Finite-dimensional Lie Algebra Research Articles

Root components for tensor product of affine Kac-Moody Lie algebra modules

Let g \mathfrak {g} be an affine Kac-Moody Lie algebra and let λ , μ \lambda , \mu be two dominant integral weights for g \mathfrak {g} . We prove that under some mild restriction, for any positive root β \beta , V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) contains V ( λ + μ − β ) V(\lambda +\mu -\beta ) as a component, where V ( λ ) V(\lambda ) denotes the integrable highest weight (irreducible) g \mathfrak {g} -module with highest weight λ \lambda . This extends the corresponding result by Kumar from the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras. One crucial ingredient in the proof is the action of Virasoro algebra via the Goddard-Kent-Olive construction on the tensor product V ( λ ) ⊗ V ( μ ) V(\lambda )\otimes V(\mu ) . Then, we prove the corresponding geometric results including the higher cohomology vanishing on the G \mathcal {G} -Schubert varieties in the product partial flag variety G / P × G / P \mathcal {G}/\mathcal {P}\times \mathcal {G}/\mathcal {P} with coefficients in certain sheaves coming from the ideal sheaves of G \mathcal {G} -sub-Schubert varieties. This allows us to prove the surjectivity of the Gaussian map.

Inner Ideals of Real Simple Lie Algebras

A classification up to automorphism of the inner ideals of the real finite-dimensional simple Lie algebras is given, jointly with precise descriptions in the case of the exceptional Lie algebras.

CR embeddings of CR manifolds

We improve results of Baouendi, Rothschild and Treves and of Hill and Nacinovich by finding a much weaker sufficient condition for a CR manifold of type (n, k) to admit a local CR embedding into a CR manifold of type (n+ell ,k-ell ). While their results require the existence of a finite dimensional solvable transverse Lie algebra of vector fields, we require only a finite dimensional extension.

Reduction and reconstruction of multisymplectic Lie systems

A Lie system is a non-autonomous system of first-order ordinary differential equations describing the integral curves of a non-autonomous vector field taking values in a finite-dimensional real Lie algebra of vector fields, a so-called Vessiot–Guldberg Lie algebra. In this work, multisymplectic forms are applied to the study of the reduction of Lie systems through their Lie symmetries. By using a momentum map, we perform a reduction and reconstruction procedure of multisymplectic Lie systems, which allows us to solve the original problem by analysing several simpler multisymplectic Lie systems. Conversely, we study how reduced multisymplectic Lie systems allow us to retrieve the form of the multisymplectic Lie system that gave rise to them. Our results are illustrated with examples from physics, mathematics, and control theory.

On Chevalley Restriction Theorem for Semi-reductive Algebraic Groups and Its Applications

An algebraic group is called semi-reductive if it is a semi-direct product of a reductive subgroup and the unipotent radical. Such a semi-reductive algebraic group naturally arises and also plays a key role in the study of modular representations of non-classical finite-dimensional simple Lie algebras in positive characteristic, and some other cases. Let G be a connected semi-reductive algebraic group over an algebraically closed field \(\mathbb{F}\) and \(\mathfrak{g}=Lie{\rm{(}}G{\rm{)}}\). It turns out that G has many same properties as reductive groups, such as the Bruhat decomposition. In this note, we obtain an analogue of classical Chevalley restriction theorem for \(\mathfrak{g}\), which says that the G-invariant ring \(\mathbb{F}{[\mathfrak{g}]^G}\) is a polynomial ring if \(\mathfrak{g}\) satisfies a certain “positivity” condition suited for lots of cases we are interested in. As applications, we further investigate the nilpotent cones and resolutions of singularities for semi-reductive Lie algebras.

Image of Lie polynomial of degree 2 evaluated on nilpotent lie algebra

We delineate the image of multilinear Lie polynomial of degree 2 evaluated on L where L is a finite-dimensional nilpotent Lie algebra over field k with

Weyl roots and equivalences of integral quadratic forms

We study integral quadratic forms in the sense of Roiter, that is, quadratic forms whose integer coefficients satisfy certain divisibility condition assuring that the associated Weyl group is integral. Such forms are known to be useful for characterizing classes of finite-dimensional algebras and Lie algebras. We present a solution of the problem considered by Roiter in 1978 of providing a characterization of Weyl roots of positive definite forms which is independent on an explicit action of the Weyl group. We also prove some new facts on the Weyl group and the Weyl roots in general. Moreover, we provide a detailed comparison of several distinct equivalence relations of integral forms appearing in the literature. In particular, we discuss how these relations behave with respect to the Weyl roots and some properties of integral forms, and we supplement some missing proofs related to the classification of positive definite and principal forms.

