The course is recommended to students in F4 specializing either in mathematical physics or in mathematics, and also to interested PhD students. The course is given in English. Recommended prerequisites: Good knowledge of linear algebra. Familiarity with abstract algebra, for example algebra course SF 5B or discrete mathematics course SF 5B Kac and A. Raina: Bombay lectures on highest weight representations of infinite-dimensional Lie algebras, World Scientific Publ. INL1 - Assignments, 4.
The examiner may apply another examination format when re-examining individual students. A combination of hand-in homework exercises INL1; 4,5 university credits and of a written or oral examination TEN1; 3 university credits. Jouko Michelsson. Iachello is a renowned physicist.
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From the reviews of the first edition: "Iachello has written a pedagogical and straightforward presentation of Lie algebras [ For instance, in the general linear group , which can be identified with the Lie algebra of complex matrices, the subalgebra of strictly upper triangular matrices is nilpotent but not abelian for , while the subalgebra of upper triangular matrices is solvable but not nilpotent for.
It is also clear that any subalgebra of a nilpotent algebra is nilpotent, and similarly for solvable or abelian algebras. From the above discussion we see that a Lie algebra is solvable if and only if it can be represented by a tower of abelian extensions, thus. Similarly, a Lie algebra is nilpotent if it is expressible as a tower of central extensions so that in all the extensions in the above factorisation, is central in , where we say that is central in if.
We also see that an extension is solvable if and only of both factors are solvable. Splitting abelian algebras into cyclic i. For our next fundamental example of using short exact sequences to split a general Lie algebra into simpler objects, we observe that every abstract Lie algebra has an adjoint representation , where for each , is the linear map ; one easily verifies that this is indeed a representation indeed, 2 is equivalent to the assertion that for all.
The kernel of this representation is the center , which the maximal central subalgebra of. We thus have the short exact sequence. For our next fundamental decomposition of Lie algebras, we need some more definitions. A Lie algebra is simple if it is non-abelian and has no ideals other than and ; thus simple Lie algebras cannot be factored into strictly smaller algebras.
In particular, simple Lie algebras are automatically perfect and centerless.
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We have the following fundamental theorem:. Theorem 1 Equivalent definitions of semisimplicity Let be a Lie algebra. Then the following are equivalent:. We review the proof of this theorem later in these notes. A Lie algebra obeying any and hence all of the properties i - iv is known as a semisimple Lie algebra. The equivalence of i and ii is easy. If and are solvable ideals of a Lie algebra , then it is not difficult to see that the vector sum is also a solvable ideal because on quotienting by we see that the derived series of must eventually fall inside , and thence must eventually become trivial by the solvability of.
As our Lie algebras are finite dimensional, we conclude that has a unique maximal solvable ideal, known as the radical of. The quotient is then a Lie algebra with trivial radical, and is thus semisimple by the above theorem, giving the Levi decomposition. Indeed, a deep theorem of Levi allows one to upgrade this decomposition to a split extension.
In view of the above decompositions, we see that we can factor any Lie algebra using a suitable combination of direct sums and extensions into a finite number of simple Lie algebras and the scalar algebra. In principle, this means that one can understand an arbitrary Lie algebra once one understands all the simple Lie algebras which, being defined over , are somewhat confusingly referred to as simple complex Lie algebras in the literature.
Amazingly, this latter class of algebras are completely classified:.
Theorem 2 Classification of simple Lie algebras Up to isomorphism, every simple Lie algebra is of one of the following forms:. The precise definition of the classical Lie algebras and the exceptional Lie algebras will be recalled later. One can extend the families of classical Lie algebras a little bit to smaller values of , but the resulting algebras are either isomorphic to other algebras on this list, or cease to be simple; see this previous post for further discussion.
This classification is a basic starting point for the classification of many other related objects, including Lie algebras and Lie groups over more general fields e. Being so fundamental to the subject, this classification is covered in almost every basic textbook in Lie algebras, and I myself learned it many years ago in an honours undergraduate course back in Australia. The proof is rather lengthy, though, and I have always had difficulty keeping it straight in my head.
Introduction to Lie Algebras | DiGMi
So I have decided to write some notes on the classification in this blog post, aiming to be self-contained though moving rapidly. In fact it seems remarkably hard to deviate from the standard routes given in the literature to the classification; I would be interested in knowing about other ways to reach the classification or substeps in that classification that are genuinely different from the orthodox route. One of the key strategies in the classification of a Lie algebra is to work with representations of , particularly the adjoint representation , and then restrict such representations to various simpler subalgebras of , for which the representation theory is well understood.
In particular, one aims to exploit the representation theory of abelian algebras and to a lesser extent, nilpotent and solvable algebras , as well as the fundamental example of the two-dimensional special linear Lie algebra , which is the smallest and easiest to understand of the simple Lie algebras, and plays an absolutely crucial role in exploring and then classifying all the other simple Lie algebras. We begin this program by recording the representation theory of abelian Lie algebras. We begin with representations of the one-dimensional algebra.
Setting , this is essentially the representation theory of a single linear transformation. Here, the theory is given by the Jordan decomposition. Firstly, for each complex number , we can define the generalised eigenspace. One easily verifies that the are all linearly independent -invariant subspaces of , and in particular that there are only finitely many the spectrum of for which is non-trivial. Thus the generalised eigenspaces span :.
On each space , the operator only has spectrum at zero, and thus again from the fundamental theorem of algebra has non-trivial kernel; similarly for any -invariant subspace of , such as the range of. Iterating this observation we conclude that is a nilpotent operator on , thus for some. If we then write to be the direct sum of the scalar multiplication operators on each generalised eigenspace , and to be the direct sum of the operators on these spaces, we have obtained the Jordan decomposition or Jordan-Chevalley decomposition.
401-3172-10L Lie Algebras
Furthermore, as we may use polynomial interpolation to find a polynomial such that vanishes to arbitrarily high order at for each and also , we see that and hence can be expressed as polynomials in with zero constant coefficient; this fact will be important later. In particular, and commute. Conversely, given an arbitrary linear transformation , the Jordan-Chevalley decomposition is the unique decomposition into commuting semisimple and nilpotent elements. Indeed, if we have an alternate decomposition into a semisimple element commuting with a nilpotent element , then the generalised eigenspaces of must be preserved by both and , and so without loss of generality we may assume that there is just a single generalised eigenspace ; subtracting we may then assume that , but then is nilpotent, and so is also nilpotent; but the only transformation which is both semisimple and nilpotent is the zero transformation, and the claim follows.
From the Jordan-Chevalley decomposition it is not difficult to then place in Jordan normal form by selecting a suitable basis for ; see e. But in contrast to the Jordan-Chevalley decomposition, the basis is not unique in general, and we will not explicitly use the Jordan normal form in the rest of this post. Given an abstract complex vector space , there is in general no canonical notion of complex conjugation on , or of linear transformations.
However, we can define the conjugate of any semisimple transformation , defined as the direct sum of on each eigenspace of. In particular, we can define the conjugate of the semisimple component of an arbitrary linear transformation , which will be the direct sum of on each generalised eigenspace of. The significance of this transformation lies in the observation that the product has trace on each generalised eigenspace since nilpotent operators have zero trace , and in particular we see that.
Thus 7 provides a test for nilpotency, which will be turn out to be quite useful later in this post.
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Note that this trick relies very much on the special structure of , in particular the fact that it has characteristic zero. In the above arguments we have used the basic fact that if two operators and commute, then the generalised eigenspaces of one operator are preserved by the other. Iterating this fact, we can now start understanding the representations of an abelian Lie algebra.