Simply-laced Case Research Articles

Combinatorial Structure of Finite Dimensional Representations of Yangians: the Simply-Laced Case

We compute the decomposition of representations of Yangians into g-modules for simply-laced g. The decomposition has an interesting combinatorial tree structure. Results depend on a conjecture of Kirillov and Reshetikhin.

Minimal Affinizations of Representations of Quantum Groups: The Simply Laced Case

If Uq(g) is a finite-dimensional complex simple Lie algebra, an affinization of a finite-dimensional irreducible representationV of Uq(g) is a finite-dimensional irreducible representation\(\hat V\) of Uq(ĝ) which containsV with multiplicity one, and is such that all other Uq(g)-types in\(\hat V\) have highest weights strictly smaller than that ofV. There is a natural partial ordering\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec } \) on the set of affinizations, defined by Chari. In this Letter, we describe the minimal affinizations, with respect to\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \prec } \), when g is not simply-laced.

Lie algebras graded by 3-graded root systems and Jordan pairs covered by grids

Let R be an irreducible root system. A Lie Algebra L is called graded by R if L is graded with grading group the root lattice of R such that the only nonzero homogeneous subspaces of L have degree 0 or a root in R , the grading is induced by the adjoint action of a split Cartan subalgebra of a finite-dimensional simple Lie subalgebra of L with root system R , and L is generated by the homogeneous subspaces of nonzero degree. This class of Lie algebras was introduced and studied by S. Berman and R. Moody in Invent. Math. 108 (1992), where, in particular, a classification up to central equivalence is given in the simply-laced case. The doubly-laced cases have recently been classified by G. Benkart and E. Zelmanov. Let R be a 3-graded root system, i.e., R is not of type E_8, F_4 or G_2. In this paper, Lie algebras graded by R are described in a unified way, without case-by-case considerations. Namely, it is shown that a Lie algebra L is 3-graded if and only if L is a central extension of the Tits-Kantor-Koecher algebra of a Jordan pair covered by a grid whose associated 3-graded root system is isomorphic to R . This result is then used to classify Lie algebras graded by R : we give the classification of Jordan pairs covered by a grid and describe their Tits-Kantor-Koecher algebras. One of the advantages of this approach is that it works over rings containing 1/2 and 1/3, and also for infinite root systems. Another application is the description of Slodowy's intersection matrix algebras arising from multiply-affinized Cartan matrices.

Thermodynamics of the U q(X r(1)) Bethe ansatz system with q a root of unity

The Bethe ansatz equations connected to the U q (X r (1))algebra are studied at q = exp(2πi (l + g)) , with l being a level and g the dual Coxeter number. Based on a “string hypothesis” in the thermodynamic limit, the central charges relevant to the X r (1) restricted solid-on-solid models and their fusion hierarchies are determined in two critical regimes. The calculation leads naturally to a generalized conjecture on the Rogers dilogarithmic function identity for an arbitrary pair (X r (1), l), which includes the earlier one for the simply-laced cases X r (1) = A r (1), D r (1) and E 6,7,8 (1).