THE NON-DEGENERACY OF THE SKEW-SYMMETRIC BILINEAR FORM OF THE FINITE DIMENSIONAL REAL FROBENIUS LIE ALGEBRA

A Frobenius Lie algebra is recognized as the Lie algebra whose stabilizer at a Frobenius functional is trivial. This condition is equivalent to the existence of a skew-symmetric bilinear form which is non-degenerate. On the other hand, the Lie algebra is Frobenius as well if its orbit on the dual vector space is open. In this paper, we study the skew-symmetric bilinear form of finite dimensional Frobenius Lie algebra corresponding to its Frobenius functional. The work aims to prove that a Lie algebra of dimension is Frobenius if and only if the -th derivation of the Frobenius functional is not equal to zero. Indeed, this condition implies that the skew-symmetric bilinear form is non-degenerate and vice versa. In addition, some properties of Frobenius functionals are obtained. Furthermore, the computations are given using the coadjoint orbits and the structure matrix. As a discussion, we can investigate these results in the algebra case whether giving rise to a left-invariant K hler structure of a Frobenius Lie group or not.

On triple homomorphisms of Lie algebras

Let [Formula: see text] and [Formula: see text] be two Lie algebras over a commutative ring with identity. In this paper, under some conditions on [Formula: see text] and [Formula: see text], it is proved that every triple homomorphism from [Formula: see text] onto [Formula: see text] is the sum of a homomorphism and an antihomomorphism from [Formula: see text] into [Formula: see text]. We also show that a finite-dimensional Lie algebra [Formula: see text] over an algebraically closed field of characteristic zero is nilpotent of class at most [Formula: see text] if and only if the sum of every homomorphism and every antihomomorphism on [Formula: see text] is a triple homomorphism.

Nondegenerate invariant symmetric bilinear forms on simple Lie superalgebras in characteristic 2

As is well-known, the dimension of the space spanned by the non-degenerate invariant symmetric bilinear forms (NISes) on any simple finite-dimensional Lie algebra or Lie superalgebra is equal to at most 1 if the characteristic of the algebraically closed ground field is not 2.We prove that in characteristic 2, the superdimension of the space spanned by NISes can be equal to 0, or 1, or 0|1, or 1|1; it is equal to 1|1 if and only if the Lie superalgebra is a queerification (defined in arXiv:1407.1695) of a simple classically restricted Lie algebra with a NIS (for examples, mainly in characteristic ≠2, see arXiv:1806.05505).We give examples of NISes on deformations (with both even and odd parameters) of several simple finite-dimensional Lie superalgebras in characteristic 2.We also recall examples of multiple NISes on simple Lie algebras over non-closed fields.

Classical dynamics from self-consistency equations in quantum mechanics

During the last three decades, Pavel Bóna developed a non-linear generalization of quantum mechanics, which is based on symplectic structures for normal states. One important application of such a generalization is a general setting that is very convenient to study the emergence of macroscopic classical dynamics from microscopic quantum processes. We propose here a new mathematical approach to Bóna’s non-linear quantum mechanics. It is based on C0-semigroup theory and has a domain of applicability that is much broader than Bóna’s original one. It highlights the central role of self-consistency. This leads to a mathematical framework in which the classical and quantum worlds are naturally entangled. In this new mathematical approach, we build a Poisson bracket for the polynomial functions on the Hermitian weak*-continuous functionals on any C*-algebra. This is reminiscent of a well-known construction for finite-dimensional Lie algebras. We then restrict this Poisson bracket to states of this C*-algebra by taking quotients with respect to Poisson ideals. This leads to densely defined symmetric derivations on the commutative C*-algebras of real-valued functions on the set of states. Up to a closure, these are proven to generate C0-groups of contractions. As a matter of fact, in generic commutative C*-algebras, even the closableness of unbounded symmetric derivations is a non-trivial issue. Some new mathematical concepts are introduced, which are possibly interesting by themselves: the convex weak* Gâteaux derivative and the state-dependent C*-dynamical systems. Our recent results on macroscopic dynamical properties of lattice-fermion and quantum-spin systems with long-range, or mean-field, interactions corroborate the relevance of the general approach we present here.

