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.
Let E be the infinite dimensional Grassmann algebra over a finite field F of characteristic not 2. In this paper we deal with the homogeneous Z_2-gradings of E. In particular, we compute an exact value for the Z_2-graded homogeneous codimensions of E, and a lower and an upper bound for the Z_2-graded (non-homogeneous) codimensions of E for each of its Z_2-homogeneous grading.
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.
We show that if $T$ is any of four semigroups of two elements that are not groups, there exists a finite dimensional associative $T$-graded algebra over a field of characteristic $0$ such that the codimensions of its graded polynomial identities have a non-integer exponent of growth. In particular, we provide an example of a finite dimensional graded-simple semigroup graded algebra over an algebraically closed field of characteristic $0$ with a non-integer graded PI-exponent, which is strictly less than the dimension of the algebra. However, if $T$ is a left or right zero band and the $T$-grad…
We study polynomial identities of finite dimensional simple color Lie superalgebras over an algebraically closed field of characteristic zero graded by the product of two cyclic groups of order $2$. We prove that the codimensions of identities grow exponentially and the rate of exponent equals the dimension of the algebra. A similar result is also obtained for graded identities and graded codimensions.
We introduce grading on certain finite dimensional simple Lie superalgebras of type $P(t)$ by elementary abelian 2-group. This grading gives rise to Pauli matrices and is a far generalization of $(\mathbb Z_2\times \mathbb Z_2)$-grading on Lie algebra of $(2\times 2)$-traceless matrices.We use this grading for studying numerical invariants of polyomial identities of Lie superalgebras. In particular, we compute graded PI-exponent corresponding to Pauli grading.
We study polynomial identities of nonassociative algebras constructed by using infinite binary words and their combinatorial properties. Infinite periodic and Sturmian words were first applied for constructing examples of algebras with arbitrary real PI-exponent greater than one. Later we used these algebras for confirmation of the conjecture that PI-exponent increases precisely by one after adjoining an external unit to a given algebra. Here we prove the same result for these algebras for graded identities and graded PI-exponent, provided that the grading group is cyclic of order two.
Let W be an affine PI algebra over a field of characteristic zero graded by a finite group G. We show that there exist $α_{1},α_{2}\in\mathbb{R}, β\in\frac{1}{2}\mathbb{Z}$, and $l\in\mathbb{N}$ such that $α_{1}n^βl^{n}\leq c_{n}^{G}(W)\leqα_{2}n^βl^{n}$. Furthermore, if W has a unit then the asymptotic behavior of $c_{n}^{G}(W)$ is $αn^βl^{n}$ where $α\in\mathbb{R}, β\in\frac{1}{2}\mathbb{Z}, l\in\mathbb{N}$.
Consider a finite dimensional Lie algebra L with an action of a finite group G over a field of characteristic 0. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial G-identities of L. As a consequence, we prove the analog of Amitsur's conjecture for graded codimensions of any finite dimensional Lie algebra graded by a finite Abelian group.
A.Regev proved that the codimension growth of an associative PI-algebra is at most exponential. The author established a scale for the codimension growth of Lie PI-algebras, which includes a series of functions between exponential and factorial functions. We prove that the same scale stratifies the ordinary codimension growth of Poisson PI-algebras satisfying Lie identical relations. As a byproduct, we obtain a new bound on the codimension growth of Lie PI-algebras in terms of complexity functions. We also study very fast codimension growth of some Poisson algebras without Lie identities. We e…
In this work we consider extensions of solvable Lie algebras with naturally graded filiform nilradicals. Note that there exist two naturally graded filiform Lie algebras $n_{n, 1}$ and $Q_{2n}.$ We find all one-dimensional central extensions of the algebra $n_{n, 1}$ and show that any extension of $Q_{2n}$ is split. After that we find one-dimensional extensions of solvable Lie algebras with nilradical $n_{n, 1}$. We prove that there exists a unique non-split central extension of solvable Lie algebras with nilradical $n_{n, 1}$ of maximal codimension. Moreover, all one-dimensional extensions of…
We study codimension growth of infinite dimensional Lie algebras over a field of characteristic zero. We prove that if a Lie algebra $L$ is an extension of a nilpotent algebra by a finite dimensional semisimple algebra then the PI-exponent of $L$ exists and is a positive integer.
We study numerical invariants of identities of finite-dimensional solvable Lie superalgebras. We define new series of finite-dimensional solvable Lie superalgebras $L$ with non-nilpotent derived subalgebra $L'$ and discuss their codimension growth. For the first algebra of this series we prove the existence and integrality of $exp(L)$.
Suppose a finite dimensional semisimple Lie algebra $\mathfrak g$ acts by derivations on a finite dimensional associative or Lie algebra $A$ over a field of characteristic $0$. We prove the $\mathfrak g$-invariant analogs of Wedderburn - Mal'cev and Levi theorems, and the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial identities with derivations of $A$. It turns out that for associative algebras the differential PI-exponent coincides with the ordinary one. Also we prove the analog of Amitsur's conjecture for finite dimensional associative algebras with an …
We study $\mathbb{Z}_2$-graded identities of Lie superalgebras of the type $b(t), t\ge 2$, over a field of characteristic zero. Our main result is that the $n$-th codimension is strictly less than $(\dim b(t))^n$ asymptotically. As a consequence we obtain an upper bound for ordinary (non-graded) PI-exponent for each simple Lie superalgebra $b(t), t\ge 3$.
