Finite-dimensional Complex Simple Lie Algebra Research Articles

Categories of weight modules for unrolled restricted quantum groups at roots of unity

Motivated by connections to the singlet vertex operator algebra in the $\mathfrak{g}=\mathfrak{sl}_2$ case, we study the unrolled restricted quantum group $\overline{U}_q^{ H}(\mathfrak{g})$ for any finite dimensional complex simple Lie algebra $\mathfrak{g}$ at arbitrary roots of unity with a focus on its category of weight modules. We show that the braid group action naturally extends to the unrolled quantum groups and that the category of weight modules is a generically semi-simple ribbon category (previously known only for odd roots) with trivial Müger center and self-dual projective modules.

Weyl modules and Weyl functors for hyper-map algebras

We investigate the representations of the hyperalgebras associated to the map algebras [Formula: see text], where [Formula: see text] is any finite-dimensional complex simple Lie algebra and [Formula: see text] is any associative commutative unitary algebra with a multiplicatively closed basis. We consider the natural definition of the local and global Weyl modules, and the Weyl functor for these algebras. Under certain conditions, we prove that these modules satisfy certain universal properties, and we also give conditions for the local or global Weyl modules to be finite-dimensional or finitely generated, respectively.

Yangians and Baxter’s relations

We study a category O of representations of the Yangian associated to an arbitrary finite-dimensional complex simple Lie algebra. We obtain asymptotic modules as analytic continuation of a family of finite-dimensional modules, the Kirillov--Reshetikhin modules. In the Grothendieck ring we establish the three-term Baxter's TQ relations for the asymptotic modules. We indicate that Hernandez--Jimbo's limit construction can also be applied, resulting in modules over anti-dominantly shifted Yangians.

Finite-Dimensional Representations of Hyper Multicurrent and Multiloop Algebras

We investigate the categories of finite-dimensional representations of multicurrent and multiloop hyperalgebras in positive characteristic, i.e., the hyperalgebras associated to the multicurrent algebras $\mathfrak {g}\otimes \mathbb {C}[t_{1},\ldots ,t_{n}]$ and to the multiloop algebras $\mathfrak {g}\otimes \mathbb {C}[t_{1}^{\pm 1},\ldots ,t_{n}^{\pm 1}]$ , where $\mathfrak {g}$ is any finite-dimensional complex simple Lie algebra. The main results are the construction of the universal finite-dimensional highest-weight modules and a classification of the irreducible modules in each category. In the characteristic zero setting we also provide a relationship between these modules.

Bases for local Weyl modules for the hyper and truncated current [formula omitted]-algebras

We use the theory of Gröbner–Shirshov bases for ideals to construct linear bases for graded local Weyl modules for the (hyper) current and the truncated current algebras associated to the finite-dimensional complex simple Lie algebra sl2. The main result is a characteristic-free construction of bases for this important family of modules for the hyper current sl2-algebra. In the positive characteristic setting this work represents the first construction in the literature. In the characteristic zero setting, the method provides a different construction of the Chari–Pressley (also Kus–Littelmann) bases and the Chari–Venkatesh bases for local Weyl modules for the current sl2-algebra. Our construction allows us to obtain new bases for the local Weyl modules for truncated current sl2-algebras with very particular properties.

Biderivations of finite-dimensional complex simple Lie algebras

In this paper, we prove that a biderivation of a finite-dimensional complex simple Lie algebra without the restriction of being skewsymmetric is an inner biderivation. As an application, the biderivation of a general linear Lie algebra is presented. In particular, we find a class of a non-inner and non-skewsymmetric biderivations. Furthermore, we also obtain the forms of the linear commuting maps on the finite-dimensional complex simple Lie algebra or general linear Lie algebra.

Generalized Demazure modules and fusion products

Let g be a finite-dimensional complex simple Lie algebra with highest root θ and let g[t] be the corresponding current algebra. In this paper, we consider the g[t]-stable Demazure modules associated to integrable highest weight representations of the affine Lie algebra gˆ. We prove that the fusion product of Demazure modules of a given level with a single Demazure module of a different level and with highest weight a multiple of θ is a generalized Demazure module, and also give defining relations. This also shows that the fusion product of such Demazure modules is independent of the chosen parameters. As a consequence we obtain generators and relations for certain types of generalized Demazure modules. We also establish a connection with the modules defined by Chari and Venkatesh.

Spin norm, pencils, and the u-small convex hull

Let g \mathfrak {g} be any finite-dimensional complex simple Lie algebra. In this paper, we show that the spin norm increases strictly along any pencil once it goes beyond the u-small convex hull of g \mathfrak {g} .

