Comparison of 3-Dimensional ℤ-Graded and ℤ2-Graded L ∞ Algebras

This article explores L _∞ structures on 3-dimensional vector spaces with both ?- and ?₂-gradings. Since ?-graded L _∞ algebras are special cases of ?₂-graded algebras in the induced ?₂-grading, there are generally fewer ?-graded L _∞ structures on a given space. However, degree zero automorphisms (rather than even automorphisms) determine equivalence in a ?-graded space. We therefore find nontrivial examples in which the map from the ?-graded moduli space to the ?₂-graded moduli space is bijective, injective but not surjective, or surjective but not injective. Additionally, we study how the codifferentials in the moduli spaces deform into other nonequivalent codifferentials.     

Cosimplicial DGLAs in Deformation Theory

We identify ?ech cocycles in nonabelian (formal) group cohomology with Maurer–Cartan elements in a suitable L _∞-algebra. Applications to deformation theory are described.     

Rectifications of A∞-Algebras

In this article, in the setting of connected DG-modules, we prove that, for any A _∞-algebra (M, {m _i } _i≥1), there is a chain contraction from a DG-algebra A _M onto the DG-module M such that the A _∞-algebra structure induced by perturbation theory on M is precisely the original one. In fact, the mentioned DG-algebra can be considered a rectification of the A _∞-algebra in the sense of (Boardman and Vogt, 1973 Boardman , J. M. , Vogt , R. M. ( 1973 ). Homotopy Invariant Algebraic Structures on Topological Spaces . Lecture Notes in Mathematics . Springer-Verlag , pp. 347 . [Google Scholar]). Appropiate dual results are given for A _∞-coalgebras.     

Polynomial operations on complex K-theory

We adapt here the results of the author concerning polynomial operations on the 0th stable cohomotopy to the case of the 0th complex K-theory and consider polynomial operations : Kh, where h is a ring-valued contravariant functor, defined on finite CW-complexes, satisfying some properties. We construct a family of generating operations for the ring Pol(K,h) of all polynomial operations : Kh and doing so, we describe the additive structure of this ring in terms of the h(BU(n)'s. As an illustration of how polynomiality could be used to study operations in the setting of algebraic K-theory, we consider, from our point of view, the well known situation operations : KK on complex K-theory.     

On QB ∞-Rings

In this article, we introduce a new class of rings, the QB _∞-rings. We establish various properties of this concept. These show that, in several respects, QB _∞-rings behave like QB-rings. We prove that the notion of QB _∞-rings is a Morita invariant property of rings and every finite subdirect product of QB _∞-rings is a QB _∞-ring. We also exhibit examples to point out that the class of QB _∞-rings is much larger than the class of QB-rings.     

Complex structured singular value analysis using fixed-structure dynamic D-scales

Models of systems are always inexact. Hence, to better predict the performance of a system it is necessary to take into account uncertainty in a nominal model of a system. The structured singular value was developed to nonconservatively analyze robust stability and performance for systems with multiple-block uncertainty. In practice, optimization techniques are used to compute an upper bound on the structured singular value. For dynamic uncertainty with bounded magnitude and arbitrary phase (i.e., "complex uncertainty"), the standard approach to computing an upper bound involves finding diagonal scaling matrices D(jω) that minimize σ_max (D(jω)G(jω)D^-1(jω)) over a (theoretically) infinite number of frequencies. The order of the corresponding stable, minimum phase, rational function D(s) (if it exists) is hence arbitrary, which can lead to very high order controllers when D(s) is used for controller synthesis. This paper develops a fixed-structure approach to computing an upper bound for the complex structured singular value. In particular, by relying on results from mixed-norm H₂/H_∞ analysisD(s) is a priori constrained to be a rational matrix function of a chosen order and a new approach to computing an upper bound on the structured singular value is developed. The results are illustrated using two examples which clearly demonstrate the suboptimality of standard curve fitting. The proposed approach can be extended to mixed uncertainty and structured singular value controller synthesis without D — K type iteration.     

Solving ℓ1 Regularization Problems With Piecewise Linear Losses

This article is concerned with the computational aspect of ?₁ regularization problems with a certain class of piecewise linear loss functions. The problem of computing the ?₁ regularization path for a piecewise linear loss can be formalized as a parametric linear programming problem. We propose an efficient implementation method of the parametric simplex algorithm for such a problem. We also conduct a simulation study to investigate the behavior of the number of “breakpoints” of the regularization path when both the number of observations and the number of explanatory variables vary. Our method is also applicable to the computation of the regularization path for a piecewise linear loss and the blockwise ?_∞ penalty. This article has supplementary material online.     

