Dimensional Lie Algebra Research Articles

A class of nilpotent Lie algebras admitting a compact subgroup of automorphisms

The realification of the (2n+1)-dimensional complex Heisenberg Lie algebra is a (4n+2)-dimensional real nilpotent Lie algebra with a 2-dimensional commutator ideal coinciding with the centre, and admitting the compact algebra sp(n) of derivations. We investigate, in general, whether a real nilpotent Lie algebra with 2-dimensional commutator ideal coinciding with the centre admits a compact Lie algebra of derivations. This also gives us the occasion to revisit a series of classic results, with the expressed aim of attracting the interest of a broader audience.

Gradings on Lie algebras with applications to infra-nilmanifolds

In this paper, we study positive as well as non-negative and non-trivial gradings on finite dimensional Lie algebras. The existence of such a grading on a Lie algebra is invariant under taking field extensions, a result very recently obtained by Y. Cornulier and we give a different proof of this fact. Similarly, we prove that given a grading of one of these types and a finite group of automorphisms, there always exist a grading of the same type which is preserved by this group. From these results we conclude that the existence of an expanding map or a non-trivial self-cover on an infra-nilmanifold depends only on the covering Lie group. Another application is the construction of a nilmanifold admitting an Anosov diffeomorphism but no non-trivial self-covers and in particular no expanding maps, which is the first known example of this type.

A modular analogue of Morozov's theorem on maximal subalgebras of simple Lie algebras

Let G be a simple algebraic group over an algebraically closed field of characteristic p>0 and suppose that p is a very good prime for G. In this paper we prove that any maximal Lie subalgebra M of g=Lie(G) with rad(M)≠0 has the form M=Lie(P) for some maximal parabolic subgroup P of G. This means that Morozov's theorem on maximal subalgebras is valid under mild assumptions on G. We show that such assumptions are necessary by providing a counterexample to Morozov's theorem for groups of type E8 over fields of characteristic 5. Our proof relies on the main results and methods of the classification theory of finite dimensional simple Lie algebras over fields of prime characteristic.

ON THE STRUCTURE OF FACTOR LIE ALGEBRAS

The Lie algebra analogue of Schur's result which is proved by Moneyhun in 1994, states that if L is a Lie algebra such that dimL/Z(L) = n, then <TEX>$dimL_{(2)}={\frac{1}{2}}n(n-1)-s$</TEX> for some non-negative integer s. In the present paper, we determine the structure of central factor (for s = 0) and the factor Lie algebra <TEX>$L/Z_2(L)$</TEX> (for all <TEX>$s{\geq}0$</TEX>) of a finite dimensional nilpotent Lie algebra L, with n-dimensional central factor. Furthermore, by using the concept of n-isoclinism, we discuss an upper bound for the dimension of <TEX>$L/Z_n(L)$</TEX> in terms of <TEX>$dimL_{(n+1)}$</TEX>, when the factor Lie algebra <TEX>$L/Z_n(L)$</TEX> is finitely generated and <TEX>$n{\geq}1$</TEX>.

On Group Classification and Nonlocal Conservation Laws for a Multiphase Flow Model

The main objective of this study is five dimensional Lie algebra which describes the symmetry properties of a isothermal drift flux model of two phase flows. The classification provided is based on method given by Patera and Winternitz. Multi-dimensional optimal systems are obtained for the symmetry algebra of model. Further, using general theorem proved by Ibragimov we find several nonlocal conservation laws corresponding to every symmetry in one dimensional optimal system.

Onservation Laws in Group Analysis of Gas Filtration Model

One-dimensional gas filtration was described nonlinear parabolic equation as the conservation law. The potential of the conservation law satisfies a equation as the conservation law. The introduce of the second potential gives a system of equations which admits 6-dimensional Lie algebra. This extends group properties of initial model. With the help of optimal system of subalgebras are classified all invariant and partial invariant solutions which are reduced to invariant solutions of the initial model. Some time it is possible to find integral of invariant submodel.

