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Eli Aljadeff

Publications and source records attributed to Eli Aljadeff.

At least 19 recordsLinked to original sources
arXiv2022

On generic $G$-graded Azumaya algebras

Let $F$ be an algebraically closed field of characteristic zero and let $G$ be a finite group. Consider $G$-graded simple algebras $A$ which are finite dimensional and $e$-central over $F$, i.e. $Z(A)_{e} := Z(A)\cap A_{e} = F$. For any such algebra we construct a \textit{generic} $G$-graded algebra $\mathcal{U}$ which is \textit{Azumaya} in the following sense. $(1)$ \textit{$($Correspondence of ideals$)$}: There is one to one correspondence between the $G$-graded ideals of $\mathcal{U}$ and the ideals of the ring $R$, the $e$-center of $\mathcal{U}$. $(2)$ \textit{Artin-Procesi condition}: $\mathcal{U}$ satisfies the $G$-graded identities of $A$ and no nonzero $G$-graded homomorphic image of $\mathcal{U}$ satisfies properly more identities. $(3)$ \textit{Generic}: If $B$ is a $G$-graded algebra over a field then it is a specialization of $\mathcal{U}$ along an ideal $\mathfrak{a} \in spec(Z(\mathcal{U})_{e})$ if and only if it is a $G$-graded form of $A$ over its $e$-center. We apply this to characterize finite dimensional $G$-graded simple algebras over $F$ that admit a $G$-graded division algebra form over their $e$-center.

math.RA↗
arXiv2021

Semisimple Algebras and PI-Invariants of Finite Dimensional Algebras

Let $Γ$ be a $T$-ideal of identities of an affine PI-algebra over an algebraically closed field $F$ of characteristic zero. Consider the family $\mathcal{M}_Γ$ of finite dimensional algebras $Σ$ with $Id(Σ) = Γ$. By Kemer's theory it is known that such $Σ$ exists. We show there exists a semisimple algebra $U$ which satisfies the following conditions. $(1)$ There exists an algebra $A \in \mathcal{M}_Γ$ with Wedderburn-Malcev decomposition $A \cong U \oplus J_{A}$, where $J_{A}$ is the Jacobson's radical of $A$ $(2)$ If $B \in \mathcal{M}_Γ$ and $B \cong B_{ss} \oplus J_{B}$ is its Wedderburn-Malcev decomposition then $U$ is a direct summand of $B_{ss}$. We refer to $U$ as the unique minimal semisimple algebra corresponding to $Γ$. We fully extend this result to the non-affine $G$-graded setting where $G$ is a finite group. In particular we show that if $A$ and $B$ are finite dimensional $G_{2}:= \mathbb{Z}_{2} \times G$-graded simple algebras then they are $G_{2}$-graded isomorphic if and only if $E(A)$ and $E(B)$ are $G$-graded PI-equivalent, where $E$ is the unital infinite dimensional Grassmann algebra and $E(A)$ is the Grassmann envelope of $A$.

math.RA↗
arXiv2020

Division algebras graded by a finite group

Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a normal abelian subgroup $H$, a positive integer $d$ and an element of $Hom(M(H), k^\times)^G$, where $M(H)$ is the Schur multiplier of $H$. Our main theorem is the converse: Given an extension $1\rightarrow H\rightarrow G\rightarrow G/H\rightarrow 1$, where $H$ is abelian, a positive integer $d$, and an element of $Hom(M(H), k^\times)^G$, there is a division algebra with center containing $k$ that realizes these data. We apply this result to classify the $G$-simple algebras over an algebraically closed field of characteristic zero that admit a division algebra form over a field containing an algebraically closed field.

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arXiv2018

Verbally prime T-ideals and graded division algebras

Let $F$ be an algebraically closed field of characteristic zero and let $G$ be a finite group. We consider graded Verbally prime $T$-ideals in the free $G$-graded algebra. It turns out that equivalent definitions in the ordinary case (i.e. ungraded) extend to nonequivalent definitions in the graded case, namely verbally prime $G$-graded $T$-ideals and strongly verbally prime $T$-ideals. At first, following Kemer's ideas, we classify $G$-graded verbally prime $T$-ideals. The main bulk of the paper is devoted to the stronger notion. We classify $G$-graded strongly verbally prime $T$-ideals which are $T$-ideal of affine $G$-graded algebras or equivalently $G$-graded $T$-ideals that contain a Capelli polynomial. It turns out that these are precisely the $T$-ideal of $G$-graded identities of finite dimensional $G$-graded, central over $F$ (i.e. $Z(A)_{e}=F$) which admit a $G$-graded division algebra twisted form over a field $k$ which contains $F$ or equivalently over a field $k$ which contains enough roots of unity (e.g. a primitive $n$-root of unity where $n = ord(G)$).

