Finite-dimensional Complex Simple Lie Research Articles

U0-free quantum group representations

Let g be a finite dimensional complex simple Lie algebra, and let U=Uq(g) be its quantized enveloping algebra with a triangular decomposition U=U−U0U+. We classify all U-module structures on U0 with the regular action of U0 on itself, which are shown to have direct connections with the q-Weyl algebra and the bounded weight representations. We obtain that the necessary and sufficient condition for the existence of such modules is that U has to be of type An(n≥1), Bn(n≥2) or Cn(n≥3). Moreover, we study their module structures and associated weight modules for each type.

The Polynomial Modules over Quantum Group Uq(sl3)

Let [Formula: see text] be a finite dimensional complex simple Lie algebra with Cartan subalgebra [Formula: see text]. Then [Formula: see text] has a [Formula: see text]-module structure if and only if [Formula: see text] is of type [Formula: see text] or of type [Formula: see text]; this is called the polynomial module of rank one. In the quantum version, the rank one polynomial modules over [Formula: see text] have been classified. In this paper, we prove that the quantum group [Formula: see text] has no rank one polynomial module.

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. Bibliography: 44 titles.

Relations in Quantized Function Algebras

We develop a method to give presentations of quantized function algebras of complex reductive groups. In particular, we give presentations of quantized function algebras of automorphism groups of finite dimensional simple complex Lie algebras.

Level bounds for exceptional quantum subgroups in rank two

There is a long-standing belief that the modular tensor categories [Formula: see text], for [Formula: see text] and finite-dimensional simple complex Lie algebras [Formula: see text], contain exceptional connected étale algebras (sometimes called quantum subgroups) at only finitely many levels [Formula: see text]. This premise has known implications for the study of relations in the Witt group of nondegenerate braided fusion categories, modular invariants of conformal field theories, and the classification of subfactors in the theory of von Neumann algebras. Here, we confirm this conjecture when [Formula: see text] has rank 2, contributing proofs and explicit bounds when [Formula: see text] is of type [Formula: see text] or [Formula: see text], adding to the previously known positive results for types [Formula: see text] and [Formula: see text].

Drinfeld J Presentation of Twisted Yangians

We present a quantization of a Lie coideal structure for twisted half-loop algebras of finite-dimensional simple complex Lie algebras. We obtain algebra closure relations of twisted Yangians in Drinfeld J presentation for all symmetric pairs of simple Lie algebras and for simple twisted even half-loop Lie algebras. We provide the explicit form of the closure relations for twisted Yangians in Drinfeld J presentation for the ${\mathfrak{sl}}_3$ Lie algebra.

Integral Bases for the Universal Enveloping Algebras of Cartan Map Superalgebras

Let 𝔤 be a finite dimensional complex simple Lie superalgebra of Cartan type and A be a commutative, associative algebra with unity over ℂ. We refer to the Lie superalgebras of the form 𝔤 ⊗ A as Cartan map superalgebras. In this paper, following Bagci and Chamberlin (J. of Pure and Applied Algebra 218(8), 1563–1576, 2014), we define an integral form for the universal enveloping algebra of the Cartan map superalgebras, and exhibit an explicit integral basis for this integral form.

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).

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.

Cohomology rings for quantized enveloping algebras

We compute the structure of the cohomology ring for the quantized enveloping algebra (quantum group) $U_q$ associated to a finite-dimensional simple complex Lie algebra $\mathfrak{g}$. We show that the cohomology ring is generated as an exterior algebra by homogeneous elements in the same odd degrees as generate the cohomology ring for the Lie algebra $\mathfrak{g}$. Partial results are also obtained for the cohomology rings of the non-restricted quantum groups obtained from $U_q$ by specializing the parameter $q$ to a non-zero value $\epsilon \in \mathbb{C}$.

