-defined. 8) of 9 via the above scheme with Y-, y+ and yO == [Y+, Y-]. 1) and belongs to the center of 9, where A == {AI} is the eigenvector of the Cartan matrix k corresponding to the eigenvalue 0 and 2:1

Obviously, we can also consider representations g = g+g-go, g = g'-g~g~. 4) Thanks to the Levi-Malcev theorem, an element g E G factors into the product of elements from its maximal semisimple (5) and solvable (R) subgroups. If G is simply connected then the representation g = r . 5) is unique and holds for any g E G. For semisimple finite-dimensional Lie groups there are several useful decompositions . • ) [7, 13, 35, 51, 60, 61, 113] 42 Chapter 1: Background of the theory of Lie algebras and ...

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Group-Theoretical Methods for Integration of Nonlinear Dynamical Systems by Professor A. N. Leznov, Professor M. V. Saveliev (auth.)


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