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The Generalized Burnside Theorem in Noncommutative
DeformationT heory
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By
Eivind Eriksen
© 2011. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 5
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Abstract
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Let A be an associative algebra over a field k, and let
M be a finite family of right A - modules. A study of
the noncommutative deformation functor Def
M of the family M leads to the construction of the algebra OA(M)
of observables and the generalized Burnside theorem, due to Laudal (2002). In this paper, we give an overview of
aspects of noncommutative deformations closely connected to the generalized Burnside theorem. |
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2. |
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Phase Spaces and Deformation Theory
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By
Olav Arn © 2011. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 5
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We have previously introduced the notion of non-commutative phase space (algebra) associated to any
associative algebra, defined over a field. The purpose of the present paper is to prove that this construction is useful
in non-commutative deformation theory for the construction of the versal family of finite families of modules. In
particular, we obtain a muchbetter understanding of the obstruction calculus, that is, of the Massey products. |
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3. |
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Meandersand Frobenius Seaweed Lie Algebras
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By
Vincent Coll, Anthony Giaquinto, and Colton Magnan
© 2011. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 5
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Abstract
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The index of a seaweed Lie algebra can be computed from its associated meander graph. We examine this
graph in several ways with a goal of determining families of Frobenius(index zero)seaweed algebras. Ouranalysis
gives two new families of Frobenius seaweed algebras as well as elementary proofs of known families of such Lie
algebras.
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4. |
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Deformations of Complex 3-Dimensional Associative
Algebras
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By
Alice Fialowski, Michael Penkava, and Mitch Phillipson
© 2011. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 5
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1-22 Page(s)
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Abstract
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We study deformations and the moduli space of 3-dimensional complex associative algebras. We use
extensions to compute the moduli space, and then give a decomposition of this moduli space in to strata consisting of
complex projective orbifolds, glued together through jump deformations. The main purpose of this paper is to give
a logically organized description of the moduli space, and to give an explicit description of how the modulis paceis
constructed by extension. |
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5. |
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On Graded Global Dimension of Color Hopf
Algebras
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By
Yan-Hua Wang
© 2011. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 5
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Abstract
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In this paper, we prove the fundamental the orem of color Hopf modulesimilar to the fundamental the orem
of Hopf module. As an application, we prove that the graded global dimension of a color Hopf algebra coincides
with the projective dimension of the trivial module K.
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6. |
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Some Properties of Intuitionistic Fuzzy Lie
Algebras over a Fuzzy Field
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By
P. L. Antony and P. L. Lilly © 2011. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 5
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1-6 Page(s)
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Abstract
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The concept of intuitionistic fuzzy Lie algebra over a fuzzy field is introduced. We study the "necessity"
and "possibility" operators on intuitionistic fuzzy Lie algebra over a fuzzy field and give some properties of
homomorphic images. |
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7. |
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An Approachto Omni-Lie Algebroids using
Quasi-Derivations
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By
Dennise Garc © 2011. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 5
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Abstract
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We introduce the notion of left (and right) quasi-Loday algebroids and a "universal space" for them,
called a left (right) omni-Loday algebroid, in such a way that Lie algebroids, omni-Lie algebras and omni-Loday
algebroids are particular substructures.
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8. |
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Algebraic Structures Derived from Foams
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By
J.Scott Carter and Masahico Saito
© 2011. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 5
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1-9 Page(s)
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Abstract
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Foams are surfaces with branch lines at which three sheets merge. They have been used in the categorification
of sl(3) quantum knot invariants and also in physics. The 2D-TQFT of surfaces, on the other hand,
is classified by means of commutative Frobenius algebras, where saddle points correspond to multiplication and
comultiplication.In this paper,we explore algebraic operations that branch lines derive under TQFT.Inparticular,we
investigate Lie bracket and bialgebra structures. Relations to the original Frobenius algebra structures are discussed
both algebraically and diagrammatically.
