Maximal Nilpotent Subalgebra Research Articles

Rook placements in $G_2$ and $F_4$ and associated coadjoint orbits

Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Lie algebra with root system $\Phi$. A subset $D$ of the set $\Phi^+$ of positive roots is called a rook placement if it consists of roots with pairwise non-positive scalar products. To each rook placement $D$ and each map $\xi$ from $D$ to the set $\mathbb{C}^{\times}$ of nonzero complex numbers one can naturally assign the coadjoint orbit $\Omega_{D,\xi}$ in the dual space $\mathfrak{n}^*$. By definition, $\Omega_{D,\xi}$ is the orbit of $f_{D,\xi}$, where $f_{D,\xi}$ is the sum of root covectors $e_{\alpha}^*$ multiplied by $\xi(\alpha)$, $\alpha\in D$. (In fact, almost all coadjoint orbits studied at the moment have such a form for certain $D$ and $\xi$.) It follows from the results of Andr\`e that if $\xi_1$ and $\xi_2$ are distinct maps from $D$ to $\mathbb{C}^{\times}$ then $\Omega_{D,\xi_1}$ and $\Omega_{D,\xi_2}$ do not coincide for classical root systems $\Phi$. We prove that this is true if $\Phi$ is of type $G_2$, or if $\Phi$ is of type $F_4$ and $D$ is orthogonal.

Cohomology of the Lie algebra ℌ2: Experimental results and conjectures

The cohomology with trivial coefficients of the Lie algebra ℌ of Hamiltonian vector fields in the plane and of its maximal nilpotent subalgebra L 1ℌ is considered. The cohomology H 2(L 1ℌ) is calculated, and some far-reaching conjectures concerning the cohomology of the Lie algebras mentioned above and based on an extensive experimental material are formulated.

The poset of the nilpotent commutator of a nilpotent matrix

Let B be an n×n nilpotent matrix with entries in an infinite field k. The nilpotent commutator NB of B is known to be an irreducible algebraic variety. Assume that B is in Jordan canonical form with the associated Jordan block partition P and let Q(P) be the Jordan partition of a generic element of NB. P. Oblak associated to a given partition P another partition resulting from a recursive process. She conjectured that this partition is the same as Q(P). R. Basili, A. Iarrobino and the author later generalized the process introduced by Oblak. The main result of this paper states that all such processes give rise to the same partition – a partition that is defined in terms of the lengths of unions of special symmetric chains in a poset associated to NB. This poset was defined by P. Oblak as the digraph determined by the non-zero entries of a generic matrix in a maximal nilpotent subalgebra of NB.

Solvable extensions of a class of nilpotent linear Lie algebras

Let n(>2) be a positive integer, n a maximal nilpotent subalgebra of the symplectic algebra sp(2n,F) over a field F of characteristic not 2, s a solvable Lie algebra whose nilradical is isomorphic to n. The derivations of n are shown to be the sum of several types standard derivations. The dimension of s is shown at most dim(n)+n, and s is isomorphic to the standard Borel subalgebra b of sp(2n,F) if and only if dim(s)=dim(n)+n.

Tame Quivers and Affine Enveloping Algebras

Let g be an affine Kac-Moody algebra with symmetric Cartan datum, n + be the maximal nilpotent subalgebra of g. By the Hall algebra approach, we construct integral bases of the Z-form of the enveloping algebra U(n + ). In particular, the representation theory of tame quivers is essentially used in this paper.

Generalized Lie triple derivations

Let be a complex semisimple Lie algebra, be a Borel subalgebra of and denote the maximal nilpotent subalgebra of . In this article, we prove that every generalized Lie triple derivation of can be written as the sum of a Lie triple derivation and a block diagonal map. We also show that every generalized Lie triple derivation for of each classical complex simple Lie algebra is the sum of a Lie triple derivation and a homothety.

Laplacian spectrum for the nilpotent Kac–Moody Lie algebras

We prove that a maximal nilpotent subalgebra of a Kac‐Moody Lie algebra has an (essentially unique) Euclidean metric whose Laplace operator in the chain complex is scalar on each component of a given degree. Moreover, both the Lie algebra structure and the metric are uniquely determined by this property.

