Quadratic Lie Algebras

In this article, the notion of universal enveloping algebra introduced in Ardizzoni [4 Ardizzoni , A. On primitively generated braided bialgebras . Algebr. Represent. Theory , to appear. (doi:10.1007/s10468-010-9257-z)  [Google Scholar]] is specialized to the case of braided vector spaces whose Nichols algebra is quadratic as an algebra. In this setting, a classification of universal enveloping algebras for braided vector spaces of dimension not greater than 2 is handled. As an application, we investigate the structure of primitively generated connected braided bialgebras whose braided vector space of primitive elements forms a Nichols algebra, which is a quadratic algebra.     

A Note on the Linkage of Hurwitz Algebras

Let F be a field of even characteristic and H, H two Hurwitz algebras over F. We investigate the common quadratic separable and inseparable subfields of the algebras H, H.Mathematics Subject Classification (2000): 11E88, 12F10, 12F15, 16B99     

RESEARCH ANNOUNCEMENTS——Structure of Solvable Quadratic Lie Algebras

Killing form plays a key role in the theory of semisimple Lie algebras. It is natural to extend the study to Lie algebras with a nondegenerate symmetric invariant bilinear form. Such a Lie algebra is generally called a quadratic Lie algebra which occur naturally in physics^[10，12，13]. Besides semisimple Lie algebras, interesting quadratic Lie algebras include the Kac-Moody algebras and the Extended Affine Lie algebras. In this paper, we study solvable quadratic Lie algebras. In Section 1, we study quadratic solvable Lie algebras whose Cartan subalgebras consist of semi-simple elements. In Section 2,we present a procedure to construct a class of quadratic Lie algebras, and we can exhaust all solvable quadratic Lie algebras in such a way. All Lie algebras mentioned in this paper are finite dimensional Lie algebras over a field F of characteristic 0.     

Deformations of Quadratic Algebras with Antisymmetric Relations

A finitely generated quadratic algebra with antisymmetric generating relations is called a weakly symmetric algebra. The automorphism group and Calabi–Yau property of a Poincaré–Birkhoff–Witt (PBW)-deformation of a weakly symmetric algebra are discussed. It is shown that the Calabi–Yau property of a PBW-deformation of a weakly symmetric algebra is equivalent to that of the corresponding augmented PBW-deformation under some mild conditions.     

A New Invariant of Quadratic Lie Algebras

We define a new invariant of quadratic Lie algebras and give a complete study and classification of singular quadratic Lie algebras, i.e. those for which the invariant does not vanish. The classification is related to O(n)-adjoint orbits in $\mathfrak{o}(n)$ .     

Division Algebras, Galois Fields, Quadratic Residues

Intended for mathematical physicists interested in applications of the division algebras to physics, this article highlights some of their more elegant properties with connections to the theories of Galois fields and quadratic residues.     

Mutation of Auslander Generators for Selfinjective Algebras

Let Λ be an artin algebra with representation dimension equal to three and M an Auslander generator of Λ. We show how, under certain assumptions, we can mutate M to get a new Auslander generator whose endomorphism ring is derived equivalent to the endomorphism ring of M. We apply our results to selfinjective algebras with radical cube zero of infinite representation type, where we construct an infinite set of Auslander generators.     

How to Solve a Quadratic Equation In Rationals

The title alludes to a similar title of the paper [3] by Grunewaldand Segal, in which it is shown how to solve a quadratic equationin integers. This latter procedure seems to be quite difficult,and the algorithm outlined in [3] is rather involved, althoughit is completely effective in the logical sense. 1991 MathematicsSubject Classification 11D09.     

On Algebras of Skew Polynomials Generated by Quadratic Homogeneous Relations

We consider the algebras, with two generators a and b, generated by the quadratic relations ba = α² + βab + γb², where the coefficients α, β, and γ belong to an arbitrary field F of characteristic 0. We find conditions for such an algebra to be expressed as a skew polynomial algebra with generator b over the polynomial ring F [a]. These conditions are equivalent to the existence of the Poincaree-Birkhoff-Witt basis, i. e., basis of the form {a^m, bⁿ}. Bibliography: 16 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 301, 2003, pp. 144–171.     

Gerstenhaber Algebra Structure on the Hochschild Cohomology of Quadratic String Algebras

We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH^?(A) when A is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdell’s resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzell’s resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.     

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.     

Class Partition Algebras as Centralizer Algebras of Wreath Products

In this article, we study an important subalgebra of the tensor product partition algebra P _k (x)? P _k (y), denoted by P _k (x, y) and called “Class Partition Algebra.” We show that the algebra P _k (n, m) is the centralizer algebra of the wreath product S _m ? S _n . Furthermore, the algebra P _k (x, y) and the tensor product partition algebra P _k (x)? P _k (y) are subalgebras of the G-colored partition algebra P _k (x;G) and G-vertex colored partition algebra P _k (x, G) respectively, for every group G with |G|=y ≥ 2k.     

Effect Algebras as Presheaves on Finite Boolean Algebras

For an effect algebra A, we examine the category of all morphisms from finite Boolean algebras into A. This category can be described as a category of elements of a presheaf R(A) on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra A can be characterized in terms of some properties of the category of elements of the presheaf R(A). We prove that the tensor product of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. The tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.     

二次图构形的边搜索法及其Orlik-Solomon代数的上同调

本文研究了一类特殊的图构形,即二次图构形.首先,给出了弦图的边搜索法,用边搜索法可以决定弦图所对应的超平面构形中超平面的哪些排序使构形是二次均.其次,对二次图构形的Orlik-Solomon代数作为线性空间的维数进行了计算,得到了它的Poincare多项式的表达式.最后,对二次图构形Orlik-Solomon代数的上同调进行了研究,对边缘算子集中在秩为二的模元时,得到了二次图构形的Orlik-Solomon代数的上同调的维数计算公式.     

Azumaya Algebras as Galois Comodules

It is shown that Azumaya algebras (over commutative rings) are special examples of Galois comodules. This leads to a new characterization of Azumaya algebras.