Search results for "Division algebra"
showing 10 items of 18 documents
Group graded algebras and almost polynomial growth
2011
Let F be a field of characteristic 0, G a finite abelian group and A a G-graded algebra. We prove that A generates a variety of G-graded algebras of almost polynomial growth if and only if A has the same graded identities as one of the following algebras: (1) FCp, the group algebra of a cyclic group of order p, where p is a prime number and p||G|; (2) UT2G(F), the algebra of 2×2 upper triangular matrices over F endowed with an elementary G-grading; (3) E, the infinite dimensional Grassmann algebra with trivial G-grading; (4) in case 2||G|, EZ2, the Grassmann algebra with canonical Z2-grading.
Banach elements and spectrum in Banach quasi *-algebras
2006
A normal Banach quasi -algebra (X;A_0) has a distinguished Banach - algebra X_b consisting of bounded elements of X. The latter -algebra is shown to coincide with the set of elements of X having fi nite spectral radius. If the family P(X) of bounded invariant positive sesquilinear forms on X contains suffi ciently many elements then the Banach -algebra of bounded elements can be characterized via a C -seminorm defi ned by the elements of P(X).
The new results on lattice deformation of current algebra
2008
The topic “Quantum Integrable Models” was reviewed in the literature and presented to the conferences and schools many times. Only the reports of our own have been done on quite a few occasions (see, e.g., [1], [2]). So here we shall try to present a fresh approach to the description of the ingredients of construction of integrable models. It has gradually evolved in the process of our joint work. Whereas our goal was the Sugawara construction for the lattice affine algebra (known now as the St.Petersburg algebra), (see, e.g., [1]), some technical developments happen to be new and useful for the already developed subjects. Here we shall underline this development.
Generalised Deformations, Koszul Resolutions, Moyal Products
1998
We generalise Gerstenhaber's theory of deformations, by dropping the assumption that the deformation parameter should commute with the elements of the original algebra. We give the associated cohomology and construct a Koszul resolution for the polynomial algebra [Formula: see text] in the "homogeneous" case. We then develop examples in the case of [Formula: see text] and find some Moyal-like products of a new type. Finally, we show that, for any field K, matrix algebras with coefficients in K and finite degree extensions of K are rigid, as in the commutative case.
Sturmian words and overexponential codimension growth
2018
Abstract Let A be a non necessarily associative algebra over a field of characteristic zero satisfying a non-trivial polynomial identity. If A is a finite dimensional algebra or an associative algebra, it is known that the sequence c n ( A ) , n = 1 , 2 , … , of codimensions of A is exponentially bounded. If A is an infinite dimensional non associative algebra such sequence can have overexponential growth. Such phenomenon is present also in the case of Lie or Jordan algebras. In all known examples the smallest overexponential growth of c n ( A ) is ( n ! ) 1 2 . Here we construct a family of algebras whose codimension sequence grows like ( n ! ) α , for any real number α with 0 α 1 .
Group-graded algebras with polynomial identity
1998
LetG be a finite group and letR=Σg∈GRg be any associative algebra over a field such that the subspacesRg satisfyRgRh⊆Rgh. We prove that ifR1 satisfies a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the order ofG. This result implies the following: ifH is a finite-dimensional semisimple commutative Hopfalgebra andR is anyH-module algebra withRH satisfying a PI of degreed, thenR satisfies a PI of degree bounded by an explicit function ofd and the dimension ofH.
Divisible designs from semifield planes
2002
AbstractWe give a general method to construct divisible designs from semifield planes and we use this technique to construct some divisible designs. In particular, we give the case of twisted field plane as an example.
Some geometric properties of disk algebras
2014
Abstract In this paper we study some geometrical properties of certain classes of uniform algebras, in particular the ball algebra A u ( B X ) of all uniformly continuous functions on the closed unit ball and holomorphic on the open unit ball of a complex Banach space X . We prove that A u ( B X ) has k -numerical index 1 for every k , the lushness and also the AHSP. Moreover, the disk algebra A ( D ) , and more in general any uniform algebra whose Choquet boundary has no isolated points, is proved to have the polynomial Daugavet property. Most of those properties are extended to the vector valued version A X of a uniform algebra A .
Algebras with involution with linear codimension growth
2006
AbstractWe study the ∗-varieties of associative algebras with involution over a field of characteristic zero which are generated by a finite-dimensional algebra. In this setting we give a list of algebras classifying all such ∗-varieties whose sequence of ∗-codimensions is linearly bounded. Moreover, we exhibit a finite list of algebras to be excluded from the ∗-varieties with such property. As a consequence, we find all possible linearly bounded ∗-codimension sequences.
Finite-dimensional non-associative algebras and codimension growth
2011
AbstractLet A be a (non-necessarily associative) finite-dimensional algebra over a field of characteristic zero. A quantitative estimate of the polynomial identities satisfied by A is achieved through the study of the asymptotics of the sequence of codimensions of A. It is well known that for such an algebra this sequence is exponentially bounded.Here we capture the exponential rate of growth of the sequence of codimensions for several classes of algebras including simple algebras with a special non-degenerate form, finite-dimensional Jordan or alternative algebras and many more. In all cases such rate of growth is integer and is explicitly related to the dimension of a subalgebra of A. One…