Abelian ideals and cohomology of symplectic type

Let b be a Borel subalgebra of the symplectic Lie algebra sp (2n, C) and let n be the corresponding maximal nilpotent subalgebra. We find a connection between the abelian ideals of b and the cohomology of n with trivial coefficients. Using this connection, we are able to enumerate the number of abelian ideals of b with given dimension via the Poincare polynomials of Weyl groups of types A n- 1 and C n .

On central extensions of preprojective algebras

We show that the Hilbert polynomial P ( t ) of the trace space A / [ A , A ] of the centrally extended preprojective algebra A of an ADE quiver is equal to the Hilbert series of the maximal nilpotent subalgebra of the corresponding simple Lie algebra under the principal gradation. This implies that the Hilbert polynomial of the center of A is t 2 h − 4 P ( 1 / t ) , where h is the Coxeter number.

Series of Lie groups

For various series of complex semi-simple Lie algebras $\fg (t)$ equipped with irreducible representations $V(t)$, we decompose the tensor powers of $V(t)$ into irreducible factors in a uniform manner, using a tool we call {\it diagram induction}. In particular, we interpret the decompostion formulas of Deligne \cite{del} and Vogel \cite{vog} for decomposing $\fg^{\ot k}$ respectively for the exceptional series and $k\leq 4$ and all simple Lie algebras and $k\leq 3$, as well as new formulas for the other rows of Freudenthal's magic chart. By working with Lie algebras augmented by the symmetry group of a marked Dynkin diagram, we are able to extend the list \cite{brion} of modules for which the algebra of invariant regular functions under a maximal nilpotent subalgebra is a polynomial algebra. Diagram induction applied to the exterior algebra furnishes new examples of distinct representations having the same Casimir eigenvalue.

Imaginary vectors in the dual canonical basis of U q(n)

Let n be a maximal nilpotent subalgebra of a simple complex Lie algebra g. We introduce the notion of imaginary vector in the dual canonical basis of Uq(n), and we give examples of such vectors for types An(n ~ 5), Bn(n ~ 3), Cn(n ~ 3), Dn(n ~ 4) , and all exceptional types. This disproves a conjecture of Berenstein and Zelevinsky about q-commuting products of vectors of the dual canonical basis. It also shows the existence of finite-dimensional irreducible representations of quantum affine algebras whose tensor square is not irreducible.

Homology of Nilpotent Subalgebras of the Lie Superalgebra K(1,1).3

We calculate the dimensions of the second homology groups with trivial coefficients of nilpotent subalgebras of the Lie superalgebra \(K(1,1)\), which is the natural superanalog of the Witt algebra. The proof is based on direct calculations of the rank of the differential. As an application, we find deformations of the maximal nilpotent subalgebra in \(K(1,1)\).

Chains of Frobenius subalgebras of so(M) and the corresponding twists

Chains of extended Jordanian twists are studied for the universal enveloping algebras U(so(M)). The carrier subalgebra of a canonical chain FB0≺pmax cannot cover the maximal nilpotent subalgebra N+(so(M)). We demonstrate that there exist other types of Frobenius subalgebras in so(M) that can be large enough to include N+(so(M)). The problem is that the canonical chains FB0≺p do not preserve the primitivity on these new carrier spaces. We show that this difficulty can be overcome and the primitivity can be restored if one changes the basis and passes to the deformed carrier spaces. Finally, the twisting elements for the new Frobenius subalgebras are explicitly constructed. This gives rise to a new family of universal R-matrices for orthogonal algebras. For a special case of g=so(5) and its defining representation we present the corresponding matrix solution of the Yang–Baxter equation.

