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A Characterization of Varieties of Associative Algebras of Exponent two

Giambruno, A., Zaicev, M. (2000)

Serdica Mathematical Journal

∗The first author was partially supported by MURST of Italy; the second author was par- tially supported by RFFI grant 99-01-00233.It was recently proved that any variety of associative algebras over a field of characteristic zero has an integral exponential growth. It is known that a variety V has polynomial growth if and only if V does not contain the Grassmann algebra and the algebra of 2 × 2 upper triangular matrices. It follows that any variety with overpolynomial growth has exponent at least...

Centers in domains with quadratic growth

Agata Smoktunowicz (2005)

Open Mathematics

Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.

Differential smoothness of affine Hopf algebras of Gelfand-Kirillov dimension two

Tomasz Brzeziński (2015)

Colloquium Mathematicae

Two-dimensional integrable differential calculi for classes of Ore extensions of the polynomial ring and the Laurent polynomial ring in one variable are constructed. Thus it is concluded that all affine pointed Hopf domains of Gelfand-Kirillov dimension two which are not polynomial identity rings are differentially smooth.

Involution Matrix Algebras – Identities and Growth

Rashkova, Tsetska (2004)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 16R50, 16R10.The paper is a survey on involutions (anti-automorphisms of order two) of different kinds. Starting with the first systematic investigations on involutions of central simple algebras due to Albert the author emphasizes on their basic properties, the conditions on their existence and their correspondence with structural characteristics of the algebras. Focusing on matrix algebras a complete description of involutions of the first kind on Mn(F)...

Some examples of nil Lie algebras

Ivan P. Shestakov, Efim Zelmanov (2008)

Journal of the European Mathematical Society

Generalizing Petrogradsky’s construction, we give examples of infinite-dimensional nil Lie algebras of finite Gelfand–Kirillov dimension over any field of positive characteristic.

The G -graded identities of the Grassmann Algebra

Lucio Centrone (2016)

Archivum Mathematicum

Let G be a finite abelian group with identity element 1 G and L = g G L g be an infinite dimensional G -homogeneous vector space over a field of characteristic 0 . Let E = E ( L ) be the Grassmann algebra generated by L . It follows that E is a G -graded algebra. Let | G | be odd, then we prove that in order to describe any ideal of G -graded identities of E it is sufficient to deal with G ' -grading, where | G ' | | G | , dim F L 1 G ' = and dim F L g ' < if g ' 1 G ' . In the same spirit of the case | G | odd, if | G | is even it is sufficient to study only those G -gradings such that...

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