ON THE DYNKIN INDEX OF A PRINCIPAL sl2-SUBALGEBRA
نویسندگان
چکیده
The ground field k is algebraically closed and of characteristic zero. Let g be a simple Lie algebra over k. The goal of this note is to prove a closed formula for the Dynkin index of a principal sl2-subalgebra of g, see Theorem 3.2. The key step in the proof uses the “strange formula” of Freudenthal–de Vries. As an application, we (1) compute the Dynkin index any simple g-module regarded as sl2-module and (2) obtain an identity connecting the exponents of g and the dual Coxeter numbers of both g and g, see Section 4.
منابع مشابه
Annihilators of tensor density modules
We describe the two-sided ideals in the universal enveloping algebras of the Lie algebras of vector fields on the line and the circle which annihilate the tensor density modules. Both of these Lie algebras contain the projective subalgebra, a copy of sl2. The restrictions of the tensor density modules to this subalgebra are duals of Verma modules (of sl2) for Vec(R) and principal series modules...
متن کاملEf − Fe =
Let Uq(sl2) be the quantized enveloping algebra associated to the simple Lie algebra sl2. In this paper, we study the quantum double Dq of the Borel subalgebra Uq((sl2) ) of Uq(sl2). We construct an analogue of Kostant–Lusztig Z[v, v]-form for Dq and show that it is a Hopf subalgebra. We prove that, over an algebraically closed field, every simple Dq-module is the pullback of a simple Uq(sl2)-m...
متن کاملSmall Semisimple Subalgebras of Semisimple Lie Algebras
The goal of Section 2 is to provide a proof of Theorem 2.0.1. Section 3 introduces the necessary facts about Lie algebras and representation theory, with the goal being the proof of Proposition 3.5.7 (ultimately as an application of Theorem 2.0.1), and Proposition 3.3.1. In Section 4 we prove the main theorem, using Propositions 3.3.1 and 3.5.7. In Section 5, we apply the theorem to the special...
متن کاملGeneralized Drinfeld Polynomials for Highest Weight Vectors of the Borel Subalgebra of the Sl2 Loop Algebra
In a Borel subalgebra U(B) of the sl2 loop algebra, we introduce a highest weight vector Ψ. We call such a representation of U(B) that is generated by Ψ highest weight. We define a generalization of the Drinfeld polynomial for a finitedimensional highest weight representation of U(B). We show that every finitedimensional highest weight representation of the Borel subalgebra is irreducible if th...
متن کاملOn the universal sl2 invariant of ribbon bottom tangles
A bottom tangle is a tangle in a cube consisting of arc components whose boundary points are on a line in the bottom square of the cube. A ribbon bottom tangle is a bottom tangle whose closure is a ribbon link. For every n-component ribbon bottom tangle T , we prove that the universal invariant JT of T associated to the quantized enveloping algebra Uh(sl2) of the Lie algebra sl2 is contained in...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009