Constructions for Octonion and Exceptional Jordan Algebras

In this note we reverse theusual process of constructing the Lie algebras of types G ₂and F ₄ as algebras of derivations of the splitoctonions or the exceptional Jordan algebra and instead beginwith their Dynkin diagrams and then construct the algebras togetherwith an action of the Lie algebras and associated Chevalley groups.This is shown to be a variation on a general construction ofall standard modules for simple Lie algebras and it is well suitedfor use in computational algebra systems. All the structure constantswhich occur are integral and hence the construction specialisesto all fields, without restriction on the characteristic, avoidingthe usual problems with characteristics 2 and 3.     

Nijenhuis forms on Lie-infinity algebras associated to Lie algebroids

Introducing Nijenhuis forms on L_∞-algebras gives a general frame to understand deformations of the latter. We give here a Nijenhuis interpretation of a deformation of an arbitrary Lie algebroid into an L_∞-algebra. Then we show that Nijenhuis forms on L_∞-algebras also give a short and e?cient manner to understand Poisson-Nijenhuis structures and, more generally, the so-called exact Poisson quasi-Nijenhuis structures with background.     

Tits' Constructions of Jordan Algebras and F4 Bundles on the Plane

In this paper, constructions of Jordan algebras over commutative rings are given which place, within a general set-up, the classical Tits constructions of exceptional central simple Jordan algebras over fields. These are used to exhibit nontrivial Jordan algebra bundles over the affine plane with a given exceptional Jordan division algebra over k as the fibre. The associated principal F₄ bundles are shown to admit no reduction of the structure group to any proper connected reductive subgroup.     

Classical r-Matrices and Novikov Algebras

We study the existence problem for Novikov algebra structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov algebra is necessarily solvable. Conversely we present a 2-step solvable Lie algebra without any Novikov structure. We use extensions and classical r-matrices to construct Novikov structures on certain classes of solvable Lie algebras.     

Generation of groups by rank-2 tori

Let k be a field of characteristic different from 2 and 3. In this paper, we study the number of rank-2 k-tori required (in fact exhibit such tori explicitly) for the generation of groups of type A₂, G₂ and F₄ arising from division algebras and subgroups of type D₄ of Aut(A), for A an Albert division algebra, over infinite perfect fields.     

Spin Spaces, Lipschitz Groups, and Spinor Bundles

It is shown that every bundle M of complex spinormodules over the Clifford bundle Cl(g) of a Riemannian space(M, g) with local model (V, h)is associated with an lpin(Lipschitz) structure on M, this being a reduction of theO(h)-bundle of all orthonormal frames on M to the Lipschitzgroup Lpin(h) of all automorphisms of a suitably defined spinspace. An explicit construction is given of the total space of theLpin(h)-bundle defining such a structure. If the dimension mof M is even, then the Lipschitz group coincides with the complexClifford group and the lpin structure can be reduced to a pin _c structure. If m = 2n – 1, then a spinor module on M is of the Cartan type: its fibres are 2 _n -dimensional anddecomposable at every point of M, but the homomorphism of bundlesof algebras Cl(g) End globally decomposes if, andonly if, M is orientable. Examples of such bundles are given. Thetopological condition for the existence of an lpin structure on anodd-dimensional Riemannian manifold is derived and illustrated by theexample of a manifold admitting such a structure, but no pin _c structure.     

Operator algebras associated with unitary commutation relations

We define nonselfadjoint operator algebras with generators Le₁,…,Len,Lf₁,…,Lfm subject to the unitary commutation relations of the form

     

On Exact Courant Algebras

In this note, we will show that exact Courant algebras over a Lie algebra 𝔤 can be characterized via Leibniz 2-cocycles, and the automorphism group of a given exact Courant algebra is in a one-to-one correspondence with first Leibniz cohomology space of 𝔤.     

How to Play M 13?

A puzzle called “M ₁₃” J. H. Conway has described recently is explained. We report an implementation of the puzzle in the programming language Java. The program allows the human user to “play M ₁₃” interactively (and to cheat by solving it automatically). The program is an example on how to bring to life a nice piece of discrete mathematics. In this sense it presents not only a didactical way of seeing “mathematics at work”, but also displays the stabilizer chain method developed by C. Sims to solve group theoretic puzzles, the most famous of which being Rubik's cube.     

On Whittaker modules over a class of algebras similar to U(sl 2)

Motivated by the study of invariant rings of finite groups on the first Weyl algebras A ₁ and finding interesting families of new noetherian rings, a class of algebras similar to U(sl ₂) was introduced and studied by Smith. Since the introduction of these algebras, research efforts have been focused on understanding their weight modules, and many important results were already obtained. But it seems that not much has been done on the part of nonweight modules. In this paper, we generalize Kostant’s results on the Whittaker model for the universal enveloping algebras U(g) of finite dimensional semisimple Lie algebras g to Smith’s algebras. As a result, a complete classification of irreducible Whittaker modules (which are definitely infinite dimensional) for Smith’s algebras is obtained, and the submodule structure of any Whittaker module is also explicitly described.