Search results for "Commutative algebra"
showing 10 items of 127 documents
Varieties of algebras with pseudoinvolution: Codimensions, cocharacters and colengths
2022
Abstract Let A be a finitely generated superalgebra with pseudoinvolution ⁎ over an algebraically closed field F of characteristic zero. In this paper we develop a theory of polynomial identities for this kind of algebras . In particular, we shall consider three sequences that can be attached to Id ⁎ ( A ) , the T 2 ⁎ -ideal of identities of A: the sequence of ⁎-codimensions c n ⁎ ( A ) , the sequence of ⁎-cocharacter χ 〈 n 〉 ⁎ ( A ) and the ⁎-colength sequence l n ⁎ ( A ) . Our purpose is threefold. First we shall prove that the ⁎-codimension sequence is eventually non-decreasing, i.e., c n ⁎ ( A ) ≤ c n + 1 ⁎ ( A ) , for n large enough. Secondly, we study superalgebras with pseudoinvoluti…
Extending the star order to Rickart rings
2015
Star partial order was initially introduced for semigroups and rings with (proper) involution. In particular, this order has recently been studied on Rickart *-rings. It is known that the star order in such rings can be characterized by conditions not involving involution explicitly. Owing to these characterizations, the order can be extended to certain special Rickart rings named strong in the paper; this extension is the objective of the paper. The corresponding order structure of strong Rickart rings is studied more thoroughly. In particular, the most significant lattice properties of star-ordered Rickart *-rings are successfully transferred to strong Rickart rings; also several new resu…
Bounded Bi-ideals and Linear Recurrence
2013
Bounded bi-ideals are a subclass of uniformly recurrent words. We introduce the notion of completely bounded bi-ideals by imposing a restriction on their generating base sequences. We prove that a bounded bi-ideal is linearly recurrent if and only if it is completely bounded.
POLYNOMIAL GROWTH OF THE*-CODIMENSIONS AND YOUNG DIAGRAMS
2001
Let A be an algebra with involution * over a field F of characteristic zero and Id(A, *) the ideal of the free algebra with involution of *-identities of A. By means of the representation theory of the hyperoctahedral group Z 2wrS n we give a characterization of Id(A, *) in case the sequence of its *-codimensions is polynomially bounded. We also exhibit an algebra G 2 with the following distinguished property: the sequence of *-codimensions of Id(G 2, *) is not polynomially bounded but the *-codimensions of any T-ideal U properly containing Id(G 2, *) are polynomially bounded.
Asymptotics for Graded Capelli Polynomials
2014
The finite dimensional simple superalgebras play an important role in the theory of PI-algebras in characteristic zero. The main goal of this paper is to characterize the T 2-ideal of graded identities of any such algebra by considering the growth of the corresponding supervariety. We consider the T 2-ideal Γ M+1,L+1 generated by the graded Capelli polynomials C a p M+1[Y,X] and C a p L+1[Z,X] alternanting on M+1 even variables and L+1 odd variables, respectively. We prove that the graded codimensions of a simple finite dimensional superalgebra are asymptotically equal to the graded codimensions of the T 2-ideal Γ M+1,L+1, for some fixed natural numbers M and L. In particular csupn(Γk2+l2+1…
Exponential Codimension Growth of PI Algebras: An Exact Estimate
1999
Abstract LetAbe an associative PI-algebra over a fieldFof characteristic zero. By studying the exponential behavior of the sequence of codimensions {cn(A)} ofA, we prove thatInv(A)=limn→∞ c n ( A ) always exists and is an integer. We also give an explicit way for computing such integer: letBbe a finite dimensionalZ2-graded algebra whose Grassmann envelopeG(B) satisfies the same identities ofA; thenInv(A)=Inv(G(B))=dim C(0)+dim C(1)whereC(0)+C(1)is a suitableZ2-graded semisimple subalgebra ofB.
Y-proper graded cocharacters and codimensions of upper triangular matrices of size 2, 3, 4
2012
Abstract Let F be a field of characteristic 0. We consider the upper triangular matrices with entries in F of size 2, 3 and 4 endowed with the grading induced by that of Vasilovsky. In this paper we give explicit computation for the multiplicities of the Y -proper graded cocharacters and codimensions of these algebras.
Mirror symmetry and toric degenerations of partial flag manifolds
1998
In this paper we propose and discuss a mirror construction for complete intersections in partial flag manifolds $F(n_1, ..., n_l, n)$. This construction includes our previous mirror construction for complete intersection in Grassmannians and the mirror construction of Givental for complete flag manifolds. The key idea of our construction is a degeneration of $F(n_1, ..., n_l, n)$ to a certain Gorenstein toric Fano variety $P(n_1, ..., n_l, n)$ which has been investigated by Gonciulea and Lakshmibai. We describe a natural small crepant desingularization of $P(n_1, ..., n_l, n)$ and prove a generalized version of a conjecture of Gonciulea and Lakshmibai on the singular locus of $P(n_1, ..., n…
The exterior derivative as a Killing vector field
1996
Among all the homogeneous Riemannian graded metrics on the algebra of differential forms, those for which the exterior derivative is a Killing graded vector field are characterized. It is shown that all of them are odd, and are naturally associated to an underlying smooth Riemannian metric. It is also shown that all of them are Ricci-flat in the graded sense, and have a graded Laplacian operator that annihilates the whole algebra of differential forms.
Graded polynomial identities and codimensions: Computing the exponential growth
2010
Abstract Let G be a finite abelian group and A a G-graded algebra over a field of characteristic zero. This paper is devoted to a quantitative study of the graded polynomial identities satisfied by A. We study the asymptotic behavior of c n G ( A ) , n = 1 , 2 , … , the sequence of graded codimensions of A and we prove that if A satisfies an ordinary polynomial identity, lim n → ∞ c n G ( A ) n exists and is an integer. We give an explicit way of computing such integer by proving that it equals the dimension of a suitable finite dimension semisimple G × Z 2 -graded algebra related to A.