math.RA↗
arXiv2016

The Polynomial Part of the Codimension Growth of Affine PI Algebras

Let $F$ be a field of characteristic zero and $W$ be an associative affine $F$-algebra satisfying a polynomial identity (PI). The codimension sequence associated to $W$, $c_n(W)$, is known to be of the form $Θ(c n^t d^n)$, where $d$ is the well known (PI) exponent of $W$. In this paper we establish an algebraic interpretation of the polynomial part (the constant $t$) by means of Kemer's theory. In particular, we show that in case $W$ is a basic algebra, then $t = \frac{d-q}{2} + s$, where $q$ is the number of simple component in $W/J(W)$ and $s+1$ is the nilpotency degree of $J(W)$. Thus proving a conjecture of Giambruno.

math.RA↗
arXiv2014

On group gradings on PI-algebras

We show that there exists a constant K such that for any PI- algebra W and any nondegenerate G-grading on W where G is any group (possibly infinite), there exists an abelian subgroup U of G with $[G : U] \leq exp(W)^K$. A G-grading $W = \bigoplus_{g \in G}W_g$ is said to be nondegenerate if $W_{g_1}W_{g_2}... W_{g_r} \neq 0$ for any $r \geq 1$ and any $r$ tuple $(g_1, g_2,..., g_r)$ in $G^r$.

math.RA↗
arXiv2013

On regular G-grading

Let A be an associative algebra over an algebraically closed field F of characteristic zero and let G be a finite abelian group. Regev and Seeman introduced the notion of a regular G-grading on A, namely a grading A= Σ_{g in G} A_g that satisfies the following two conditions: (1) for every integer n>=1 and every n-tuple (g_1,g_2,...,g_n) in G^n, there are elements, a_i in A_{g_i}, i=1,...,n, such that a_1*a_2*...*a_n != 0. (2) for every g,h in G and for every a_g in A_g,b_h in A_h, we have a_{g}b_{h}=theta(g,h)b_{h}a_{g}. Then later, Bahturin and Regev conjectured that if the grading on A is regular and minimal, then the order of the group G is an invariant of the algebra. In this article we prove the conjecture by showing that ord(G) coincides with an invariant of A which appears in PI theory, namely exp(A) (the exponent of A). Moreover, we extend the whole theory to (finite) nonabelian groups and show that the above result holds also in that case.

math.RA↗
arXiv2012

Multialternating graded polynomials and growth of polynomial identities

Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by mean of Regev central polynomials we construct multialternating graded polynomials of arbitrarily large degree non vanishing on A. As a consequence we compute the exponential rate of growth of the sequence of graded codimensions of an arbitrary G-graded algebra satisfying an ordinary polynomial identity. In particular we show it is an integer. The result was proviously known in case G is abelian.

math.RA↗
arXiv2011

Simple G-graded algebras and their polynomial identities

Let G be any group and F an algebraically closed field of characteristic zero. We show that any G-graded finite dimensional associative G-simple algebra over F is determined up to a G-graded isomorphism by its G-graded polynomial identities. This result was proved by Koshlukov and Zaicev in case G is abelian.

math.RA↗
arXiv2010

Hilbert series of PI relatively free G-graded algebras are rational functions

Let G be a finite group, (g_{1},...,g_{r}) an (unordered) r-tuple of G^{(r)} and x_{i,g_i}'s variables that correspond to the g_i's, i=1,...,r. Let F be the corresponding free G-graded algebra where F is a field of zero characteristic. Here the degree of a monomial is determined by the product of the indices in G. Let I be a G-graded T-ideal of F which is PI (e.g. any ideal of identities of a G-graded finite dimensional algebra is of this type). We prove that the Hilbert series of F /I is a rational function. More generally, we show that the Hilbert series which corresponds to any g-homogeneous component of F /I is a rational function.

math.RA↗
arXiv2010

Group graded PI-algebras and their codimension growth

Let W be an associative PI-algebra over a field F of characteristic zero. Suppose W is G-graded where G is a finite group. Let exp(W) and exp(W_e) denote the codimension growth of W and of the identity component W_e, respectively. The following inequality had been conjectured by Bahturin and Zaicev: exp(W)\leq |G|^2 exp(W_e). The inequality is known in case the algebra W is affine (i.e. finitely generated). Here we prove the conjecture in general.

math.RA↗
arXiv2009

On the codimension growth of G-graded algebras

Let W be an associative PI-affine algebra over a field F of characteristic zero. Suppose W is G-graded where G is a finite group. Let exp(W) and exp(W_e) denote the codimension growth of W and of the identity component W_e, respectively. We prove: exp(W) \leq |G|^2 exp(W_e). This inequality had been conjectured by Bahturin and Zaicev.

