Search results for "Codimension"
showing 10 items of 112 documents
Bifurcations of Regular Limit Periodic Sets
1998
In this chapter, (X λ ) will be a smooth or analytic (in Section 3) family of vector fields on a phase space S, with parameter λ ∈ P, as in Chapter 1. Periodic orbits and elliptic singular points which are limits of sequences of limit cycles are called regular limit periodic sets. The reason for this terminology is that for such a limit periodic set Γ one can define local return maps on transversal segments, which are as smooth as the family itself. The limit cycles near Γ will be given by a smooth equation and the theory of bifurcations of limit cycles from Γ will reduce to the theory of unfoldings of differentiable functions. In fact, we will just need the Preparation Theorem and not the …
Algebras with intermediate growth of the codimensions
2006
AbstractLet F be a field of characteristic zero and let A be an F-algebra. The polynomial identities satisfied by A can be measured through the asymptotic behavior of the sequence of codimensions and the sequence of colengths of A. For finite dimensional algebras we show that the colength sequence of A is polynomially bounded and the codimension sequence cannot have intermediate growth. We then prove that for general nonassociative algebras intermediate growth of the codimensions is allowed. In fact, for any real number 0<β<1, we construct an algebra A whose sequence of codimensions grows like nnβ.
Asymptotics for Capelli polynomials with involution
2021
Let F be the free associative algebra with involution ∗ over a field F of characteristic zero. We study the asymptotic behavior of the sequence of ∗- codimensions of the T-∗-ideal Γ∗ M+1,L+1 of F generated by the ∗-Capelli polynomials Cap∗ M+1[Y, X] and Cap∗ L+1[Z, X] alternanting on M + 1 symmetric variables and L + 1 skew variables, respectively. It is well known that, if F is an algebraic closed field of characteristic zero, every finite dimensional ∗-simple algebra is isomorphic to one of the following algebras: · (Mk(F ), t) the algebra of k × k matrices with the transpose involution; · (M2m(F ), s) the algebra of 2m × 2m matrices with the symplectic involution; · (Mh(F ) ⊕ Mh(F )op, e…
Central polynomials of graded algebras: Capturing their exponential growth
2022
Let G be a finite abelian group and let A be an associative G-graded algebra over a field of characteristic zero. A central G-polynomial is a polynomial of the free associative G-graded algebra that takes central values for all graded substitutions of homogeneous elements of A. We prove the existence and the integrability of two limits called the central G-exponent and the proper central G-exponent that give a quantitative measure of the growth of the central G-polynomials and the proper central G-polynomials, respectively. Moreover, we compare them with the G-exponent of the algebra.
ON THE ASYMPTOTICS OF CAPELLI POLYNOMIALS
2021
Abstract. We present old and new results about Capelli polynomials, Z2-graded Capelli polynomials and Capelli polynomials with involution and their asymptotics. Let Capm = Pσ2Sm (sgnσ)tσ(1)x1tσ(2) · · · tσ(m−1)xm−1tσ(m) be the m-th Capelli polynomial of rank m. In the ordinary case (see [33]) it was proved the asymptotic equality between the codimensions of the T -ideal generated by the Capelli polynomial Capk2+1 and the codimensions of the matrix algebra Mk(F ). In [9] this result was extended to superalgebras proving that the Z2-graded codimensions of the T2-ideal generated by the Z2-graded Capelli polynomials Cap0 M+1 and Cap1 L+1 for some fixed M, L, are asymptotically equal to the Z2-g…
Trace Codimensions of Algebras and Their Exponential Growth
2022
The trace codimensions give a quantitative description of the identities satisfied by an algebra with trace. Here we study the asymptotic behaviour of the sequence of trace codimensions c tr n(A) and of pure trace codimensions c ptr n (A) of a finite-dimensional algebra A over a field of characteristic zero. We find an upper and lower bound of both codimensions and as a consequence we get that the limits limn→∞ctrn(A)√n and limn→∞cptrn(A) √n always exist and are integers. This result gives a positive answer to a conjecture of Amitsur in this setting. Finally we characterize the varieties of algebras whose exponential growth is bounded by 2
*-Graded Capelli polynomials and their asymptotics
2022
Let [Formula: see text] be the free *-superalgebra over a field [Formula: see text] of characteristic zero and let [Formula: see text] be the [Formula: see text]-ideal generated by the set of the *-graded Capelli polynomials [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] alternating on [Formula: see text] symmetric variables of homogeneous degree zero, on [Formula: see text] skew variables of homogeneous degree zero, on [Formula: see text] symmetric variables of homogeneous degree one and on [Formula: see text] skew variables of homogeneous degree one, respectively. We study the asymptotic behavior of the sequence of *-graded codimensions of [Formula: se…
On central polynomials and codimension growth
2022
Let A be an associative algebra over a field of characteristic zero. A central polynomial is a polynomial of the free associative algebra that takes central values of A. In this survey, we present some recent results about the exponential growth of the central codimension sequence and the proper central codimension sequence in the setting of algebras with involution and algebras graded by a finite group.
Polynomial codimension growth of graded algebras
2009
We study associative $G$-graded algebras with 1 of polynomial $G$-codimension growth, where $G$ is a finite group. For any fixed $k\geq 1,$ we construct associative $G$-graded algebras of upper triangular matrices whose $G$-codimension sequence is given asymptotically by a polynomial of degree $k$ whose leading coefficient is the largest or smallest possible.
Capelli identities on algebras with involution or graded involution
2022
We present recent results about Capelli polynomials with involution or graded involution and their asymptotics. In the associative case, the asymptotic equality between the codimensions of the T -ideal generated by the Capelli polynomial of rank k2 + 1 and the codimensions of the matrix algebra Mk(F) was proved. This result was extended to superalgebras. Recently, similar results have been determined by the authors in the case of algebras with involution and superalgebras with graded involution.