Dual Lie bialgebra structures of loop and current Virasoro\\ algebras

The loop and current Virasoro algebras are respectively the tensor product Lie algebras of the Virasoro algebra with the polynomial and Laurent polynomial algebra. In the present paper, the structures of dual Lie bialgebras of loop and current Virasoro algebras are investigated. As a result, a series of new infinite dimensional Lie algebras are obtained.

Current algebra formulation of M-theory based on E11 Kac–Moody algebra

Quantum M-theory is formulated using the current algebra technique. The current algebra is based on a Kac–Moody algebra rather than usual finite dimensional Lie algebra. Specifically, I study the [Formula: see text] Kac–Moody algebra that was shown recently[Formula: see text] to contain all the ingredients of M-theory. Both the internal symmetry and the external Lorentz symmetry can be realized inside [Formula: see text], so that, by constructing the current algebra of [Formula: see text], I obtain both internal gauge theory and gravity theory. The energy–momentum tensor is constructed as the bilinear form of the currents, yielding a system of quantum equations of motion of the currents/fields. Supersymmetry is incorporated in a natural way. The so-called “field-current identity” is built in and, for example, the gravitino field is itself a conserved supercurrent. One unanticipated outcome is that the quantum gravity equation is not identical to the one obtained from the Einstein–Hilbert action.

On Degenerations of Algebras over an Arbitrary Field

For each $n\ge2$ we classify all $n$-dimensional algebras over an arbitrary infinite field which have the property that the $n$-dimensional abelian Lie algebra is their only proper degeneration.

Classification of $${{\mathfrak{sl}}_3}$$ sl 3 Relations in the Witt Group of Nondegenerate Braided Fusion Categories

The Witt group of nondegenerate braided fusion categories $\mathcal{W}$ contains a subgroup $\mathcal{W}_\text{un}$ consisting of Witt equivalence classes of pseudo-unitary nondegenerate braided fusion categories. For each finite-dimensional simple Lie algebra $\mathfrak{g}$ and positive integer $k$ there exists a pseudo-unitary category $\mathcal{C}(\mathfrak{g},k)$ consisting of highest weight integerable $\hat{g}$-modules of level $k$ where $\hat{\mathfrak{g}}$ is the corresponding affine Lie algebra. Relations between the classes $[\mathcal{C}(\mathfrak{sl}_2,k)]$, $k\geq1$ have been completely described in the work of Davydov, Nikshych, and Ostrik. Here we give a complete classification of relations between the classes $[\mathcal{C}(\mathfrak{sl}_3,k)]$, $k\geq1$ with a view toward extending these methods to arbitrary simple finite dimensional Lie algebras $\mathfrak{g}$ and positive integer levels $k$.

DERIVATIONS OF TERNARY LIE ALGEBRAS AND GENERALIZATIONS

We study derivations of ternary Lie algebras. Precisely, we investigate the relation between derivations of Lie algebras and the induced ternary Lie algebras. We also explore the spaces of quasi-derivations, the centroid and the quasi-centroid and give some properties. Finally, we compute these spaces for low dimensional ternary Lie algebras g.

Rota-Baxter Operators on 3-Dimensional Lie Algebras and the Classical R-Matrices

Our aim is to classify the Rota-Baxter operators of weight 0 on the 3-dimensional Lie algebra whose derived algebra’s dimension is 2. We explicitly determine all Rota-Baxter operators (of weight zero) on the 3-dimensional Lie algebras g. Furthermore, we give the corresponding solutions of the classical Yang-Baxter equation in the 6-dimensional Lie algebras g ⋉ad⁎ g⁎ and the induced left-symmetry algebra structures on g.

Additional symmetries and string equations of the noncommutative B and C type KP hierarchies

In this paper, we construct the noncommutative B and C type KP hierarchies using pseudo-differential operators and reducing conditions. Further a series of additional flows of the noncommutative B and C type KP hierarchies will be defined and the additional symmetries constitute the B and C type infinite dimensional Lie algebra W1+∞. In addition, the generating function of the additional symmetries can also be proved to have a nice form in terms of wave functions. Further, the string equations of the noncommutative B and C type KP hierarchies are derived.

