Lie Algebras Of Classical Type Research Articles

Multicomponent KP type hierarchies and their reductions, associated to conjugacy classes of Weyl groups of classical Lie algebras

This, to a large extent, expository paper describes the theory of multicomponent hierarchies of evolution equations of XKP type, where X = A, B, C, or D, and AKP = KP and their reductions, associated with the conjugacy classes of the Weyl groups of classical Lie algebras of type X. As usual, the main tool is the multicomponent boson–fermion correspondence, which leads to the corresponding tau-functions, wave functions, dressing operators, and Lax operators.

On simple modules with singular highest weights for so2l+1(K)

In this paper, we study formal characters of simple modules with singular highest weights over classical Lie algebras of type B over an algebraically closed field of characteristic p ≥ h, where h is the Coxeter number. Assume that the highest weights of these simple modules are restricted. We have given a description of their formal characters. In particular, we have obtained some new examples of simple Weyl modules. In the restricted region, the representation theory of algebraic groups and its Lie algebras are equivalent. Therefore, we can use the tools of the representation theory of semisimple and simply-connected algebraic groups in positive characteristic. To describe the formal characters of simple modules, we construct Jantzen filtrations of Weyl modules of the corresponding highest weights.

Yang-Yang functions, monodromy and knot polynomials

We derive a structure of ℤ[t, t−1]-module bundle from a family of Yang-Yang functions. For the fundamental representation of the complex simple Lie algebra of classical type, we give explicit wall-crossing formula and prove that the monodromy representation of the ℤ[t, t−1]-module bundle is equivalent to the braid group representation induced by the universal R-matrices of Uh(g). We show that two transformations induced on the fiber by the symmetry breaking deformation and respectively the rotation of two complex parameters commute with each other.

On the Gaudin model associated to Lie algebras of classical types

We derive explicit formulas for solutions of the Bethe ansatz equations of the Gaudin model associated to the tensor product of one arbitrary finite-dimensional irreducible module and one vector representation for all simple Lie algebras of classical type. We use this result to show that the Bethe ansatz is complete in any tensor product where all but one factor are vector representations and the evaluation parameters are generic.

Extremal part of the PBW-filtration and nonsymmetric Macdonald polynomials

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit t→∞ and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric q-Whittaker function. These coefficients are pure q-powers and their degrees are expected to coincide in the untwisted setting with the extremal degrees of the so-called PBW-filtration in the corresponding finite-dimensional irreducible representations of the simple Lie algebras for any root systems. This is a particular case of a general conjecture in terms of the level-one Demazure modules. We prove this coincidence for all Lie algebras of classical type and for G2, and also establish the relations of our extremal degrees to minimal q-degrees of the extremal terms of the Kostant q-partition function; they coincide with the latter only for some root systems.

Conformal Embeddings and Simple Current Extensions

In this paper we investigate the structure of intermediate vertex algebras associated with a maximal conformal embedding of a reductive Lie algebra in a semisimple Lie algebra of classical type.

Littlewood identity and crystal bases

We give a new combinatorial model for the crystals of integrable highest weight modules over the classical Lie algebras of type B and C in terms of classical Young tableaux. We then obtain a new description of its Littlewood–Richardson rule and a maximal Levi branching rule in terms of classical Littlewood–Richardson tableaux, which extends in a bijective way the well-known stable formulas at large ranks. We also show that this tableau model admits a natural superization and it produces the characters of irreducible highest weight modules over orthosymplectic Lie superalgebras, which correspond to the integrable highest weight modules over the classical Lie algebras of type B and C under the Cheng–Lam–Wangʼs super duality.

