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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics.

6787 Results for the subject "Algebra":

On AFS algebra––Part II

AbstractThis study is a sequel of Part I of this paper [On AFS Algebra – Part I, submitted to Information Sciences]. In this paper, we will further study the properties and structures of sub-algebra of EI algebra and EIn algebra.

Yunjie Zhang Publication date: 2004/12/02
The Tetrahedron algebra, the Onsager algebra, and the sl2 loop algebra

AbstractLet K denote a field with characteristic 0 and let T denote an indeterminate. We give a presentation for the three-point loop algebra sl2⊗K[T,T−1,(T−1)−1] via generators and relations. This presentation displays S4-symmetry. Using this presentation we obtain a decomposition of the above loop algebra into a direct sum of three subalgebras, each of which is isomorphic to the Onsager algebra.

Brian Hartwig Publication date: 2007/02/15
Regular ArticleNilpotency Property in Table Algebras☆

AbstractTable algebras form an important class of C-algebras. The dual of a table algebra may not be a table algebra, but just a C-algebra. It is not known under what conditions the dual of a table algebra is also a table algebra. In this paper we prove that if a table algebra has nilpotency property then its dual also is a table algebra. Finally we conclude that if (A, B) is a nilpotent table algebra then its dual also is a nilpotent table algebra.

A.Rahnamai Barghi Publication date: 2000/04/15
Generalized Koszul properties for augmented algebras

AbstractUnder certain conditions, a filtration on an augmented algebra A admits a related filtration on the Yoneda algebra E(A):=ExtA(K,K). We show that there exists a bigraded algebra monomorphism grE(A)↪EGr(grA), where EGr(grA) is the graded Yoneda algebra of grA. This monomorphism can be applied in the case where A is connected graded to determine that A has the K2 property recently introduced by Cassidy and Shelton.

Christopher Phan Publication date: 2009/03/01
The W3(2) algebra and the related hierarchies

AbstractWe show that there exists a scalar Lax pair representation for several integrable differential hierarchies which are associated to the W3(2) algebra. These integrable hierarchies are linked through Miura maps. For each of them, the second Hamiltonian structure gives a realization of the W3(2) algebra. In particular, we show that the W3(2) algebra can be constructed from the sl(2) Kac-Moody current algebra with level k=−1. Furthermore, we also give a new type of free field representations of the W3(2) algebra and the quantized W3(2) algebra.

Q.P. Liu Publication date: 1994/05/19
Left-symmetric algebra structures on the W-algebra W(2, 2)

AbstractWe study the compatible left-symmetric algebra structures on the W-algebra W(2, 2) with some natural grading conditions. The results of earlier work on left-symmetric algebra structures on the Virasoro algebra play an essential role in determining these compatible structures. As a corollary, any such left-symmetric algebra contains an infinite-dimensional trivial subalgebra that is also a submodule of the regular module.

Hongjia Chen Publication date: 2012/10/01
On vertex Leibniz algebras

AbstractIn this paper, we study a notion of what we call vertex Leibniz algebra. This notion naturally extends that of vertex algebra without vacuum, which was previously introduced by Huang and Lepowsky. We show that every vertex algebra without vacuum can be naturally extended to a vertex algebra. On the other hand, we show that a vertex Leibniz algebra can be embedded into a vertex algebra if and only if it admits a faithful module. To each vertex Leibniz algebra we associate a vertex algebra without vacuum which is universal to the forgetful functor. Furthermore, from any Leibniz algebra g we construct a vertex Leibniz algebra Vg and show that Vg can be embedded into a vertex algebra if and only if g is a Lie algebra.

Haisheng Li Publication date: 2013/12/01
Review articleMinimax algebra and applications

AbstractWe consider theories of linear and of polynomial algebra, over two scalar systems, often called max-algebra and min-algebra. Here, max-algebra is the system M = (R ∪ {−∞}, ⊛, ⊗) where x ⊛ y = max(x, y) and x ⊛ y = x + y. Min-algebra is the dual system M′ = R ∪ {+∞}, ⊛, ⊗′ with x ⊛′ y = min(x, y) and x ⊗′ y = x + y. Towards the end we also consider minimax algebra, the system M″ = (R ∪ {−∞, +∞}, ⊛, ⊗, ⊛′, ⊗′). Application fields discussed include location problems, machine scheduling, cutting and packing problems, discrete-event systems and path-finding problems.

R.A. Cuninghame-Green Publication date: 1991/06/14
The corona algebra of the stabilized Jiang–Su algebra

AbstractLet Z be the Jiang–Su algebra and K the C⁎-algebra of compact operators on an infinite dimensional separable Hilbert space. We prove that the corona algebra M(Z⊗K)/Z⊗K has real rank zero. We actually prove a more general result.

Huaxin Lin Publication date: 2016/02/01
Gradings on walled Brauer algebras and Khovanov’s arc algebra

AbstractWe introduce some Z-graded versions of the walled Brauer algebra Br,s(δ), working over a field of characteristic zero. This allows us to prove that Br,s(δ) is Morita equivalent to an idempotent truncation of a certain infinite dimensional version of Khovanov’s arc algebra. We deduce that the walled Brauer algebra is Koszul whenever δ≠0.