Differentiating the Weyl generic dimension formula with applications to support varieties

The authors compute the support varieties of all irreducible modules for the small quantum group uζ(g), where g is a finite-dimensional simple complex Lie algebra, and ζ is a primitive ℓ-th root of unity with ℓ larger than the Coxeter number of g. The calculation employs the prior calculations and techniques of Ostrik and of Nakano, Parshall, and Vella, as well as deep results involving the validity of the Lusztig character formula for quantum groups and the positivity of parabolic Kazhdan–Lusztig polynomials for the affine Weyl group. Analogous support variety calculations are provided for the first Frobenius kernel G1 of a reductive algebraic group scheme G defined over the prime field Fp.

New r-Matrices for Lie Bialgebra Structures over Polynomials

For a finite dimensional simple complex Lie algebra $${\mathfrak{g}}$$ , Lie bialgebra structures on $${\mathfrak{g}\left[\left[u \right]\right]}$$ and $${\mathfrak{g}\left[u\right]}$$ were classified by Montaner, Stolin and Zelmanov. In our paper, we provide an explicit algorithm to produce r-matrices which correspond to Lie bialgebra structures over polynomials.

Lie Algebras with a Coalgebra Splitting

In their recent article [5], the authors endow every finite-dimensional simple complex Lie algebra g with a coalgebra structure such that the composition μ◦δ of the two structure maps δ : g −→ g⊗Cg and μ : g⊗C g −→ g coincides with the identity. Moreover, the dual algebra (g∗, δ∗) associated to the Lie coalgebra is isomorphic to (g, μ). The coalgebra map δ is given explicitly for sl(n), those for the other types are obtained via embeddings g ↪→ sl(n). The purpose of the present short note is to elicit the conceptual sources of [5], starting from the observation that the coalgebra maps defined in [5] are in fact homomorphisms of g-modules. For Lie algebras affording non-degenerate symmetric associative forms, such coalgebra maps naturally arise by dualizing the multiplication. This immediately implies the abovementioned duality, and the formulae displayed in [5, §4] can also be subsumed under our general approach. By demanding that δ be g-linear, we depart from the usual compatibility condition of a Lie bialgebra, which requires δ to be a derivation. In that case, μ ◦ δ is a derivation of g. For fields of characteristic zero, only nilpotent Lie algebras afford invertible derivations (cf. [7]), so that non-zero Lie bialgebras over such fields never satisfy μ ◦ δ = idg. Let k be a field. Given a finite-dimensional k-vector space V , we consider the k-linear maps

Fixed point subalgebras of extended affine Lie algebras

It is a well-known result that the fixed point subalgebra of a finite dimensional complex simple Lie algebra under a finite order automorphism is a reductive Lie algebra so it is a direct sum of finite dimensional simple Lie subalgebras and an abelian subalgebra. We consider this for the class of extended affine Lie algebras and are able to show that the fixed point subalgebra of an extended affine Lie algebra under a finite order automorphism (which satisfies certain natural properties) is a sum of extended affine Lie algebras (up to existence of some isolated root spaces), a subspace of the center and a subspace which is contained in the centralizer of the core. Moreover, we show that the core of the fixed point subalgebra modulo its center is isomorphic to the direct sum of the cores modulo centers of the involved summands.

The universal central extension of the holomorphic current algebra

We identify the universal differential module Ω1(A) for the Frechet algebra A of holomorphic functions on a complex Stein manifold X, and more generally on a Riemannian domain R over X and for the algebra of germs of holomorphic functions on a compact subset K⊂ℂn. It turns out to be isomorphic to the Frechet space of holomorphic 1-forms on X, resp. R, resp. to the space Ω1(K) of germs of holomorphic 1-forms in K. This determines the center of the universal central extension of the Lie algebra 𝒪(R,𝔨 of holomorphic maps from R to a finite-dimensional simple complex Lie algebra 𝔨.

