Classical Simple Lie Algebras Research Articles

Shapovalov elements of classical and quantum groups

Shapovalov elements θβ,m of the classical or quantized universal enveloping algebra of a simple Lie algebra g are parameterized by a positive root β and a positive integer m. They relate the highest vector of a reducible Verma module with highest vectors of its submodules. We obtain a factorization of θβ,m to a product of θβ,1 and calculate θβ,1 as a residue of a matrix element of the inverse Shapovalov form via a generalized Nagel-Moshinsky algorithm. This way we explicitly express θβ,m of a classical simple Lie algebra through the Cartan-Weyl basis in g. In the case of quantum groups, we give an analogous formulation through the entries of the R-matrix (quantum L-operator) in fundamental representations.

Index of seaweed subalgebras of classical Lie algebras

We generalize the results in [3] giving a reduction algorithm allowing to compute the index of seaweed subalgebras of classical simple Lie algebras. We thus are able to obtain the index of some interesting families of seaweed subalgebras and to give new examples of large classes of Frobenius Lie algebras among them.

5-Graded simple Lie algebras, structurable algebras, and Kantor pairs

Relying on the classification of simple Lie algebras over algebraically closed fields of characteristic >3, we show that any finite-dimensional central simple 5-graded Lie algebra over a field k of characteristic ≠2,3 is a simple Lie algebra of Chevalley type, i.e. a central quotient of the Lie algebra of a simple algebraic k-group. As a consequence, we prove that all central simple structurable algebras and Kantor pairs over k arise from 5-gradings on simple Lie algebras of Chevalley type.

Birational geometry for the covering of a nilpotent orbit closure

Let \({\mathfrak {g}}\) be a complex classical simple Lie algebra and let O be a nilpotent orbit of \({\mathfrak {g}}\). The fundamental group \(\pi _1(O)\) is finite. Take the universal covering \(\pi ^0: X^0 \rightarrow O\). Then \(\pi ^0\) extends to a finite cover \(\pi : X \rightarrow {\bar{O}}\). By using the Kirillov–Kostant form \(\omega _{KK}\) on O, the normal affine variety X becomes a conical symplectic variety. In this article we give an explicit construction of a Q-factorial terminalization of X.

Gradings on associative algebras with involution and real forms of classical simple Lie algebras

We study gradings by abelian groups on associative algebras with involution over an arbitrary field. Of particular importance are the fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism (although not in a unique way). We classify up to equivalence the fine gradings on simple associative algebras with involution over the field of real numbers (or any real closed field) and, as a consequence, on the real forms of classical simple Lie algebras.

Classical simple Lie 2-algebras of odd toral rank and a contragredient Lie 2-algebra of toral rank 4

After the classification of simple Lie algebras over a field of characteristic p > 3, the main problem not yet solved in the theory of finite dimensional Lie algebras is the classification of simple Lie algebras over a field of characteristic 2. The first result for this classification problem ensures that all finite dimensional Lie algebras of absolute toral rank 1 over an algebraically closed field of characteristic 2 are soluble. Describing simple Lie algebras (respectively, Lie 2-algebras) of finite dimension of absolute toral rank (respectively, toral rank) 3 over an algebraically closed field of characteristic 2 is still an open problem. In this paper we show that there are no classical type simple Lie 2-algebras with toral rank odd and furthermore that the simple contragredient Lie 2-algebra G(F4,a) of dimension 34 has toral rank 4. Additionally, we give the Cartan decomposition of G(F4,a).

On the classification of simple Lie algebras of dimension seven over fields of characteristic 2

This paper is the second part of paper (Grishkov and Guerreiro in Sao Paulo J Math Sci v4(1):93–107, 2010) about simple 7-dimensional Lie algebras over an algebraically closed field k of characteristic two. In this paper we prove that all simple 7-dimensional Lie algebras over k of absolute toral rank three are isomorphic to the Cartan algebra $$W_1$$ or the Hamilton algebra $$H_2.$$ We hope to prove that those algebras are the unique simple 7-dimensional Lie algebras over the field k. Observe that in the case of absolute toral rank 2 this fact was proved in [2].

Centrally generated primitive ideals of [formula omitted] for exceptional types

Let g be a complex semisimple Lie algebra, let b be a Borel subalgebra of g, let n be the nilradical of b, and let U(n) be the universal enveloping algebra of n. We study primitive ideals of U(n). Almost all primitive ideals are centrally generated, i.e., are generated by their intersections with the center Z(n) of U(n). We present an explicit characterization of the centrally generated primitive ideals of U(n) in terms of the Dixmier map and the Kostant cascade in the case when g is a simple algebra of exceptional type. (For classical simple Lie algebras, a similar characterization was obtained by Ivan Penkov and the first author.) As a corollary, we establish a classification of centrally generated primitive ideals of U(n) for an arbitrary semisimple algebra g.

