Lie Algebras Of Type Research Articles

Combinatorial Howe duality of symplectic type

We give a new combinatorial interpretation of Howe dual pairs of the form (g,Sp2ℓ), where g is a Lie (super)algebra of classical type. This is done by establishing a symplectic analogue of the RSK algorithm associated to this pair, in a uniform way which does not depend on g. We introduce an analogue of jeu de taquin sliding for spinor model of irreducible characters of a Lie superalgebra g to define a P-tableau and show that the associated Q-tableau is given by a symplectic tableau due to King.

Symmetrisation and the Feigin–Frenkel centre

For complex simple Lie algebras of types B, C, and D, we provide new explicit formulas for the generators of the commutative subalgebra $\mathfrak z(\hat {\mathfrak g})\subset {{\mathcal {U}}}(t^{-1}\mathfrak g[t^{-1}])$ known as the Feigin–Frenkel centre. These formulas make use of the symmetrisation map as well as of some well-chosen symmetric invariants of $\mathfrak g$. There are some general results on the rôle of the symmetrisation map in the explicit description of the Feigin–Frenkel centre. Our method reduces questions about elements of $\mathfrak z(\hat {\mathfrak g})$ to questions on the structure of the symmetric invariants in a type-free way. As an illustration, we deal with type G$_2$ by hand. One of our technical tools is the map ${\sf m}\!\!: {{\mathcal {S}}}^{k}(\mathfrak g)\to \Lambda ^{2}\mathfrak g \otimes {{\mathcal {S}}}^{k-3}(\mathfrak g)$ introduced here. As the results show, a better understanding of this map will lead to a better understanding of $\mathfrak z(\hat {\mathfrak g})$.

The isomorphism problem for universal enveloping algebras of solvable Lie algebras

This paper is a contribution to the isomorphism problem for universal enveloping algebras of finite-dimensional Lie algebras. We focus on solvable Lie algebras of small dimensions over fields of arbitrary characteristic. We prove, over an arbitrary field, that the isomorphism type of a metabelian Lie algebra whose derived subalgebra has codimension one is determined by its universal enveloping algebra. As an application of the results in this paper, we solve the isomorphism problem for solvable Lie algebras of dimension four over fields of characteristic zero and also point out the problems that occur in prime characteristic.

Reeder’s Conjecture for Even Orthogonal Lie Algebras

In the paper we complete a case by case proof of Reeder’s Conjecture started in our previous work, proving the conjecture for simple Lie algebras of type D and for the exceptional cases.

On quantum toroidal algebra of type A1

In this paper we introduce a new quantum algebra which specializes to the $2$-toroidal Lie algebra of type $A_1$. We prove that this quantum toroidal algebra has a natural triangular decomposition, a (topological) Hopf algebra structure and a vertex operator realization.

Nakajima quiver varieties, affine crystals and combinatorics of Auslander-Reiten quivers

We obtain an explicit crystal isomorphism between two realizations of crystal bases of finite dimensional irreducible representations of simple Lie algebras of type A and D. The first realization we consider is a geometric construction in terms of irreducible components of certain Nakajima quiver varieties established by Saito and the second is a realization in terms of isomorphism classes of quiver representations obtained by Reineke. We give a homological description of the irreducible components of Lusztig's quiver varieties which correspond to the crystal of a finite dimensional representation and describe the promotion operator in type A to obtain a geometric realization of Kirillov-Reshetikhin crystals.

