Search results for "Identity"
showing 10 items of 1751 documents
Polynomial identities with involution, superinvolutions and the Grassmann envelope
2017
Let A be an algebra with involution ∗ over a field of characteristic zero. We prove that in case A satisfies a non-trivial ∗-identity, then A has the same ∗-identities as the Grassmann envelope of a finite dimensional superalgebra with superinvolution. As a consequence we give a positive answer to the Specht problem for algebras with involution, i.e., any T-ideal of identities of an algebra with involution is finitely generated as a T-ideal.
Polynomial growth and identities of superalgebras and star-algebras
2009
Abstract We study associative algebras with 1 endowed with an automorphism or antiautomorphism φ of order 2, i.e., superalgebras and algebras with involution. For any fixed k ≥ 1 , we construct associative φ -algebras whose φ -codimension sequence is given asymptotically by a polynomial of degree k whose leading coefficient is the largest or smallest possible.
Asymptotics for the Amitsur's Capelli - Type Polynomials and Verbally Prime PI-Algebras
2006
We consider associativePI-algebras over a field of characteristic zero. The main goal of the paper is to prove that the codimensions of a verbally prime algebra [11] are asymptotically equal to the codimensions of theT-ideal generated by some Amitsur's Capelli-type polynomialsEM,L* [1]. We recall that two sequencesan,bnare asymptotically equal, and we writean≃bn,if and only if limn→∞(an/bn)=1.In this paper we prove that\(c_n \left( {M_k \left( G \right)} \right) \simeq c_n \left( {E_{k^2 ,k^2 }^ * } \right) and c_n \left( {M_{k,l} \left( G \right)} \right) \simeq c_n \left( {E_{k^2 + l^2 ,2kl}^ * } \right) \)% MathType!End!2!1!, whereG is the Grassmann algebra. These results extend to all v…
PI-algebras with slow codimension growth
2005
Let $c_n(A),\ n=1,2,\ldots,$ be the sequence of codimensions of an algebra $A$ over a field $F$ of characteristic zero. We classify the algebras $A$ (up to PI-equivalence) in case this sequence is bounded by a linear function. We also show that this property is closely related to the following: if $l_n(A), \ n=1,2,\ldots, $ denotes the sequence of colengths of $A$, counting the number of $S_n$-irreducibles appearing in the $n$-th cocharacter of $A$, then $\lim_{n\to \infty} l_n(A)$ exists and is bounded by $2$.
Entire Functions of Bounded Type on Fréchet Spaces
1993
We show that holomorphic mappings of bounded type defined on Frechet spaces extend to the bidual. The relationship between holomorphic mappings of bounded type and of uniformly bounded type is discussed and some algebraic and topological properties of the space of all entire mappings of (uniformly) bounded type are proved, for example a holomorphic version of Schauder's theorem.
On the equivalence of McShane and Pettis integrability in non-separable Banach spaces
2009
Abstract We show that McShane and Pettis integrability coincide for functions f : [ 0 , 1 ] → L 1 ( μ ) , where μ is any finite measure. On the other hand, assuming the Continuum Hypothesis, we prove that there exist a weakly Lindelof determined Banach space X, a scalarly null (hence Pettis integrable) function h : [ 0 , 1 ] → X and an absolutely summing operator u from X to another Banach space Y such that the composition u ○ h : [ 0 , 1 ] → Y is not Bochner integrable; in particular, h is not McShane integrable.
A Vector Approach to Euler's Line of a Triangle
1992
Among the many interesting properties that triangles possess there is one that quickly attracts our curiosity and stays easily in our mind: The centroid, circumcentre and orthocentre all lie in a common line (Euler's Line). An elementary simple proof can be obtained using metric and affine properties of the points involved, [1]. Our aim here is to illustrate a proof using vectors. We identify points in the plane with their position vectors. It is easy to see that the centroid G of the triangle ABC is given by the identity
Polynomial codimension growth and the Specht problem
2017
Abstract We construct a continuous family of algebras over a field of characteristic zero with slow codimension growth bounded by a polynomial of degree 4. This is achieved by building, for any real number α ∈ ( 0 , 1 ) a commutative nonassociative algebra A α whose codimension sequence c n ( A α ) , n = 1 , 2 , … , is polynomially bounded and lim log n c n ( A α ) = 3 + α . As an application we are able to construct a new example of a variety with an infinite basis of identities.
Standard polynomials are characterized by their degree and exponent
2011
Abstract By the Giambruno–Zaicev theorem (Giambruno and Zaicev, 1999) [5] , the exponent exp ( A ) of a p.i. algebra A exists, and is always an integer. In Berele and Regev (2001) [2] it was shown that the exponent exp ( St n ) of the standard polynomial St n of degree n is not smaller than the exponent of any polynomial of degree n. Here it is proved that exp ( St n ) is strictly larger than the exponent of any other polynomial of degree n which is not a multiple of St n .
Polynomial identities on superalgebras: Classifying linear growth
2006
Abstract We classify, up to PI-equivalence, the superalgebras over a field of characteristic zero whose sequence of codimensions is linearly bounded. As a consequence we determine the linear functions describing the graded codimensions of a superalgebra.