The second cohomology groups of nilpotent orbits in classical Lie algebras

The second de Rham cohomology groups of nilpotent orbits in noncompact real forms of classical complex simple Lie algebras are explicitly computed. Furthermore, the first de Rham cohomology groups of nilpotent orbits in noncompact classical simple Lie algebras are computed, and we prove them to be zero for nilpotent orbits in all the complex simple Lie algebras. A key component in these computations is a description of the second and first cohomology groups of homogeneous spaces of general connected Lie groups which is obtained here. This description, which generalizes a previous theorem of the first two authors, may be of independent interest.

Jet modules for the centerless Virasoro-like algebra

In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a [Formula: see text]-torus. The jet modules are certain natural modules over the Lie algebra of semi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over [Formula: see text]. To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of [Irreducible representations for toroidal Lie algebras, J. Algebras 221 (1999) 188–231; Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004) 23–42]. Consequently, indecomposable jet modules are described using modules over the algebra [Formula: see text], which is the “positive part” of a Block type algebra studied first by [Some infinite-dimensional simple Lie algebras in characteristic [Formula: see text] related to those of Block, J. Pure Appl. Algebra 127(2) (1998) 153–165] and recently by [A [Formula: see text]-graded generalization of the Witt-algebra, preprint; Classification of simple Lie algebras on a lattice, Proc. London Math. Soc. 106(3) (2013) 508–564]).

Gradings on classical central simple real Lie algebras

For any abelian group G, we classify up to isomorphism all G-gradings on the classical central simple Lie algebras, except those of type D4, over the field of real numbers (or any real closed field).

Positive Systems of Kostant Roots

Let \(\mathfrak {g}\) be a simple complex Lie algebra and let \(\mathfrak {t} \subset \mathfrak {g}\) be a toral subalgebra of \(\mathfrak {g}\). As a \(\mathfrak {t}\)-module \(\mathfrak {g}\) decomposes as $$\mathfrak{g} = \mathfrak{s} \oplus \left( \oplus_{\nu \in \mathcal{R}}~ \mathfrak{g}^{\nu}\right)$$ where \(\mathfrak {s} \subset \mathfrak {g}\) is the reductive part of a parabolic subalgebra of \(\mathfrak {g}\) and \(\mathcal {R}\) is the Kostant root system associated to \(\mathfrak {t}\). When \(\mathfrak {t}\) is a Cartan subalgebra of \(\mathfrak {g}\) the decomposition above is nothing but the root decomposition of \(\mathfrak {g}\) with respect to \(\mathfrak {t}\); in general the properties of \(\mathcal {R}\) resemble the properties of usual root systems. In this note we study the following problem: “Given a subset \(\mathcal {S} \subset \mathcal {R}\), is there a parabolic subalgebra \(\mathfrak {p}\) of \(\mathfrak {g}\) containing \(\mathcal {M} = \oplus _{\nu \in \mathcal {S}} \mathfrak {g}^{\nu }\) and whose reductive part equals \(\mathfrak {s}\)?”. Our main results is that, for a classical simple Lie algebra \(\mathfrak {g}\) and a saturated \(\mathcal {S} \subset \mathcal {R}\), the condition \((\text {Sym}^{\cdot }(\mathcal {M}))^{\mathfrak {s}} = \mathbb {C}\) is necessary and sufficient for the existence of such a \(\mathfrak {p}\). In contrast, we show that this statement is no longer true for the exceptional Lie algebras F4,E6,E7, and E8. Finally, we discuss the problem in the case when \(\mathcal {S}\) is not saturated.

Diophantine equations, Platonic solids, McKay correspondence, equivelar maps and Vogel’s universality

We notice that one of the Diophantine equations, $knm=2kn+2km+2nm$, arising in the universality originated Diophantine classification of simple Lie algebras, has interesting interpretations for two different sets of signs of variables. In both cases it describes "regular polyhedrons" with $k$ edges in each vertex, $n$ edges of each face, with total number of edges $|m|$, and Euler characteristics $\chi=\pm 2$. In the case of negative $m$ this equation corresponds to $\chi=2$ and describes true regular polyhedrons, Platonic solids. The case with positive $m$ corresponds to Euler characteristic $\chi=-2$ and describes the so called equivelar maps (charts) on the surface of genus $2$. In the former case there are two routes from Platonic solids to simple Lie algebras - abovementioned Diophantine classification and McKay correspondence. We compare them for all solutions of this type, and find coincidence in the case of icosahedron (dodecahedron), corresponding to $E_8$ algebra. In the case of positive $k$, $n$ and $m$ we obtain in this way the interpretation of (some of) the mysterious solutions (Y-objects), appearing in the Diophantine classification and having some similarities with simple Lie algebras.

