### Lie algebras and their representations Aug--Nov 2016 Institute of Mathematical Sciences (IMSc)

Main reference book for the course: Introduction to Lie algebras and Representation Theory, by James Humphreys, which fortunately is available in Indian paperback edition. While we will follow the general scheme of Humphreys's development of the subject, we may not always proceed linearly nor adhere closely on even such matters as notation. Other standard texts will be freely consulted.

To be on the emailing list for the course, send an email to the instructor, K N Raghavan.

The course meets in Room 217 on Mondays, Wednesdays, and Fridays during 1530-1700 hrs. The first meeting is on Wednesday 03 August.

The webpage of the AIS on Lie algebras held at CMI in the summer of 2011 has useful material (e.g., lecture videos).

Detailed list of topics will appear here as the course progresses.

• Basic definitions and examples: Algebra in general: vector space V with a bilinear map V x V to V. Associative algberas and Lie algebras. Any associative algebra is a Lie algebra with commutator as bracket. Derivation of an algebra: a linear endomorphism satisfying Liebniz rule. Derivations of any algebra A form a Lie subalgebra of the Lie algebra gl(A). adX is a derivation; inner derivations; ad is a Lie algebra homomorphism from L to Der(L) with kernel the centre Z(L).
Structure constants for gl(V) in terms of the standard basis (with respect to a given basis of V). Linear Lie algebra: statement of Ado's theorem that every finite dimensional Lie algebra is a linear Lie algebra.
Homework: exercises at the end of section 1 in Humphreys
• Ideals and quotients; ideals as kernels of homomorphisms; abelian Lie algebras and simple Lie algebras; statement of the classification of complex simple Lie algebras; the special linear Lie algebra; commutation relations for sl_n; simplicity of sl_2 in characteristic not 2;
• Nilpotent and solvable Lie algebras. The Lie algebra of upper triangular matrices is solvable, that of strictly upper triangular ones is nilpotent. Subalgebras and quotients of solvable (respectively nilpotent) Lie algebras are solvable (respectively nilpotent). If an ideal and the quotient by it are both solvable, then the Lie algebra is solvable. Caveat: The last statement with "solvable" replaced by "nilpotent" is false. However, if the quotient by a central ideal is nilpotent, then the Lie algebra is so.
• Engel's theorem: If a Lie algebra acts on a finite dimensional vector space V such that each of its elements acts nilpotently, then there exists a basis of V with respect to which the matrices of all elements of the Lie algebra are strictly upper triangular. In particular, the image of the Lie algebra in gl(V) is nilpotent. Observation (used in the proof): If X is a nilpotent endomorphism of a finite dimensional vector space V, then the action of ad(X) on gl(V) is also nilpotent (because ad(x) = l(x)-r(X), where l(X) and r(X) are operators of multiplication on the left and right respectively by X; these commute with each other and are nilpotent if X is so); Corollary: If every inner derivation of a finite dimensional Lie algebra is nilpotent, then the Lie algebra is nilpotent. Notes.
• Jordan decomposition (JD) : (over an algebraically closed base field) Given a linear operator T on a finite dimensional vector space, there exist polynomials s(X) and n(X) (with coefficients in the base field) without constant term such that s(T) is semisimple, n(T) is nilpotent, and T=s(T)+n(T). In particular, s(T) and n(T) commute with T and hence with each other. Uniqueness: if T=S+N such that S is semisimple, N is nilpotent, and SN=NS, then S=s(T) and N=n(T).
If T=S+N be the JD of a linear transformation T on a finite dimensional vector space V, then ad(T)=ad(X)+ad(N) is the JD of ad(T). Indeed, ad preserves semisimplicity, nilpotency, and commutativity.
If D is a derivation of a finite dimensional algebra, then so are its Jordan components: formula for the expansion of divided powers of D-a-b.
• (Finite dimensional) Representations of sl(2,C): Let V be a finite dimensional representation.
• Weights and weight vectors: H acts semisimply on V (by preservation of JD). The eigenvalues of H are called the weights of V. By a weight vector we mean a (non-zero) eigenvector for H. By a weight space we mean an eigenspace of H.
• Action of X and Y on weight vectors: If v is a weight vector of weight k, then Xv is a weight vector of weight k+2, for HXv=[H,X]v+XHv=2Xv+kXv=(k+2)Xv; similarly Yv is a weight vector of weight k-2. Given a weight k, the sum of all weight spaces corresponding to weights k+2n, n an integer, is invariant. Thus, if V is irreducible, any two weights differ from each other by an even integer.
• Maximal vectors: A weight vector v is called maximal if Xv=0. Such vectors exist if V is not 0: by finite dimensionality there exists a weight k such that k+2 is not a weight, and any weight vector of weight k is then maximal.
• Basic identities determining the action: Let v0 be a maximal vector of weight k, and set vj=Y(j)v0. For all non-negative integers j (with the understanding that v-1=0), we have Hvj=(k-2j)vj; Yvj=(j+1)vj+1; and Xvj=(k+1-j)vj-1. (The last identity is proved by induction.)
• Conclusions: Suppose V is non-zero. Let v0 be a maximal vector of weight k. Then k is a non-negative integer and, with notation as in the previous item, vk+1=0. If V is irreducible, then V is spanned by vj, for j between 0 and k. Thus the weights of V are k, k-2, ..., 2-k, -k. In particular, the dimension of V is k+1. Any two irreducible representations of the same dimension are isomorphic.
• Realization of the irreducible representations, one for every non-negative integer: Consider the action on the polynomial ring in two variables u and v. Let X act as uDv, Y as vDu, and H as uDu-vDv, where Du and Dv are the derivations with respect to u and v respectively. Then the homogeneous polynomials of degree k form an irreducible representation of dimension k+1.
• Root systems: as covered in class from the text