61.
Quadri-algebras
- Aguiar, Marcelo; Loday, Jean-Louis
We introduce the notion of quadri-algebras. These are associative algebras
for which the multiplication can be decomposed as the sum of four operations in
a certain coherent manner. We present several examples of quadri-algebras: the
algebra of permutations, the shuffle algebra, tensor products of dendriform
algebras. We show that a pair of commuting Baxter operators on an associative
algebra gives rise to a canonical quadri-algebra structure on the underlying
space of the algebra. The main example is provided by the algebra End(A) of
linear endomorphisms of an infinitesimal bialgebra A. This algebra carries a
canonical pair of commuting Baxter operators: $\beta(T)=T\ast\id$ and
$\gamma(T)=\id\ast T$, where $\ast$ denotes the convolution...
62.
Homogeneous algebras
- Berger, Roland; Dubois-Violette, Michel; Wambst, Marc
Various concepts associated with quadratic algebras admit natural
generalizations when the quadratic algebras are replaced by graded algebras
which are finitely generated in degree 1 with homogeneous relations of degree
N. Such algebras are referred to as {\sl homogeneous algebras of degree N}.
In particular it is shown that the Koszul complexes of quadratic algebras
generalize as N-complexes for homogeneous algebras of degree N.
63.
Hopfish algebras
- Tang, Xiang; Weinstein, Alan; Zhu, Chenchang
We introduce a notion of "hopfish algebra" structure on an associative
algebra, allowing the structure morphisms (coproduct, counit, antipode) to be
bimodules rather than algebra homomorphisms. We prove that quasi-Hopf algebras
are examples of hopfish algebras. We find that a hopfish structure on the
commutative algebra of functions on a finite set G is closely related to a
"hypergroupoid" structure on G. The Morita theory of hopfish algebras is also
discussed.
64.
RALL: Machine-supported Proofs
- For Relation Algebra; David Von Oheimb; Thomas F. Gritzner
Wepresent a theorem proving system for abstract relation algebra called RALL (= Relation-Algebraic Language and Logic), based on the generic theorem prover Isabelle. On the one hand, the system is an advanced case study for Isabelle/HOL, and on the other hand, a quite mature proof assistant for research on the relational calculus. RALL is able to deal with the full language of heterogeneous relation algebra including higher-order operators and domain constructions, and checks the type-correctness of all formulas involved. It offers both an interactive proof facility, with special support for substitutions and estimations, and an experimental automatic prover. The automatic...
65.
Álgebras de Koszul e resoluções projetivas
- Francisco Batista de Medeiros
Neste trabalho estudamos algumas características das álgebras de Koszul, como por exemplo, a maneira como elas se relacionam com suas respectivas álgebras de Yoneda. Descrevemos a álgebra de Yoneda de uma álgebra monomial e como aplicação construímos uma família de álgebras: as chamadas homologicamente auto-duais. Uma álgebra de Koszul pode ser definida a partir da existência de resoluções lineares dos módulos simples. Por isso faz-se necessário a dedicação de parte de nossa atenção ao estudo destas resoluções. Além disso, achamos interessante estudar métodos para a construção de resoluções projetivas de módulos sobre quocientes de álgebras de caminhos. Para tal construção...
66.
Hopf Algebra -- from MathWorld
- Weisstein, Eric W.
Let a graded module A have a multiplication \phi and a co-multiplication \psi. Then if \phi and \psi have the unity of k as unity and \psi:(A,\phi)\to(A,\phi)\otimes(A,\phi) is an algebra homomorphism, then (A,\phi,\psi) is called a Hopf algebra.
67.
Involutive Algebra -- from MathWorld
- Weisstein, Eric W.
An involutive algebra is an algebra A together with a map a\mapsto a^* of A into A (a so-called involution), satisfying the following properties: 1. (a^*)^*=a. 2. (ab)^*=b^*a^*. 3. (\lambda a+b)^*=\bar \lambda a^*+b^*.
68.
Cayley Algebra -- from MathWorld
- Weisstein, Eric W.
The only nonassociative division algebra with real scalars. There is an 8-square identity corresponding to this algebra. The elements of a Cayley algebra are called Cayley numbers or octonions, and the multiplication table for any Cayley algebra over a field F with field characteristic p\not=2 may be taken as shown in the following table, where u_1, u_2, ..., u_8 are a bases over F and \mu_1, \mu_2, and \mu_3 are nonzero elements of F (Schafer 1996, pp. 5-6). ...
69.
Tame Algebra -- from MathWorld
- Weisstein, Eric W.
Let A denote an \mathbb{R}-algebra, so that A is a vector space over R and A\times A\to A (x,y)\mapsto x\cdot y, where x\cdot y is vector multiplication which is assumed to be bilinear. Now define Z\equiv\{x\in a: x\cdot y=0{\rm\ for\ some\ nonzero\ } y\in A\}, where 0\in Z. A is said to be tame if Z is a finite union of subspaces of A. A two-dimensional 0-associative algebra is tame, but a four-dimensional 4-associative algebra and a three-dimensional 1-associative algebra need not be...
70.
Umbral Algebra -- from MathWorld
- Weisstein, Eric W.
