Natural Lie Algebra Research Articles

Toward Jordan decompositions for tensors

We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra that acts on it and embed their direct sum into an auxiliary algebra. Viewed as endomorphisms of this algebra, we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which means that the invariants of the adjoint operator of a tensor, such as the Jordan decomposition, are invariants of the tensor. We show that there is an essentially unique algebra structure that preserves the tensor structure and has a meaningful Jordan decomposition. We utilize aspects of these adjoint operators to study orbit separation and classification in examples relevant to tensor decomposition and quantum information.

Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of {mathcal {S}}({mathfrak g})

Let {mathfrak g} be a semisimple Lie algebra, vartheta in textsf{Aut}({mathfrak g}) a finite order automorphism, and {mathfrak g}_0 the subalgebra of fixed points of vartheta . Recently, we noticed that using vartheta one can construct a pencil of compatible Poisson brackets on {mathcal {S}}({mathfrak g}), and thereby a ‘large’ Poisson-commutative subalgebra {mathcal {Z}}({mathfrak g},vartheta ) of {mathcal {S}}({mathfrak g})^{{mathfrak g}_0}. In this article, we study invariant-theoretic properties of ({mathfrak g},vartheta ) that ensure good properties of {mathcal {Z}}({mathfrak g},vartheta ). Associated with vartheta one has a natural Lie algebra contraction {mathfrak g}_{(0)} of {mathfrak g} and the notion of a good generating system (=g.g.s.) in {mathcal {S}}({mathfrak g})^{mathfrak g}. We prove that in many cases the equality mathrm{ind,}{mathfrak g}_{(0)}=mathrm{ind,}{mathfrak g} holds and {mathcal {S}}({mathfrak g})^{mathfrak g} has a g.g.s. According to V. G. Kac’s classification of finite order automorphisms (1969), vartheta can be represented by a Kac diagram, mathcal {K}(vartheta ), and our results often use this presentation. The most surprising observation is that {mathfrak g}_{(0)} depends only on the set of nodes in mathcal {K}(vartheta ) with nonzero labels, and that if vartheta is inner and a certain label is nonzero, then {mathfrak g}_{(0)} is isomorphic to a parabolic contraction of {mathfrak g}.

The primitive filtration of the Leibniz complex

Pirashvili exhibited a small subcomplex of the Leibniz complex $$(T(s {\mathfrak {g}}), d_{\textrm{Leib}})$$ of a Leibniz algebra $${\mathfrak {g}}$$ . The main result of this paper generalizes this result to show that the primitive filtration of $$T(s{\mathfrak {g}})$$ provides an increasing, exhaustive filtration of the Leibniz complex by subcomplexes, thus establishing a conjecture due to Loday. The associated spectral sequence is used to give a new proof of Pirashvili’s conjecture that, when $${\mathfrak {g}}$$ is a free Leibniz algebra, the homology of the Pirashvili complex is zero except in degree one. This result is then used to show that the desuspension of the Pirashvili complex carries a natural $$L_\infty $$ -structure that induces the natural Lie algebra structure on the homology of the complex in degree zero.

The natural lie algebra brackets on couples of vector fields and p-forms

If m ? p + 1 ? 2 or m = p ? 3, all natural Lie algebra brackets on couples of vector fields and p-forms on m-manifolds are described.

Dual Lie bialgebra structures of Poisson types

Let $\mathcal{A} = \mathbb{F}[x,y]$ be the polynomial algebra on two variables x, y over an algebraically closed field $\mathbb{F}$ of characteristic zero. Under the Poisson bracket, $\mathcal{A}$ is equipped with a natural Lie algebra structure. It is proven that the maximal good subspace of $\mathcal{A}*$ induced from the multiplication of the associative commutative algebra $\mathcal{A}$ coincides with the maximal good subspace of $\mathcal{A}*$ induced from the Poisson bracket of the Poisson Lie algebra $\mathcal{A}$ . Based on this, structures of dual Lie bialgebras of the Poisson type are investigated. As by-products, five classes of new infinite-dimensional Lie algebras are obtained.

The Lie algebra of rooted planar trees

We study a natural Lie algebra structure on the free vector space generated by all rooted planar trees as the associated Lie algebra of the nonsymmetric operad (non-Σ operad, preoperad) of rooted planar trees. We determine whether the Lie algebra and some related Lie algebras are finitely generated or not, and prove that a natural surjection called the augmentation homomorphism onto the Lie algebra of polynomial vector fields on the line has no splitting preserving the units.

Group Algebras of Finite Groups as Lie Algebras

We consider the natural Lie algebra structure on the (associative) group algebra of a finite group G, and show that the Lie subalgebras associated to natural involutive antiautomorphisms of this group algebra are reductive ones. We give a decomposition in simple factors of these Lie algebras, in terms of the ordinary representations of G.

HERMITIAN VECTOR FIELDS AND SPECIAL PHASE FUNCTIONS

We start by analyzing the Lie algebra of Hermitian vector fields of a Hermitian line bundle. Then, we specify the base space of the above bundle by considering a Galilei, or an Einstein spacetime. Namely, in the first case, we consider, a fibred manifold over absolute time equipped with a spacelike Riemannian metric, a spacetime connection (preserving the time fibring and the spacelike metric) and an electromagnetic field. In the second case, we consider a spacetime equipped with a Lorentzian metric and an electromagnetic field. In both cases, we exhibit a natural Lie algebra of special phase functions and show that the Lie algebra of Hermitian vector fields turns out to be naturally isomorphic to the Lie algebra of special phase functions. Eventually, we compare the Galilei and Einstein cases.

Lie algebra of anomalously scaled fluctuations

For quantum lattice systems, it is proven that anomalously scaled fluctuations have a natural Lie algebra structure. The harmonic lattice in the ground state is given as an illustration of the general theorem.

L'algèbre de Lie des gradients itérés d'un générateur markovien---développements de moyennes et entropies

We define a natural Lie algebra structure associated to a Markov generator which extends the carre du champ operator and its iterated gradients. While the iterated gradients may be used in expansions ofthe variance of a function of the demain, this more general structure is needed in formai expansions of the entropy of a function.

Limits of tame automorphisms of k[ x1, …, Xn

Let J consist of all endomorphisms ψ of S = k[ x 1, …, x N ] whose Jacobtan determinant, det( ∂ψ(x i) ∂ j) , is a non-zero constant. Let G be the group of tame automorphisms of S. In this paper we prove that G is dense in J in the “formal power series topology.” In particular, every automorphism of S is a limit of tame automorphisms. We also demonstrate that certain subquotients of G have a natural Lie algebra structure.