Commuting Derivations and Automorphisms of Certain Nilpotent Lie Algebras Over Commutative Rings

Let L be a finite-dimensional complex simple Lie algebra, L ℤ be the ℤ-span of a Chevalley basis of L, and L R = R ⊗ℤ L ℤ be a Chevalley algebra of type L over a commutative ring R. Let 𝒩(R) be the nilpotent subalgebra of L R spanned by the root vectors associated with positive roots. A map ϕ of 𝒩(R) is called commuting if [ϕ(x), x] = 0 for all x ∈ 𝒩(R). In this article, we prove that under some conditions for R, if Φ is not of type A 2, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a central derivation (resp., automorphism), and if Φ is of type A 2, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism).

Demazure Modules, Chari-Venkatesh Modules and Fusion Products

Let $\mathfrak{g}$ be a finite-dimensional complex simple Lie algebra with highest root $\theta$. Given two non-negative integers $m$, $n$, we prove that the fusion product of $m$ copies of the level one Demazure module $D(1,\theta)$ with $n$ copies of the adjoint representation ${\rm ev}_0 V(\theta)$ is independent of the parameters and we give explicit defining relations. As a consequence, for $\mathfrak{g}$ simply laced, we show that the fusion product of a special family of Chari-Venkatesh modules is again a Chari-Venkatesh module. We also get a description of the truncated Weyl module associated to a multiple of $\theta$.

Fusion Product Structure of Demazure Modules

Let 𝔤 be a finite–dimensional complex simple Lie algebra. Given a non–negative integer ℓ, we define $\mathcal {P}^{+}_{\ell }$ to be the set of dominant weights λ of 𝔤 such that ℓΛ0+λ is a dominant weight for the corresponding untwisted affine Kac–Moody algebra $\widehat {{\mathfrak {g}}}$ . For the current algebra 𝔤[t] associated to 𝔤, we show that the fusion product of an irreducible 𝔤–module V(λ) such that $\lambda \in \mathcal {P}^{+}_{\ell }$ and a finite number of special family of 𝔤–stable Demazure modules of level ℓ (considered in Fourier and Littelmann, Nagoya Math. J. 182, 171–198 (2006), Adv. Math. 211(2), 566–593 2007) again turns out to be a Demazure module. This fact is closely related with several important conjectures. We use this result to construct the 𝔤[t]–module structure of the irreducible ${\widehat {\mathfrak {g}}}$ –module V(ℓ Λ0 + λ) as a semi–infinite fusion product of finite dimensional 𝔤[t]–modules as conjectured in Fourier and Littelmann, Adv. Math. 211(2), 566–593 (2007). As a second application we give further evidence to the conjecture on the generalization of Schur positivity (see Chari, Fourier and Sagaki 2013).

Tilting Modules in Truncated Categories

We begin the study of a tilting theory in certain truncated categories of modules $\mathcal G(\Gamma)$ for the current Lie algebra associated to a finite-dimensional complex simple Lie algebra, where $\Gamma = P^+ \times J$, $J$ is an interval in $\mathbb Z$, and $P^+$ is the set of dominant integral weights of the simple Lie algebra. We use this to put a tilting theory on the category $\mathcal G(\Gamma')$ where $\Gamma' = P' \times J$, where $P'\subseteq P^+$ is saturated. Under certain natural conditions on $\Gamma'$, we note that $\mathcal G(\Gamma')$ admits full tilting modules.

Invertible Linear Maps on Simple Lie Algebras Preserving Solvability

Let 𝔤 be a (finite-dimensional) complex simple Lie algebra of rank l. An invertible linear map ϕ on 𝔤 is said to preserve solvability in both directions if ϕ, as well as ϕ−1, sends every solvable subalgebra to some solvable one. In this article, it is shown that an invertible linear map ϕ on 𝔤 preserves solvability in both directions if and only if it can be decomposed into the product of an inner automorphism, a graph automorphism, a scalar multiplication map and a diagonal automorphism.

Automorphisms and Derivations of Certain Solvable Lie Algebras Over Commutative Rings

Let L be a finite-dimensional complex simple Lie algebra, L ℤ be the ℤ-span of a Chevalley basis of L, and L R = R ⊗ℤ L ℤ be a Chevalley algebra of type L over a commutative ring R with identity. Let ℬ(R) be the solvable subalgebra of L R spanned by the basis elements of the maximal toral subalgebra and the root vectors associated with positive roots. In this article, we prove that under some conditions for R, any automorphism of ℬ(R) is uniquely decomposed as a product of a graph automorphism, a diagonal automorphism and an inner automorphism, and any derivation of ℬ(R) is uniquely decomposed as a sum of an inner derivation induced by root vectors and a diagonal derivation. Correspondingly, the automorphism group and the derivation algebra of ℬ(R) are determined.