The algebra structure of linear functionals on polynomials of a single variable (Roman 1984, pp. 2-3). See also: Umbral Calculus
71.
Universal Algebra -- from MathWorld
- Weisstein, Eric W.
Universal algebra studies common properties of all algebraic structures, including groups, rings, fields, lattices, etc. A universal algebra is a pair \mathbb{A}=\left({A,(f_i^{\mathbb{A}})_{i\in I}}\right), where A and I are sets and for each i\in I, f_i^{\mathbb{A}} is an operation on A. The algebra \mathbb{A} is finitary if each of its operations is finitary. A set of function symbols (or operations) of degree n \geq 0 is called a signature (or type). Let \Sigma be a signature. An algebra A...
72.
Robbins Algebra -- from MathWorld
- Weisstein, Eric W.
Building on work of Huntington (1933), Robbins conjectured that the equations for a Robbins algebra, commutativity, associativity, and the Robbins axiom !(!(x\lor y)\lor !(x\lor !y)) = x, where !x denotes NOT and x\lor y denotes OR, imply those for a Boolean algebra. The conjecture was finally proven using a computer (McCune 1997). See also: Boolean Algebra, Huntington Axiom, Robbins Conjecture, Robbins Axiom, Winkler Conditions
73.
Graded Algebra -- from MathWorld
- Weisstein, Eric W.
If A is a graded module and there exists a degree-preserving linear map \phi:A\otimes A\to A, then (A,\phi) is called a graded algebra. Cohomology is a graded algebra. In addition, the grading set is monoid having a compatibility relation such that if A is in the a grading of the algebra M, and B is in the b grading of the algebra M, then AB is in the ab grading of the algebra (where A and B are multiplied in M, and a and b are multiplied in the index monoid). For example, cohomology of a...
74.
Triangular Algebra -- from MathWorld
- Weisstein, Eric W.
Suppose that A and B are two algebras and M is a unital A-B-bimodule. Then \left[{\matrix{A & M\cr 0 & B\cr}}\right] = \left\{{\left[{\... ...ix{a & m\cr 0 & b\cr}}\right]: a\in A, m\in M, b\in B}\right\} with the usual 2\times 2 matrix-like addition and matrix-like multiplication is an algebra. An algebra \mathcal{T} is called a triangular algebra if there exist algebras A and B and an A-B-bimodule M such that \mathcal{T} is (algebraically) isomorphic to \left[{\matrix{A &...
75.
On Griess algebras
- Roitman, Michael
In this paper we prove that for any commutative (but in general
non-associative) algebra A with an invariant symmetric bilinear form there is
an OZ vertex algebra V = k 1 + V_2 + V_3 + ... (k is a ground field of
characteristic 0), such that the Griess algebra V_2 of V contains A. This shows
that the are no Griess identities, i.e. for any non-trivial commutative algebra
identity there is a Griess algebra in which this identity does not hold.
76.
Baxter Algebras and Differential Algebras
- Guo, Li
A Baxter algebra is a commutative algebra $A$ that carries a generalized
integral operator. In the first part of this paper we review past work of
Baxter, Miller, Rota and Cartier in this area and explain more recent work on
explicit constructions of free Baxter algebras that extended the constructions
of Rota and Cartier. In the second part of the paper we will use these explicit
constructions to relate Baxter algebras to Hopf algebras and give applications
of Baxter algebras to the umbral calculus in combinatorics.
77.
Host Algebras
- Grundling, Hendrik
A host algebra generalises the concept of a group algebra as follows. Let F
be a unital C*-algebra, and let S_0 be a proper subset of its states within
which one wants to keep the analysis (e.g. F is the group algebra of a discrete
group G, and S_0 is the set of states continuous w.r.t. some nondiscrete
topology of G). Then a host algebra is a C*-algebra L for which we have
embeddings of F and L into a larger C*-algebra E, such that the states on L
extend uniquely to F, and this extension defines a norm continuous affine
bijection between S_0 and the whole...
78.
OPE-Algebras
- Rosellen, Markus
In hep-th/0010293 Kapustin and Orlov introduce the notion of an OPE-algebra
and propose that it formalizes conformal field theories in the same way as
vertex algebras formalize chiral algebras, i.e. the subalgebras of holomorphic
fields of conformal field theories. In this thesis we study the question which
concepts and results of the general theory of vertex algebras can be extended
to OPE-algebras.
79.
Characterisations of Nelson algebras
- Spinks,M.; Veroff,R.
Nelson algebras arise naturally in algebraic logic as the algebraic models of Nelson's constructive logic with strong negation. This note gives two characterisations of the variety of Nelson algebras up to term equivalence, together with a characterisation of the finite Nelson algebras up to polynomial equivalence. The results answer a question of Blok and Pigozzi and clarify some earlier work of Brignole and Monteiro.
80.
Characterisations of Nelson algebras
- Spinks,M.; Veroff,R.
Nelson algebras arise naturally in algebraic logic as the algebraic models of Nelson's constructive logic with strong negation. This note gives two characterisations of the variety of Nelson algebras up to term equivalence, together with a characterisation of the finite Nelson algebras up to polynomial equivalence. The results answer a question of Blok and Pigozzi and clarify some earlier work of Brignole and Monteiro.