The classification of locally finite split simple Lie algebras

One of the great early achievements in Lie theory is the classification of the finite dimensional simple complex Lie algebras by W. Killing and E. Cartan. More than 50 years later new aspects were added to this classification by the theory of Coxeter groups and the visualization of the classification in terms of Dynkin diagrams. Furthermore Serre’s description of the simple Lie algebras by generators and relations provided a direct way to construct the Lie algebras from the Cartan matrix corresponding to the choice of a root base, i.e., a system of simple roots.

The centralizer algebras of lie color algebras

In his 1977 paper, V.G. Kac classified the finite-dimensional simple complex Lie superalgebras. After Kac’s paper, M. Scheunert initiated the study of a generalization of Lie superalgebras - the Lie color algebras. We construct some new families of simple Lie color algebras. Following the work of A. Berele and A. Regev and A.N. Sergeev, who studied the general linear and sq(n)-series superalgebra cases, and the work of G. Benkart, C. Lee Shader, and A. Ram, who studied the orthosymplectic cases, we examine the centralizer algebras of some classical Lie superalgebras and their Lie color algebra counterparts acting on tensor space and derive Schur-Weyl duality results for these algebras and their centralizers.

Quantum Lie Algebras, Their Existence, Uniqueness and q-Antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules of the quantized enveloping algebras. On them the quantum Lie bracket is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra g an abstract quantum Lie algebra g_h independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras are isomorphic to an abstract quantum Lie algebra g_h. In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras associated to the same g are isomorphic, 2) the quantum Lie bracket of any quantum Lie algebra is $q$-antisymmetric. We also describe a construction of quantum Lie algebras which establishes their existence.

Representations of Quantum Groups at Even Roots of Unity

Let g be a finite dimensional complex simple Lie algebra and U(g) its enveloping algebra. The quantum group of Drinfeld and Jimbo is a Hopf algebra denoted Uq(g) defined on Chevalley-like generators over C[q, q−1]. Through "specialization" of q at different ϵ ∈ C one obtains a parameterized family of Hopf algebras. When ϵ = 1 one recovers the classical universal enveloping algebra. Moreover, when ϵ is not a root of unity Lusztig (Adv. Math.70 (1988), 237-249) and Rosso, (Comm. Math. Phys. 117 (1988), 581-593) have shown independently that the representation theory of Uϵ(= Uϵ(g)) is analogous to that of U(g). When we fix ϵ a primitive root of unity the situation changes considerably. At odd roots of unity the representations of Uϵ are partitioned by conjugacy classes of the algebraic group G with Lie algebra g [DC-K]. This paper relates the representations of Uϵ at even roots of unity to conjugacy classes in the group G∨-the Langlands dual of G. The partition is somewhat finer than that at odd roots of unity and requires a more detailed analysis. This correspondence is then used to study the representations at even roots of unity. In particular, we obtain a "triangulability" result which allows us to calculate the degree of Uϵ using a deformation argument.

The coradical filtration of U Q (S(2)) Q at roots of unity

Let u q (s[(2)) denote the quantum enveloping algebra of s((2) over a field k. In this note we show that if q is a root of unity, then the coradical filtration {H n}n ≥0 of u q (s[(2)) is the filtration by a certain degree function depending on the order of q 4 and on the characteristic of k. The degree function will be explicitly described in Section 2 for char s = 0 and in Section 3 for char k = s > 0. We use the fact that u q (g) is pointed for any finite-dimensional semisimple Lie algebra (proved in [R], [Ml] and remarked in [Tl]). The skew-primitives of u q (g[(n)) and u q (s[(n)) are described in [T2], based en [Tl]. The theorem in [T2] motivated the present result, although we did not need its full strength in the proof. The filtration of u q (g) over Q(q), where g is a finite dimensional complex simple Lie algebra, and q is an indeterminate over Q, is described in [CM]. The case of u q (s[(2)) with char k = 0 and where q is not a root of unity was first solved in [Ml].