Harmonic analysis of boxed hyperoctahedral Hall-Littlewood polynomials

For positive integers n and c with c≥2n (= the Coxeter number of the hyperoctahedral group of signed permutations of degree n), we present a finite-dimensional discrete orthogonality relation for Macdonald's three-parameter hyperoctahedral Hall-Littlewood polynomials of degree at most c in each of the n variables. These polynomials are labeled by partitions λ that fit inside a rectangular box of shape cn, i.e. the partitions in question are of length ≤n and have parts of size ≤c. We employ coordinate patches around the vertices of the alcove of boxed partitions Λ(n,c)={λ⊆cn} to establish the self-adjointness of a one-parameter family of commuting discrete difference operators ▪ acting on functions f:Λ(n,c)→C. By construction, the basis of hyperoctahedral Hall-Littlewood polynomials constitutes a joint eigenbasis for ▪ with simple spectrum, which gives rise to our discrete orthogonality relation. From the point of view of quantum integrable particle dynamics, the present geometric construction establishes the orthogonality of the Bethe Ansatz eigenfunctions for a recently studied q-boson system on a finite lattice with integrable open-end boundary conditions. The Bethe Ansatz equations enter in the geometric picture as compatibility conditions between coordinate patches stemming from distinct vertices. Two applications of the orthogonality relations are highlighted: (i) a cubature rule for the integration of symmetric functions with respect to the Haar measure on the compact symplectic group Sp(n), and (ii) a Verlinde formula for the structure constants of a deformation of the Wess-Zumino-Witten fusion ring associated with the affine Lie algebra of type Cˆn (= Cn(1)).

A generalized Abhyankar’s conjecture for simple Lie algebras in characteristic p>5

In the present paper, we study a purely inseparable counterpart of Abhyankar’s conjecture for the affine line in positive characteristic, and prove its validity for all the finite local non-abelian simple group schemes in characteristic $$p>5$$ . The crucial point is how to deal with finite local group schemes which cannot be realized as the Frobenius kernel of a smooth algebraic group. Such group schemes appear as the ones associated with Cartan type Lie algebras. We settle the problem for such Lie algebras by making use of natural gradations or filtrations on them.

Affine Pieri rule for periodic Macdonald spherical functions and fusion rings

Let gˆ be an untwisted affine Lie algebra or the twisted counterpart thereof (which excludes the affine Lie algebras of type BCˆn=A2n(2)). We present an affine Pieri rule for a basis of periodic Macdonald spherical functions associated with gˆ. In type Aˆn−1=An−1(1) the formula in question reproduces an affine Pieri rule for cylindric Hall-Littlewood polynomials due to Korff, which at t=0 specializes in turn to a well-known Pieri formula in the fusion ring of genus zero slˆ(n)c-Wess-Zumino-Witten conformal field theories.

The Allison–Faulkner construction of

AbstractWe show that the Tits index $E_{8,1}^{133}$ cannot be obtained by means of the Tits construction over a field with no odd degree extensions. The proof uses a general method coming from the theory of symmetric spaces. We construct two cohomological invariants, in degrees $6$ and $8$ , of the Tits construction and the more symmetric Allison–Faulkner construction of Lie algebras of type $E_8$ and show that these invariants can be used to detect the isotropy rank.

On the Semisimple Orbits of Restricted Cartan Type Lie Algebras W, S and H

In this short note, we give a description of semisimple orbits in the restricted Cartan type Lie algebras W, S, H.

Bethe Ansatz Equations for Orthosymplectic Lie Superalgebras and Self-dual Superspaces

We study solutions of the Bethe ansatz equations associated to the orthosymplectic Lie superalgebras $\mathfrak{osp}_{2m+1|2n}$ and $\mathfrak{osp}_{2m|2n}$. Given a solution, we define a reproduction procedure and use it to construct a family of new solutions which we call a population. To each population we associate a symmetric rational pseudo-differential operator $\mathcal R$. Under some technical assumptions, we show that the superkernel $W$ of $\mathcal R$ is a self-dual superspace of rational functions, and the population is in a canonical bijection with the variety of isotropic full superflags in $W$ and with the set of symmetric complete factorizations of $\mathcal R$. In particular, our results apply to the case of even Lie algebras of type D${}_m$ corresponding to $\mathfrak{osp}_{2m|0}=\mathfrak{so}_{2m}$.

