simple Lie algebras of classical and Cartan types (as in [BlWil]), many of the same questions could be raised for the restricted simple Melikian algebra, as well as for the minimal p-envelopes of the non-restricted Cartan-type and Melikian Lie algebras. For some related problems, we refer the reader to [Sk4, Questions 1{3]. 2. Background and A modern pedagogical introduction to QFT including the Weinberg-Salam model and other selected topics. Mandl, F., and G. Shaw. Quantum Field Theory. New York, NY: John Wiley & Sons, 1984. ISBN: 9780471105091. A clear and concise introduction to the basic computations in quantum field theory. tive algebras, Jordan algebras, alternative algebras, composition algebras, Hopf algebras, and Lie algebras|these last being the primary focus of our study. (All the others will be discussed at least brie y.) In most cases these algebra types naturally form subcategories of the additive category KAlg of K-algebras, the maps 'of Hom positive characteristic: interesting open problems. Helmut Strade 1 The classification of simple finite dimensional Lie algebras over algebraically closed fields of characteristic p > 3 Many of the present results and open problems in locally finite dimensional Lie algebra theory concern those algebras which are built out of simple finite this relationship. Many problems about Lie groups can be "linearized", i.e., formulated on the level of Lie algebras, which often makes them more accessible. We also give an introduction in this lecture to representation theory of Lie algebras. We study highest weight modules for complex semisimple Lie algebras. The basic case here is the 1.3. Inertia Lie algebras 8 2. Lie algebroids: de nition and basic pieces of structure 9 2.1. Lie algebroids and the main forgetful functor 9 2.2. From Lie algebroids to Lie algebras 10 3. Examples of Lie algebroids 12 3.1. The tangent and zero Lie algebroids 12 3.2. The Lie algebroid attached to a map 13 3.3. Digression: the universal 1.1. Lie algebras from Algebraic Groups. An algebraic group over a eld F is a collection of polynomials over F in the variables corresponding to entries of a matrix over F. Condition is that the set of invertible solutions for every base extension (algebra over F) is closed under multiplication and inversion in the group of matrices. Use the De nition. Let g and h be two Lie algebras. A Lie (algebra) homomorphism is a linear map ˚: g ! h which commutes with the brackets, i.e. ˚([x;y]) = [˚(x);˚(y)]: It is called an isomorphism if ˚ 1 (exists and) is also a Lie algebra homo-morphism. If ˚has an inverse as a linear map, then the inverse is automatically a Lie homomorphism. To 2. Prove that the Lie algebra from Example 2 is isomorphic to o(3) by comparing the structure constants. 3. Let k= Ror C. The Lie algebras sl(2), o(3), t(2), n(3) all have dimension 3. Are any of these isomorphic? (The answer will depend on k.) Definition 1.2The derived algebra g0of a Lie algebra g is the ideal [gg] gener-ated by [ab], for all LIE ALGEBRAS 3 Since gl 2 is only one more dimension, choose the basis x;y;and has above, and I 1 0 0 1: Since IPZpEndpC2qqwe have rx;Is ry;Is rh;Is 0; So as a Lie algebra gl 2 C 'sl :In general, gl n is one more dimension than sl n, with extra basis element I. Since Iis central, gl n C'sl : The center of a Lie algebra g is the maximal subspace Z—g such that rZ;gs 0. Lie Algebras Contents 4.1 Why Bother? 61 4.2 How to Linearize a