Jonathan Brundan Publication date: 2012/10/01
Symplectic structures on quadratic Lie algebras

AbstractWe study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T∗-extension of a nilpotent Lie algebra admitting an invertible derivation and also as the double extension of another quadratic symplectic Lie algebra by the one-dimensional Lie algebra. Finally, we prove that every symplectic quadratic Lie algebra is a special symplectic Manin algebra and we give an inductive description in terms of symplectic quadratic double extensions.

Ignacio Bajo Publication date: 2007/10/01
Gradings on the Lie algebra D4 revisited

AbstractWe classify group gradings on the simple Lie algebra L of type D4 over an algebraically closed field of characteristic different from 2: fine gradings up to equivalence and G-gradings, with a fixed group G, up to isomorphism. For each G-grading on L, we also study graded L-modules (assuming characteristic 0).

Alberto Elduque Publication date: 2015/11/01
12 - Algebra of logic (Boolean algebra)

Publisher SummaryThis chapter provides an overview of algebra of logic. The basis of this algebra is the Boolean algebra, which was created as a calculus (calculus of logic) for the description of logical relations that mathematically symbolizes the statements as true or false. Its development into a two-value Boolean algebra as switching algebra is based on the fact that also in digital techniques with storage-free binary elements only the states “on” and “off” are possible: (1) x = L when x ≠ 0 and (2) x = 0 when x ≠ L. The chapter discusses basic logical interconnections.

HANS-JOCHEN BARTSCH Publication date: 1974/01/01
W3-algebra constructed from the SU(3) parafermion

AbstractA construction of a W3-algebra for the SU(3) parafermion is proposed. The details of the calculation are given, in which the Z-algebra technique is used instead of the popular free field realization. We find that the W3-algebra is closed at level k = 3, and the central charge of the underlying algebra is different from known series of Fateev-Lykyanov W-algebras; as a by-product we get a field T(4)(z), whose conformal dimension is 4, and is null at k = 3.

Xiang-Mao Ding Publication date: 1994/07/04
Regular ArticleAlgèbres de Lie kählériennes et double extension

RésuméA Kähler Lie algebra is a real Lie algebra carrying a symplectic 2-cocycle ω and an integrable complex structurejsuch that ω(x, j(y)) is a scalar product. We give a process, called Kähler double extension, which realizes a Kähler Lie algebra as the Kähler reduction of another one. We show that every Kähler algebra is obtained by a sequence of such a process from {0} or a flat Kähler algebra; it is obtained from {0} iff it contained a lagrangian sub-algebra. These methods allow us to prove that any completely solvable and unimodular Kähler algebra is commutative.

Jean-Michel Dardié Publication date: 1996/11/01
Derivations of Leavitt path algebras

AbstractIn this paper, we describe the K-module HH1(LK(Γ)) of outer derivations of the Leavitt path algebra LK(Γ) of a row-finite graph Γ with coefficients in an associative commutative ring K with unit. We explicitly describe a set of generators of HH1(LK(Γ)) and relations among them. We also describe a Lie algebra structure of outer derivation algebra of the Toeplitz algebra. We prove that every derivation of a Leavitt path algebra can be extended to a derivation of the corresponding C⁎-algebra.

Viktor Lopatkin Publication date: 2019/02/15
Vertex algebras generated by Lie algebras

AbstractIn this paper we introduce a notion of vertex Lie algebra U, in a way a “half” of vertex algebra structure sufficient to construct the corresponding local Lie algebra L(U) and a vertex algebra L(U). We show that we may consider U as a subset U ⊂ V (U) which generates V(U) and that the vertex Lie algebra structure on U is induced by the vertex algebra structure on V(U). Moreover, for any vertex algebra V a given homomorphism U → V of vertex Lie algebras extends uniquely to a homomorphism V(U) → V of vertex algebras. In the second part of paper we study under what conditions on structure constants one can construct a vertex Lie algebra U by starting with a given commutator formula for fields.

Mirko Primc Publication date: 1999/02/26
Infinite-Dimensional Prolongation Structures for the Robinson–Trautman Type III Metric

The universal covering symmetry algebra of the Robinson–Trautman equations of Petrov Type III is shown to include the infinite-dimensional Lie algebra A2⊕C[λ−1, λ], the loop algebra over A2. This algebra has slower growth than the contragradient algebra K2 obtained previously for this metric by other authors.

E.O. Ifidon Publication date: 2013/06/01
A-generators for ideals in the Dickson algebra

AbstractThe Dickson Algebra on q-variables is the algebra of invariants of the action of the mod-2 general linear group on a polynomial algebra in q-variables. We study the structure of certain ideals in this algebra as a module over the Steenrod Algebra A, and develop methods to determine which elements are hit by Steenrod operations. This allows us to display a very small set of A-generators for these ideals and show that the set is minimal in some cases.

V. Giambalvo Publication date: 2001/04/24
Free Malcev algebra of rank three

AbstractWe find a basis for the free Malcev algebra on three free generators over a field of characteristic zero. The semiprimity and speciality of this algebra are proved. Also, we prove the decomposability of this algebra into a subdirect sum of the free Lie algebra of rank three and the free algebra of rank three of the variety generated by a simple seven-dimensional Malcev algebra. These results were announced in [1].

Alexandr I. Kornev Publication date: 2014/03/01
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