Borel subalgebras of restricted Cartan -Type Lie algebras

It is still an open problem to determine the conjugacy classes of Borel subalgebras of non-classical type Lie algebras. In this paper, we prove that there are at least 2 conjugacy classes of Borel subalgebras as well as maximal triangulable subalgebras of restricted Cartan type Lie algebras of type W, S and H. We are particularly interested in maximal triangulable subalgebras of [Formula: see text] under some conditions which is called [Formula: see text]-subalgebras (Definition 3.1). We classify the conjugacy classes of [Formula: see text]-subalgebras for [Formula: see text] and determine their representatives. This paper and its sequel [Z. Lin, K. Ou and B. Shu, Geometric Setting of Jacobson–Witt Algebras, preprint] attempt to establish both algebraic and geometric setting for geometric representation theory of [Formula: see text]

Combinatorics of canonical bases revisited: type A

We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type A. Using geometric objects called rhombic tilings we derive a “Crossing Formula” for the action of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein–Fomin–Zelevinsky we prove an enhanced version of the Anderson–Mirković conjecture for the crystal structure on MV polytopes. We establish a duality between Kashiwara’s string and Lusztig’s parametrization, revealing that each of them is controlled by the crystal structure of the other. We identify the potential functions of the unipotent radical of a Borel subgroup of \(SL_n\) defined by Berenstein–Kazhdan and Gross–Hacking–Keel–Kontsevich, respectively, with a function arising from the crystal structure on Lusztig data.

Cohomology for the Lie algebra of type $A_2$ over a field of characteristic $2$

Cohomology for the Lie algebra of type $A_2$ over a field of characteristic $2$

On a Class of Infinite Simple Lie Conformal Algebras

In this paper, we study a class of infinite simple Lie conformal algebras associated to a class of generalized Block type Lie algebras. The central extensions by a one-dimensional center \(\mathbb {C}\mathfrak {c}\), conformal derivations and free intermediate series modules of this class of Lie conformal algebras are determined. Moreover, we also show that these Lie conformal algebras do not have any non-trivial finite conformal modules. Consequently, these Lie conformal algebras cannot be embedded into gcN for any positive integer N.

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.

The Global Sections of Chiral de Rham Complexes on Compact Ricci-Flat Kähler Manifolds

The space of the global sections of the chiral de Rham complex on a compact Ricci-flat Kähler manifold is calculated and it is expressed as the subspace of invariant elements in a \(\beta \gamma -bc\) system under the action of certain Lie algebra of Cartan type.

CLASSIFICATION OF 3-GRADED CAUSAL SUBALGEBRAS OF REAL SIMPLE LIE ALGEBRAS

Let ( mathfrak{g} , τ) be a real simple symmetric Lie algebra and let W ⊂ mathfrak{g} be an invariant closed convex cone which is pointed and generating with τ(W) = −W. For elements h ∈ mathfrak{g} with τ(h) = h, we classify the Lie algebras mathfrak{g} (W, τ, h) which are generated by the closed convex cones {C}_{pm}left(W,tau, hright):= left(pm Wright)cap {mathfrak{g}}_{pm 1}^{-tau }(h) , where {mathfrak{g}}_{pm 1}^{-tau }(h):= left{xin mathfrak{g}:tau (x)=-xleft[h,xright]=pm xright} . These cones occur naturally as the skew-symmetric parts of the Lie wedges of endomorphism semigroups of certain standard subspaces. We prove in particular that, if mathfrak{g} (W, τ, h) is non-trivial, then it is either a hermitian simple Lie algebra of tube type or a direct sum of two Lie algebras of this type. Moreover, we give for each hermitian simple Lie algebra and each equivalence class of involutive automorphisms τ of mathfrak{g} with τ(W) = −W a list of possible subalgebras mathfrak{g} (W, τ, h) up to isomorphy.

Perfect Integrability and Gaudin Models

We suggest the notion of perfect integrability for quantum spin chains and conjecture that quantum spin chains are perfectly integrable. We show the perfect integrability for Gaudin models associated to simple Lie algebras of all finite types, with periodic and regular quasi-periodic boundary conditions.