Enumeration in the Chevalley algebras

We consider the maximal nilpotent subalgebra NΦ( K ) of the Chevalley algebra of the type Φ over an arbitrary field K . Our purpose is enumeration of ideals of NΦ( K ) that are invariant under the subgroup D of all diagonal automorphisms. Connections between ideals of NΦ( K ) and normal subgroups of the unipotent subgroup of the Chevalley group of the type Φ over K are used. Finally, we use methods of integral representation of sums and of listing lattice paths. We also discuss some other problems of the enumeration of ideals.

Realization of Lie algebras and superalgebras in terms of creation and annihilation operators: I

For every finite-dimensional nilpotent complex Lie algebra or superalgebran, we offer three algorithms for realizing it in terms of creation and annihilation operators. We use these algorithms to realize Lie algebras with a maximal subalgebra of finite codimension. For a simple finite-dimensionalg whose maximal nilpotent subalgebra isn, this gives its realization in terms of first-order differential operators on the big open cell of the flag manifold corresponding to the negative roots ofg. For several examples, we executed the algorithms using the MATHEMATICA-based package SUPERLie. These realizations form a preparatory step in an explicit construction and description of an interesting new class of simple Lie (super) algebras of polynomial growth, generalizations of the Lie algebra of matrices of complex size.

Regular Nilpotent Elements and Quantum Groups

We suggest new realizations of quantum groups corresponding to complex simple Lie algebras, and of affine quantum groups. These new realizations are labeled by Coxeter elements of the corresponding Weyl group and have the following key feature: The natural counterparts of the subalgebras U(n), where n is a maximal nilpotent subalgebra in g, possess non--singular characters.

Catenarity in quantum algebras

Various quantum algebras are shown to be catenary, i.e., all saturated chains of prime ideals between any two fixed primes have the same length. Further, Tauvel's formula relating the height of a prime ideal to the Gelfand-Kirillov dimension of the corresponding factor ring is established. These results are obtained for coordinate rings of quantum affine spaces, for quantized Weyl algebras, and for coordinate rings of complex quantum general linear groups, as well as for quantized enveloping algebras of maximal nilpotent subalgebras of semisimple complex Lie algebras.

Study of q-commutations in Uq(n+) Algebra

Let g be a semi-simple Lie algebra with maximal nilpotent subalgebra n+. Let Ǔq(g) and Uq(n+) be the quantized enveloping algebras. We describe the set of normal elements of Uq(n+). We prove that each ideal of Uq(n+) can be generated by a normal sequence.

GENERALIZED HIROTA EQUATIONS AND REPRESENTATION THEORY I: THE CASE OF SL(2) AND SLq(2)

This paper begins investigation of the concept of a “generalized τ function,” defined as a generating function of all the matrix elements of a group element g∈G in a given highest weight representation of a universal enveloping algebra [Formula: see text]. In the generic situation, the time variables correspond to the elements of maximal nilpotent subalgebras rather than Cartanian elements. Moreover, in the case of quantum groups such τ “functions” are not c numbers but take their values in noncommutative algebras (of functions on the quantum group G). Despite all these differences from the particular case of conventional τ functions of integrable (KP and Toda lattice) hierarchies [which arise when G is a Kac-Moody (one-loop) algebra of level k=1], these generic τ functions also satisfy bilinear Hirota-like equations, which can be deduced from manipulations with intertwining operators. The most important applications of the formalism should be to k>1 Kac-Moody and multiloop algebras, but this paper contains only illustrative calculations for the simplest case of ordinary (zero-loop) algebra SL (2) and its quantum counterpart SL q(2), as well as for the system of fundamental representations of SL (n).

Maximal Abelian subalgebras of complex Euclidean Lie algebras

Maximal Abelian subalgebras (MASAs) of the complex Euclidean Lie algebra [Formula: see text] are classified into conjugacy classes under the action of the Lie group [Formula: see text] Use is made of an earlier classification of MASAs of the orthogonal Lie algebra [Formula: see text] These are then extended to nonsplitting MASAs of [Formula: see text] in which maximal Abelian nilpotent subalgebras of [Formula: see text] are coupled with translations in a nontrivial manner. The methods presented are applicable to the classification of MASAs of any affine Lie algebra.