Nilpotent Lie algebras of class 3 with 1-dimensional center and 3-dimensional derived subalgebra

In this paper, we obtain the structure of all non-isomorphic finite-dimensional nilpotent Lie algebras L of class 3 with derived subalgebra of dimension 3 when

The Weakly Solvable Radical and Locally Strongly Algebraic Derivations of Locally Generalized Special Lie Algebras

In this paper, the classical theorem on the image of the solvable radical of a finite-dimensional Lie algebra over a field of characteristic zero under the action of its derivation is generalized to locally generalized special Lie algebras.

Triple derivations on parabolic subalgebras of Kac-Moody algebras

A linear map on a Lie algebra is called a triple derivation if for all Let be a Kac-Moody algebra over an algebraically closed field of characteristic 0, and an arbitrary parabolic subalgebra of In this paper, we prove that any triple derivation of is a derivation, which generalizes the corresponding existing result for finite-dimensional simple Lie algebra.

A commutative algebra approach to multiplicative Hom-Lie algebras

Let be a finite-dimensional complex Lie algebra and be the affine variety of all multiplicative Hom-Lie algebras on . We use a method of computational ideal theory to describe , showing that consists of two 1-dimensional and one 3-dimensional irreducible components and for . We construct a new family of multiplicative Hom-Lie algebras on the Heisenberg Lie algebra and characterize the affine varieties and . We also study the derivation algebra of a multiplicative Hom-Lie algebra D on and, under some hypotheses on D, we prove that the Hilbert series is a rational function.

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.

On the problem of classifying solvable Lie algebras having small codimensional derived algebras

In this paper, we study the classification of finite-dimensional real solvable Lie algebras whose derived algebras are of codimension 1 or 2. We present an effective method to classify -dimensional real solvable Lie algebras having 1-codimensional derived algebras provided that a full classification of n-dimensional nilpotent Lie algebras is given. In addition, the problem of classifying -dimensional real solvable Lie algebras having 2-codimensional derived algebras is proved to be wild. In this case, we classify a subclass of the considered Lie algebras which are extended from their derived algebras by a pair of derivations containing at least one inner derivation.

Isomorphisms among quantum Grothendieck rings and propagation of positivity

Abstract Let ( 𝔤 , 𝗀 ) {\mathfrak{g},\mathsf{g})} be a pair of complex finite-dimensional simple Lie algebras whose Dynkin diagrams are related by (un)folding, with 𝗀 {\mathsf{g}} being of simply-laced type. We construct a collection of ring isomorphisms between the quantum Grothendieck rings of monoidal categories 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} and 𝒞 𝗀 {\mathscr{C}_{\mathsf{g}}} of finite-dimensional representations over the quantum loop algebras of 𝔤 {\mathfrak{g}} and 𝗀 {\mathsf{g}} , respectively. As a consequence, we solve long-standing problems: the positivity of the analogs of Kazhdan–Lusztig polynomials and the positivity of the structure constants of the quantum Grothendieck rings for any non-simply-laced 𝔤 {\mathfrak{g}} . In addition, comparing our isomorphisms with the categorical relations arising from the generalized quantum affine Schur–Weyl dualities, we prove the analog of Kazhdan–Lusztig conjecture (formulated in [D. Hernandez, Algebraic approach to q , t q,t -characters, Adv. Math. 187 2004, 1, 1–52]) for simple modules in remarkable monoidal subcategories of 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} for any non-simply-laced 𝔤 {\mathfrak{g}} , and for any simple finite-dimensional modules in 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} for 𝔤 {\mathfrak{g}} of type B n {\mathrm{B}_{n}} . In the course of the proof we obtain and combine several new ingredients. In particular, we establish a quantum analog of T-systems, and also we generalize the isomorphisms of [D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras, J. reine angew. Math. 701 2015, 77–126, D. Hernandez and H. Oya, Quantum Grothendieck ring isomorphisms, cluster algebras and Kazhdan–Lusztig algorithm, Adv. Math. 347 2019, 192–272] to all 𝔤 {\mathfrak{g}} in a unified way, that is, isomorphisms between subalgebras of the quantum group of 𝗀 {\mathsf{g}} and subalgebras of the quantum Grothendieck ring of 𝒞 𝔤 {\mathscr{C}_{\mathfrak{g}}} .

Geometric nilpotent Lie algebras and zero-dimensional simple complete intersection singularities

Abstract The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

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.