math.RA↗
arXiv2009

Representability and Specht problem for G-graded algebras

Let W be an associative PI algebra over a field F of characteristic zero, graded by a finite group G. Let id_{G}(W) denote the T-ideal of G-graded identities of W. We prove: 1. {[G-graded PI equivalence]} There exists a field extension K of F and a finite dimensional Z/2ZxG-graded algebra A over K such that id_{G}(W)=id_{G}(A^{*}) where A^{*} is the Grassmann envelope of A. 2. {[G-graded Specht problem]} The T-ideal id_{G}(W) is finitely generated as a T-ideal. 3. {[G-graded PI-equivalence for affine algebras]} Let W be a G-graded affine algebra over F. Then there exists a field extension K of F and a finite dimensional algebra A over K such that id_{G}(W)=id_{G}(A).

math.RA↗
arXiv2009

Nilpotency of Bocksteins, Kropholler's hierarchy and a conjecture of Moore

A conjecture of Moore claims that if G is a group and H a finite index subgroup of G such that G - H has no elements of prime order (e.g. G is torsion free), then a G-module which is projective over H is projective over G. The conjecture is known for finite groups. In that case, it is a direct consequence of Chouinard's theorem which is based on a fundamental result of Serre on the vanishing of products of Bockstein operators. It was observed by Benson, using a construction of Baumslag, Dyer and Heller, that the analog of Serre's Theorem for infinite groups is not true in general. We prove that the conjecture is true for groups which satisfy the analog of Serre's theorem. Using a result of Benson and Goodearl, we prove that the conjecture holds for all groups inside Kropholler's hierarchy LHF, extending a result of Aljadeff, Cornick, Ginosar, and Kropholler. We show two closure properties for the class of pairs of groups (G,H) which satisfy the conjecture, the one is closure under morphisms, and the other is a closure operation which comes from Kropholler's construction. We use this in order to exhibit cases in which the analog of Serre's theorem does not hold, and yet the conjecture is true. We will show that in fact there are pairs of groups (G,H) in which H is a perfect normal subgroup of prime index in G, and the conjecture is true for (G,H). Moreover, we will show that it is enough to prove the conjecture for groups of this kind only.

math.GR↗
arXiv2009

Graded polynomial identities and exponential growth

Let A be a finite dimensional algebra over a field of characteristic zero graded by a finite abelian group G. Here we study a growth function related to the graded polynomial identities satisfied by A by computing the exponential rate of growth of the sequence of graded codimensions of A. We prove that the G-exponent of A exists and is an integer related in an explicit way to the dimension of a suitable semisimple subalgebra of A.

math.RA↗
arXiv2008

On cohomology rings of infinite groups

Let R be any ring (with 1), Γa group and RΓthe corresponding group ring. Let Ext_{RΓ}^{*}(M,M) be the cohomology ring associated to the RΓ-module M. Let H be a subgroup of finite index of Γ. The following is a special version of our main Theorem: Assume the profinite completion of Γis torsion free. Then an element ζin Ext_{RΓ}^{*}(M,M) is nilpotent (under Yoneda's product) if and only if its restriction to Ext_{RH}^{*}(M,M)$ is nilpotent. In particular this holds for the Thompson group. There are torsion free groups for which the analogous statement is false.

math.KT↗
arXiv2007

Graded identities of matrix algebras and the universal graded algebra

We consider fine G-gradings on M_n(C) (i.e. gradings of the matrix algebra over the complex numbers where each component is 1 dimensional). Groups which provide such a grading are known to be solvable. We consider the T-ideal of G-graded identities and show that it is generated by a special type of binomial identities which we call elementary. In particular we show that the ideal of graded identities is finitely generated as a T-ideal. Next, given such grading we construct a universal algebra U_{G,c} in two different ways: one by means of polynomial identities and the other one by means of a generic two-cocycle (this parallels the classical constructions in the non-graded case). We show that a suitable central localization of U_{G,c} is Azumaya over its center and moreover, its homomorphic images are precisely the G-graded forms of M_n(C). Finally, we consider the ring of central quotients Q(U_{G,c}) (this is an F-central simple algebra where F=Frac(Z) and Z is the center of of U_{G,c}). Using an earlier results of the authors (see E. Aljadeff, D. Haile and M. Natapov, Projective bases of division algebras and groups of central type, Israel J. Math.146 (2005) 317-335 and M. Natapov arXiv:0710.5468v1 [math.RA]) we show that this is a division algebra for a very explicit (and short) family of nilpotent groups. As a consequence, for groups G such that Q(U_{G,c}) is not a division algebra, one can find a non identity polynomial p(x_{i,g}) such that p(x_{i,g})^r is a graded identity for some integer r. We illustrate this phenomenon with a fine G-grading of M_6(C) where G is a semidirect product of S_3 and C_6.

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