Weyl groups for non-classical restricted Lie algebras and the Chevalley Restriction Theorem

Abstract Let ( 𝔤 , [ p ] ) $(\mathfrak {g},[p])$ be a finite-dimensional restricted Lie algebra, defined over an algebraically closed field k of characteristic p > 0. The scheme of tori of maximal dimension of 𝔤 gives rise to a finite group S(𝔤) that coincides with the Weyl group of 𝔤 in case 𝔤 is a Lie algebra of classical type. In this paper, we compute the group S(𝔤) for Lie algebras of Cartan type and provide applications concerning weight space decompositions, the existence of generic tori and polynomial invariants.

Covariant Dirac operators on quantum groups

We give a construction of a Dirac operator on a quantum group based on any simple Lie algebra of classical type. The Dirac operator is an element in the vector space Uq(g)⊗ cl q(g), where the second tensor factor is a q-deformation of the classical Clifford algebra. The tensor space Uq(g)⊗ cl q(g) is given by a structure of the adjoint module of the quantum group and the Dirac operator is invariant under this action. The purpose of this approach is to construct equivariant Fredholm modules and K-homology cycles. This work generalizes the operator introduced by P. N. Bibikov and P. P. Kulish [J. Math. Sci. (N.Y.) 100, 2039–2050 (2000)].

Combinatorics of the Springer correspondence for classical Lie algebras and their duals in characteristic 2

We give a combinatorial description of the Springer correspondence for classical Lie algebras of type B, C or D and their duals in characteristic 2. The combinatorics used here is of the same kind as those appearing in the description of (generalized) Springer correspondence for unipotent case of classical groups by Lusztig in odd characteristic and by Lusztig and Spaltenstein in characteristic 2.

Symplectic resolutions for nilpotent orbits (III)

We prove that two symplectic resolutions of a nilpotent orbit closures in a simple complex Lie algebra of classical type are related by Mukai flops in codimension 2. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 342 (2006).

Weight multiplicities for affine Lie algebras of type [formula omitted] and Kostka numbers

In this paper, we prove the conjecture on the polynomial behavior of weight multiplicities for affine Lie algebras of type A r ( 1 ) , which was originally due to Benkart and Kass. More precisely, we prove that the degree of the weight multiplicity function m λ ( μ ) is equal to the depth associated with dominant integral weights λ and μ, and compute the leading coefficient of m λ ( μ ) explicitly in terms of a Kostka number. As applications, we verify other conjectures on the leading coefficient of m λ ( μ ) , and the polynomial behavior of weight multiplicities for classical Lie algebras of type A r .

On some noncommutative algebras related to K-theory of flag varieties. Part I

For any Lie algebra of classical type or type G2 we define a K-theoretic analog of Dunkl's elements, the so-called truncated Ruijsenaars-Schneider-Macdonald elements, RSM-elements for short, in the corresponding Yang-Baxter group, which form a commuting family of elements in the latter. For the root systems of type A we prove that the subalgebra of the bracket algebra generated by the RSM-elements is isomorphic to the Grothendieck ring of the flag variety. In general, we prove that the subalgebra generated by the images of the RSM-elements in the corresponding Nichols-Woronowicz algebra is canonically isomorphic to the Grothendieck ring of the corresponding flag varieties of classical type or of type G2. In other words, we construct the “Nichols-Woronowicz algebra model” for the Grothendieck calculus on Weyl groups of classical type or type G2, providing a partial generalization of some recent results by Y. Bazlov. We also give a conjectural description (theorem for type A) of a commutative subalgebra generated by the truncated RSM-elements in the bracket algebra for the classical root systems. Our results provide a proof and generalizations of recent conjecture and result by C. Lenart and A. Yong for the root system of type A.

Quantum zonal spherical functions and Macdonald polynomials

A unified theory of quantum symmetric pairs is applied to q-special functions. Previous work characterized certain left coideal subalgebras in the quantized enveloping algebra and established an appropriate framework for quantum zonal spherical functions. Here a distinguished family of such functions, invariant under the Weyl group associated to the restricted roots, is shown to be a family of Macdonald polynomials, as conjectured by Koornwinder and Macdonald. Our results place earlier work for Lie algebras of classical type in a general context and extend to the exceptional cases.