MSC 2010: 57M25, 16T10, 17B37, 81R50 |
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9. |
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Abstract
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Generalized polar decomposition method (or briefly GPD method) has been introduced by Munthe-Kaas
and Zanna [5] to approximate the matrix exponential. In this paper, we investigate the numerical stability of that
method with respect to round off propagation. The numerical GPD method includes two parts: splitting of a matrix
Z ∈ g, a Lie algebra of matrices and computing exp(Z)v for a vector v.We show that the former is stable provided
that (Z) is not so large, while the latter is not stable in general except with some restrictions on the entries of the
matrix Z and the vector v.
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10. |
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C(α,β)-fuzzy Lie Algebras Over an
(α,β)-fuzzy Field
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By
P. L. Antony a and p. L. Lilly © 2010. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 4
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Table of Contents
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1-8 Page(s)
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Abstract
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The concept of (α,β)-fuzzy Lie algebras over an (α,β)-fuzzy field is introduced. We
provide characterizations of an (∈,∈ V q)-fuzzy Lie algebra over an (∈,∈ V q)-fuzzy field.
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11. |
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Double Cross Bi-product and Bi-cycle
Bicrossproduct Lie Bi-algebras
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By
Tao Zhang
© 2010. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 4
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1-16 Page(s)
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Abstract
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We construct double cross biproduct and bi-cycle bicrossproduct Lie bialgebras from
braided Lie bialgebras. The main results generalize Majid's matched pair of Lie algebras
and Drinfeld's quantum double and Masuoka's cross product Lie bialgebras.
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12. |
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Classications of Some Classes of Zinbiel Algebras
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By
J. Q. Adashev, A. Kh. Khudoyberdiyev, and B. A. Omirov © 2010. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 4
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Abstract
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In this work, the nul-filiform and filiform Zinbiel algebras are described up to isomorphism.
Moreover, the classification of complex Zinbiel algebras dimensions
≤ 3 is
extended up to dimension 4. |
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13. |
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On Hom-type Algebras
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By
Yael Fregier and Aron Gohr © 2010. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 4
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Abstract
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Hom-algebras are generalizations of algebras obtained using a twisting by a linear
map.But there is a priori a freedom on where to twist.We enumerate here all the possible
choices in the Lie and associative types and study the relations between the obtained
algebras. The associative case is richer since it admits the notion of unit element. We use
this fact to find sufficient conditions for Hom-associative algebras to be associative and
classify the implications between the Hom-associative types of unital algebras. |
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14. |
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Cheban Loops
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By
J. D. Phillips and V. A. Shcherbacov © 2010. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 4
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Abstract
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Left Cheban loops are loops that satisfy the identity x(xy ·z)=yx ·xz. Right
Cheban loops satisfy the mirror identity (z ·yx)x =zx·xy. Loops that are both left
and right Cheban are called Cheban loops. Cheban loops can also be characterized as
those loops that satisfy the identityx(xy·z)=(y·zx)x. These loops were introduced by
A. M. Cheban. Here we initiate a study of their structural properties. Left Cheban loops
are left conjugacy closed. Cheban loops are weak inverse property, power associative,
conjugacy closed loops; they are centrally nilpotent of class at most two.
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15. |
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Canonical Endomorphism Field on a Lie Algebra
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By
Jerzy Kocik © 2010. Ashdin Publishing
Journal of Generalized Lie Theory and Applications (JGLTA), Volume 4
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Abstract
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We show that every Lie algebra is equipped with a natural (1,1)-variant tensor field,
the "canonical endomorphism field", determined by the Lie structure, and satisfying a
certain Nijenhuis bracket condition. This observation may be considered as complementary to the Kirillov-Kostant-Souriau theorem on symplectic geometry of coadjoint orbits.
We show its relevance for classical mechanics, in particular for Lax equations. We show
that the space of Lax vector fields is closed under Lie bracket and we introduce a new
bracket for vector fields on a Lie algebra. This bracket defines a new Lie structure on the
space of vector fields. |
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