Application of H infinite control to ship steering system

On solvable Lie and Leibniz superalgebras with maximal codimension of nilradical

We discuss a generalization of Chern-Simons theory in three dimensions based on Leibniz (or Loday) algebras, which are generalizations of Lie algebras. Special cases of such theories appear in gauged supergravity, where the Leibniz algebra is defined in terms of the global (Lie) symmetry algebra of the ungauged limit and an embedding tensor. We show that the Leibniz algebra of generalized diffeomorphisms in exceptional field theory can similarly be obtained from a Lie algebra that describes the enhanced symmetry of an `ungauged phase' of the theory. Moreover, we show that a `topological phase' of ${\rm E}_{8(8)}$ exceptional field theory can be interpreted as a Chern-Simons theory for an algebra unifying the three-dimensional Poincar\'e algebra and the Leibniz algebra of ${\rm E}_{8(8)}$ generalized diffeomorphisms.

This paper presents the classification over the fields of real and complex numbers, of the minimal Z2×Z2-graded Lie algebras and Lie superalgebras spanned by four generators and with no empty graded sector. The inequivalent graded Lie (super)algebras are obtained by solving the constraints imposed by the respective graded Jacobi identities. A motivation for this mathematical result is to systematically investigate the properties of dynamical systems invariant under graded (super)algebras. Recent works only paid attention to the special case of the one-dimensional Z2×Z2-graded Poincaré superalgebra. As applications, we are able to extend certain constructions originally introduced for this special superalgebra to other listed Z2×Z2-graded (super)algebras. We mention, in particular, the notion of Z2×Z2-graded superspace and of invariant dynamical systems (both classical worldline sigma models and quantum Hamiltonians). As a further by-product, we point out that, contrary to Z2×Z2-graded superalgebras, a theory invariant under a Z2×Z2-graded algebra implies the presence of ordinary bosons and three different types of exotic bosons, with exotic bosons of different types anticommuting among themselves.

In this paper we introduce generalized Spencer cohomology for finite depth Z-graded Lie (super)algebras. We develop a method of finding filtered deformations of such Z-graded Lie (super)algebras based on this cohomology. As an application we determine all simple filtered deformations of certain Z-graded Lie superalgebras classified in [K3], thus completing the last step in the classification of simple infinite-dimensional linearly compact Lie superalgebras.

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.

Lie algebras graded by finite irreducible reduced root systems have been classified up to central extensions by Berman and Moody, Benkart and Zelmanov, and Neher. In this paper we determine the central extensions of these Lie algebras and hence describe them completely up to isomorphism.

We introduce the notion of ``local system of $\Bbb{Z}_{T}$-twisted vertex operators'' on a $\Bbb{Z}_{2}$-graded vector space $M$, generalizing the notion of local system of vertex operators [Li]. First, we prove that any local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$ has a vertex superalgebra structure with an automorphism $\sigma$ of order $T$ with $M$ as a $\sigma$-twisted module. Then we prove that for a vertex (operator) superalgebra $V$ with an automorphism $\sigma$ of order $T$, giving a $\sigma$-twisted $V$-module $M$ is equivalent to giving a vertex (operator) superalgebra homomorphism from $V$ to some local system of $\Bbb{Z}_{T}$-twisted vertex operators on $M$. As applications, we study the twisted modules for vertex operator (super)algebras associated to some well-known infinite-dimensional Lie (super)algebras and we prove the complete reducibility of $\Bbb{Z}_{T}$-twisted modules for vertex operator algebras associated to standard modules for an affine Lie algebra.

We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes to the (simplicial) generalized associahedra or, equivalently, to the cluster complexes for the cluster algebras of finite type. Our computation of the face numbers and h-vectors of these complexes produces the enumerative invariants defined in other contexts by C.A.Athanasiadis, suggesting links to a host of well studied problems in algebraic combinatorics of finite Coxeter groups, root systems, and hyperplane arrangements. Recurrences satisfied by the face numbers of our complexes lead to combinatorial algorithms for determining Coxeter-theoretic invariants. That is, starting with a Coxeter diagram of a finite Coxeter group, one can compute the Coxeter number, the exponents, and other classical invariants by a recursive procedure that only uses most basic graph-theoretic concepts applied to the input diagram. In types A and B, we rediscover the constructions and results obtained by E.Tzanaki .

For a commutative algebra A over a commutative ring k satisfying certain conditions, we construct the universal central extension of \( {\frak g}_k \otimes_k A \), regarded as a Lie superalgebra over k, where \( {\frak g}_k \) denotes a basic classical Lie superalgebra over k. To consider basic classical Lie superalgebras over an ring k, we also show the existence of their Chevalley basis. Our results contain not only the descriptions of the untwisted affine Lie superalgebras but also those of the toroidal Lie superalgebras.

Lusztig isomorphisms for Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type

In this paper we construct a minimal faithful representation of the (2 m+ 2) -dimensional complex general Diamond Lie algebra, Dm(C) , which is isomorphic to a subalgebra of the special linear Lie algebra sl(m+ 2 , C). We also construct a faithful representation of the real general Diamond Lie algebra Dm which is isomorphic to a subalgebra of the special symplectic Lie algebra sp(2 m+ 2 , R). Furthermore, we describe Leibniz algebras with corresponding (2 m+ 2) -dimensional real general Diamond Lie algebra Dm and such that the ideal generated by the squares of elements provides a faithful representation of Dm.

Parabolic subgroup orbits on finite root systems

This article gives a local answer to the coquecigrue problem for Leibniz algebras, that is, the problem of finding a generalization of the (Lie) group structure such that Leibniz algebras are the corresponding tangent algebra structure. Using links between Leibniz algebra cohomology and Lie rack cohomology, we generalize the integration of a Lie algebra into a Lie group by proving that every Leibniz algebra is isomorphic to the tangent Leibniz algebra of a local Lie rack. This article ends with an example of a Leibniz algebra integration in dimension 5.

ABSTRACTLeibniz algebras are certain generalization of Lie algebras. Recently, analyzing the structure of subalgebras, David Towers gave some criteria for the solvability and supersolvability of Lie algebras. In this paper we define analogues concepts for Leibniz algebras and extend some of these results on solvability and supersolvability to that of Leibniz algebras.

We canonically associate a Leibniz algebra with every Nambu algebra. We show how various homological and cohomological complexes for a Nambu algebra can be naturally obtained from its structure as a module over the Leibniz algebra. We also present a generalization of a classical Lie--Berezin construction for Nambu algebras and extend these results for Nambu superalgebras.