Symplectic resolutions for nilpotent orbits (II)

Based on our previous work, Fu (Invent. Math. 151 (2003) 167–186), we prove that, given any two projective symplectic resolutions Z 1 and Z 2 of a nilpotent orbit closure in a complex simple Lie algebra of classical type, Z 1 is deformation equivalent to Z 2. This provides support for a ‘folklore’ conjecture on symplectic resolutions for symplectic singularities. To cite this article: B. Fu, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Homology stability for classical regular semisimple varieties

We use techniques from homotopy theory, in particular the connection between configuration spaces and iterated loop spaces, to give geometric explanations of stability results for the cohomology of the varieties of regular semisimple elements in the simple complex Lie algebras of classical type A, B or C, as well as in the group \(GL_n({\Bbb C})\). We show that the cohomology spaces of stable versions of these varieties have an algebraic stucture, which identifies them as “free Poisson algebras” with suitable degree shifts. Using this, we are able to give explicit formulae for the corresponding Poincare series, which lead to power series identities by comparison with earlier work. The cases of type B and C involve ideas from equivariant homotopy theory. Our results may be interpreted in terms of the actions of a Weyl group on its coinvariant algebra (i.e. the coordinate ring of the affine space on which it acts, modulo the invariants of positive degree; this space coincides with the cohomology ring of the flag variety of the associated Lie group) and on the cohomology of its associated complex discriminant variety.

Extended matrix Gelfand - Dickey hierarchies: reduction to classical Lie algebras

The Drinfeld-Sokolov reduction method has been used to associate with $gl_n$ extensions of the matrix r-KdV system. Reductions of these systems to the fixed point sets of involutive Poisson maps, implementing reduction of $gl_n$ to classical Lie algebras of type $B, C, D$, are here presented. Modifications corresponding, in the first place to factorisation of the Lax operator, and then to Wakimoto realisations of the current algebra components of the factorisation, are also described.

Some $q$-beta and Mellin-Barnes integrals on compact Lie groups and Lie algebras

Multidimensional generalizations of beta type integrals of Bames, Ramanujan, Askey-Wilson, and others are evaluated. These integrals are analogues of the summation theorems for multilateral hypergeometric series associated to the simple Lie algebras of classical type and type G2 . Many of these integrals can also be written as group integrals over a compact Lie group or conjugation invariant integrals over the corresponding Lie algebra.

A construction of modular representations of classical Lie algebras

In this paper, we construct and analyze new classes of modular representations of classical Lie algebras of type C and type A. These representations include a class of pointed torsion free representations, a class of irreducible nonrestricted representations, and a class of indecomposable representations of arbitrary high dimension. The construction is based on the realization of these Lie algebras in the modular Weyl algebras acting on truncated polynomial algebras. We also classify all the irreducible representations of the modular Weyl algebra.

Finite Cartan matrices and Lie algebras of classical type

The Mills-Seligman result is that a classical Lie algebra L is a direct sum of simple ideals, each of which is again classical; that the simple classical Lie algebras are determined by their Cartan matrices; and that the possible indecomposable Cartan matrices are given by the usual classification, i.e., are of types A,,, B, ,..., G,. The proof of this result, in [3], is surprisingly hard. The difficulty arises from the fact that the root system of L is contained in H*, a vector space over F, while on the other hand, the Cartan matrix must be a matrix over H. Deducing information about the Cartan matrix (over Z) from information about the root system (over F) is difftcult. (See, for example, Lemmas 115.3 and 11.5.4 of [3].) In this paper we give a simpler proof of the Mills-Seligman theorem. Our technique is to associate to L a certain subset, A, of H’ which plays the role of root system and apply standard characteristic zero results (of [ 11) to A. These transference methods were used before by Seligman (see [4]), but only after much more theory in characteristic p was developed.