Deformations of current Lie algebras. I. Small algebras in characteristic 2

We compute low-degree cohomology of current Lie algebras extended over the 3-dimensional simple algebra, compute deformations of related semisimple Lie algebras, and apply these results to classification of simple Lie algebras of absolute toral rank 2 and having a Cartan subalgebra of toral rank one. Everything is in characteristic 2.

Nilpotent orbits of certain simple Lie algebras over truncated polynomial rings

Let F be an algebraically closed field of positive characteristic p>3, and A the divided power algebra in one indeterminate, which, as a vector space, coincides with the truncated polynomial ring of F[T] by Tpn. Let g be the special derivation algebra over A which is a simple Lie algebra, and additionally non-restricted as long as n>1. Let N be the nilpotent cone of g, and G=Aut(g), the automorphism group of g. In contrast with only finitely many nilpotent orbits in a classical simple Lie algebra, there are infinitely many nilpotent orbits in g. In this paper, we parameterize all nilpotent orbits, and obtain their dimensions. Furthermore, the nilpotent cone N is proven to be reducible and not normal. There are two irreducible components in N. The dimension of N is determined.

Borel subalgebras of the Witt algebra

Let $\mathbb{F}$ be an algebraically closed field of characteristic p > 3, and $\mathfrak{g}$ be the Witt algebra over $\mathbb{F}$ . Let $\mathcal{N}$ be the nilpotent cone of $\mathfrak{g}$ . An explicit description of $\mathcal{N}$ is given, so that the conjugacy classes of Borel subalgebras of $\mathfrak{g}$ under the automorphism group of $\mathfrak{g}$ are determined. In contrast with only one conjugacy class of Borel subalgebras in a classical simple Lie algebra, there are two conjugacy classes of Borel subalgebras in g. The representatives of conjugacy classes of Borel subalgebras, i.e., the so-called standard Borel subalgebras, are precisely given.

Local Weyl modules and cyclicity of tensor products for Yangians

We provide a sufficient condition for the cyclicity of an ordered tensor product L=Va1(ωb1)⊗Va2(ωb2)⊗…⊗Vak(ωbk) of fundamental representations of the Yangian Y(g). When g is a classical simple Lie algebra, we make the cyclicity condition concrete, which leads to an irreducibility criterion for the ordered tensor product L. In the case when g=sll+1, a sufficient and necessary condition for the irreducibility of the ordered tensor product L is obtained. The cyclicity of the ordered tensor product L is closely related to the structure of the local Weyl modules of Y(g). We show that every local Weyl module is isomorphic to an ordered tensor product of fundamental representations of Y(g).

Graded modules over classical simple Lie algebras with a grading

Given a grading by an abelian group G on a semisimple Lie algebra L over an algebraically closed field of characteristic 0, we classify up to isomorphism the simple objects in the category of finite-dimensional G-graded L-modules. The invariants appearing in this classification are computed in the case when L is simple classical (except for type D 4, where a partial result is given). In particular, we obtain criteria to determine when a finite-dimensional simple L-module admits a G-grading making it a graded L-module.

Wei–Norman equations for classical groups

We show that the nonlinear autonomous Wei–Norman equations, expressing the solution of a linear system of non-autonomous equations on a Lie algebra, can be reduced to the hierarchy of matrix Riccati equations in the case of all classical simple Lie algebras. The result generalizes our previous one concerning the complex Lie algebra of the special linear group. We show that it cannot be extended to all simple Lie algebras, in particular to the exceptional G2 algebra.

COUNTING FINE GRADINGS ON MATRIX ALGEBRAS AND ON CLASSICAL SIMPLE LIE ALGEBRAS

Known classification results allow us to find the number of (equivalence classes of) fine gradings on matrix algebras and on classical simple Lie algebras over an algebraically closed field 𝔽 (assuming char 𝔽 ≠ 2 in the Lie case). The computation is easy for matrix algebras and especially for simple Lie algebras of type Br(the answer is just r + 1), but involves counting orbits of certain finite groups in the case of Series A, C and D. For X ∈ {A, C, D}, we determine the exact number of fine gradings, NX(r), on the simple Lie algebras of type Xrwith r ≤ 100 as well as the asymptotic behavior of the average, [Formula: see text], for large r. In particular, we prove that there exist positive constants b and c such that [Formula: see text]. The analogous average for matrix algebras Mn(𝔽) is proved to be a ln n + O(1) where a is an explicit constant depending on char 𝔽.