The polynomiality of the Poisson center and semi-center of a Lie algebra and Dixmier's fourth problem

Let g be a finite dimensional Lie algebra over an algebraically closed field k of characteristic zero. We provide necessary and also some sufficient conditions in order for its Poisson center and semi-center to be polynomial algebras over k.This occurs for instance if g is quadratic of index 2 with [g,g]≠g and also if g is nilpotent of index at most 2. The converse holds for filiform Lie algebras of type Ln, Qn, Rn and Wn.We show how Dixmier's fourth problem for an algebraic Lie algebra g can be reduced to that of its canonical truncation gΛ. Moreover, Dixmier's statement holds for all Lie algebras of dimension at most eight. The nonsolvable ones among them possess a polynomial Poisson center and semi-center.

Finite dimensional Hopf actions on algebraic quantizations

Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra [arxiv.org/abs/1409.1644, arxiv.org/abs/1509.01165], we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra k[z_1,...,z_s] factors through a group action, and this in fact holds for any finite dimensional Hopf action if s=0. We also generalized these results to finite dimensional Hopf actions on algebras of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations of commutative domains. This includes universal enveloping algebras of finite dimensional Lie algebras, spherical symplectic reflection algebras, quantum Hamiltonian reductions of Weyl algebras (in particular, quantized quiver varieties), finite W-algebras and their central reductions, quantum polynomial algebras, twisted homogeneous coordinate rings of abelian varieties, and Sklyanin algebras. The generalization in the last three cases uses a result from algebraic number theory, due to A. Perucca.

The Nash–Moser theorem of Hamilton and rigidity of finite dimensional nilpotent Lie algebras

We apply the Nash–Moser theorem for exact sequences of R. Hamilton to the context of deformations of Lie algebras and we discuss some aspects of the scope of this theorem in connection with the polynomial ideal associated to the variety of nilpotent Lie algebras. This allows us to introduce the space Hk-nil2(g,g), and certain subspaces of it, that provide fine information about the deformations of g in the variety of k-step nilpotent Lie algebras.Then we focus on degenerations and rigidity in the variety of k-step nilpotent Lie algebras of dimension n with n≤7 and, in particular, we obtain rigid Lie algebras and rigid curves in the variety of 3-step nilpotent Lie algebras of dimension 7. We also recover some known results and point out a possible error in a published article related to this subject.

Lie algebra type noncommutative phase spaces are Hopf algebroids

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way, therefore obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.

On the theta completions for maximal subalgebras

ABSTRACTLet L be a finite dimensional Lie algebra. Then for a maximal subalgebra M of L, a 𝜃-completion for M is a subalgebra C of L such that CM and ML⊆C and C∕ML contains no non-zero ideal of L∕ML, properly. And a 𝜃-completion C of M is said to be a strong 𝜃-completion, if C = L or there exists a subalgebra B of L such that C be maximal in B and B is not a 𝜃-completion for M. These are analogous to the concepts of 𝜃-completion and strong 𝜃-completion of a maximal subgroup of a finite group. Now, we consider the influence of these concepts on the structure of a finite dimensional Lie algebra.

Finite-dimensional integrable systems: A collection of research problems

This article suggests a series of problems related to various algebraic and geometric aspects of integrability. They reflect some recent developments in the theory of finite-dimensional integrable systems such as bi-Poisson linear algebra, Jordan–Kronecker invariants of finite dimensional Lie algebras, the interplay between singularities of Lagrangian fibrations and compatible Poisson brackets, and new techniques in projective geometry.

SCAP-subalgebras of Lie algebras

A subalgebra H of a finite dimensional Lie algebra L is said to be a SCAP-subalgebra if there is a chief series 0 = L 0 ⊂ L 1 ⊂... ⊂ L t = L of L such that for every i = 1, 2,..., t, we have H + L i = H + L i-1 or H ∩ L i = H ∩ L i-1. This is analogous to the concept of SCAP-subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its SCAP-subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.