Derivations of Certain Nilpotent Lie Algebras Over Commutative Rings

Let L be a finite-dimensional complex simple Lie algebra, L ℤ be the ℤ-span of a Chevalley basis of L, and L R = R ⊗ℤ L ℤ be a Chevalley algebra of type L over a commutative ring R. Let N(R) be the nilpotent subalgebra of L R spanned by the root vectors associated with positive roots. In this article, we give an explicit description of any derivation of N(R). We prove that under some conditions for R, any derivation of N(R) can be expressed as a sum of inner, central, diagonal, extremal derivations of N(R).

Ringel–Hall algebras and two-parameter quantized enveloping algebras

Let g be a finite-dimensional complex simple Lie algebra and A be the finite-dimensional hereditary algebra associated to g. Let U + r,s (g) (respectively U ≥0 r,s (g)) denote the two-parameter quantized enveloping algebra of the positive maximal nilpotent (respectively Borel) Lie subalgebra of g. We study the two-parameter quantized enveloping algebras U + r,s (g) and U ≥0 r,s (g) using the approach of Ringel-Hall algebras. First of all, we show that U + r,s (g) is isomorphic to a certain two-parameter twisted Ringel-Hall algebra H r,s (A), which generalizes a result of Reineke. Based on detailed computations in H r,s (Λ), we show that H r,s (Λ) can be presented as an iterated skew polynomial ring. As an result, we obtain a PBW-basis for H r,s (A), which can be further used to construct a PBW-basis for the two-parameter quantized enveloping algebra U r,s (g). We also show that all prime ideals of U + r,s (g) are completely prime under some mild conditions on the parameters r, s. Second, we study the two-parameter extended Ringel-Hall algebra H r,s (Λ). In particular, we define a Hopf algebra structure on H r,s (Λ); and we prove that U ≥0 r,s (g) is isomorphic as a Hopf algebra to the two-parameter extended Ringel-Hall algebra H r,s (Λ).

Cauchon diagrams for quantized enveloping algebras

Let g be a finite-dimensional complex simple Lie algebra, K a commutative field and q a nonzero element of K which is not a root of unity. To each reduced decomposition of the longest element w 0 of the Weyl group W corresponds a PBW basis of the quantised enveloping algebra U q + ( g ) , and one can apply the theory of deleting-derivation to this iterated Ore extension. In particular, for each decomposition of w 0 , this theory constructs a bijection between the set of prime ideals in U q + ( g ) that are invariant under a natural torus action and certain combinatorial objects called Cauchon diagrams. In this paper, we give an algorithmic description of these Cauchon diagrams when the chosen reduced decomposition of w 0 corresponds to a good ordering (in the sense of Lusztig (1990) [Lus90]) of the set of positive roots. This algorithmic description is based on the constraints that are coming from Lusztig's admissible planes Lusztig (1990) [Lus90]: each admissible plane leads to a set of constraints that a diagram has to satisfy to be Cauchon. Moreover, we explicitly describe the set of Cauchon diagrams for explicit reduced decomposition of w 0 in each possible type. In any case, we check that the number of Cauchon diagrams is always equal to the cardinal of W. In Cauchon and Mériaux (2008) [CM08], we use these results to prove that Cauchon diagrams correspond canonically to the positive subexpressions of w 0 . So the results of this paper also give an algorithmic description of the positive subexpressions of any reduced decomposition of w 0 corresponding to a good ordering.

AUTOMORPHISMS OF CERTAIN NILPOTENT LIE ALGEBRAS OVER COMMUTATIVE RINGS

Let L be a finite-dimensional complex simple Lie algebra, Lℤ be the ℤ-span of a Chevalley basis of L and LR = R⊗ℤLℤ be a Chevalley algebra of type L over a commutative ring R. Let [Formula: see text] be the nilpotent subalgebra of LR spanned by the root vectors associated with positive roots. The aim of this paper is to determine the automorphism group of [Formula: see text].

Spherical orbits and representations of Uε(g)

Let Ue(g) be the simply connected quantized enveloping algebra at roots of one associated to a finite dimensional complex simple Lie algebra g. The De Concini-Kac-Procesi conjecture on the dimension of the irreducible representations of Ue(g) is proved for the representations corresponding to the spherical conjugacy classes of the simply connected algebraic group G with Lie algebra g. We achieve this result by means of a new characterization of the spherical conjugacy classes of G in terms of elements of the Weyl group.

A combinatorial identity for the derivative of a theta series of a finite type root lattice

AbstractLet be a (not necessarily simply laced) finite-dimensional complex simple Lie algebra with the Cartan subalgebra and Q ⊂ * the root lattice. Denote by ΘQ(q) the theta series of the root lattice Q of . We prove a curious “combinatorial” identity for the derivative of ΘQ(q), i.e. for by using the representation theory of an affine Lie algebra.