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.
Let $L$ be a finite dimensional Lie $F$-algebra endowed with a generalized action by an associative algebra $H$. We investigate the exponential growth rate of the sequence of $H$-graded codimensions $c_n^H(L)$ of $L$ which is a measure for the number of non-polynomial $H$-identities of $L$. More precisely, we construct the first example of an $S$-graded Lie algebra having a non-integer, even irrational, exponential growth rate $\lim_{n\rightarrow \infty} \sqrt[n]{c_n^{S}(L)}$. Hereby $S$ is a semigroup and an exact value is given. On the other hand, returning to general $H$, if $L$ is semisimp…
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.
We classify a class of infinite-dimensional simple graded pre-Lie algebras on the graded vector space underlying the algebra of Laurent polynomials, with a specific form for the product.
Let K be a field of characteristic 0 and L be a G-graded Lie PI-algebra, where G is a finite group. We define the graded Gelfand-Kirillov dimension of L. Then we measure the growth of the Z_n-graded polynomial identities of the Lie algebra of n x n traceless matrices sl_n(K) giving an exact value of its Z_n-graded Gelfand-Kirillov dimension.
The asymptotic behaviour is studied of exponentially bounded sequences of codimensions of identities of algebras with unity. A series of algebras is constructed for which the base of the exponential increases by exactly one when an outer unity is adjoined to the original algebra. We show that the PI-exponents of unital algebras can take any value greater than two, and the exponents of finite-dimensional unital algebras form a dense subset of the domain [2,$\infty$).
In the 80's, Regev, using results of Formanek, Procesi and Razmyslov in invariant theory and Hilbert series', determined asymptotically the codimension sequence of mXm matrices over an algebraically closed field of characteristic zero. Inspired by Regev's ideas, we found that the asymptotics of $c_{n}^{G}(A)$, the G graded codimension sequence of a finite dimensional G simple algebra A, is equal to $αn^{\frac{1-\dim(A_{e})}{2}}(\dim(A)^{n} $ (this was conjectured by E.Aljadeff, D.Haile and M. Natapov), where αis not yet determined number. Moreover, in the case where A is the algebra of mXm mat…
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 provi…
A nilpotent Lie algebra ${\mathfrak g}$ is said to be naturally graded if it is isomorphic to its associated graded Lie algebra ${\rm gr} \mathfrak{g}$ with respect to filtration by ideals of the lower central series. This concept is equivalent to the concept of the Carnot algebra arising in sub-Riemannian geometry and the geometric control theory. We classify finite-dimensional and infinite-dimensional naturally graded Lie algebras (Carnot algebras) ${\mathfrak g}=\oplus_{i=1}^{{+}\infty}{\mathfrak g}_i$ with properties $$ [{\mathfrak g}_1, {\mathfrak g}_i]={\mathfrak g}_{i{+}1}, \; \dim{{\ma…
Let $S$ be a finite semigroup and let $A$ be a finite dimensional $S$-graded algebra. We investigate the exponential rate of growth of the sequence of graded codimensions $c_n^S(A)$ of $A$, i.e $\lim\limits_{n \rightarrow \infty} \sqrt[n]{c_n^S(A)}$. For group gradings this is always an integer. Recently in [20] the first example of an algebra with a non-integer growth rate was found. We present a large class of algebras for which we prove that their growth rate can be equal to arbitrarily large non-integers. An explicit formula is given. Surprisingly, this class consists of an infinite family…
An algebra with identities $a\circ(b\circ c-c\circ b)=(a\circ b)\circ c-(a\circ c)\circ b$ and $a\circ(b\circ c)=b\circ(a\circ c)$ is called Novikov. We construct free Novikov base in terms of Young diagrams. We show that codimensions exponent for a variety of Novikov algebras exists and is equal 4.
Consider a finite dimensional H-module Lie algebra L over a field of characteristic 0 where H is a Hopf algebra. We prove the analog of Amitsur's conjecture on asymptotic behavior for codimensions of polynomial H-identities of L under some assumptions on H. In particular, the conjecture holds when H is finite dimensional semisimple. As a consequence, we obtain the analog of Amitsur's conjecture for graded codimensions of any finite dimensional Lie algebra graded by an arbitrary group and for G-codimensions of any finite dimensional Lie algebra with a rational action of a reductive affine algeb…
Suppose $A$ is a graded associative algebra over a field, $I$ is its ideal generated by a set $α$ of homogeneous elements, and B = A/I. In this note, some inequalities between Hilbert series of algebras $A,B$ and the number of elements of the set $α$ are announced. As in the Golod--Shafarevich inequality as in our case the equality in every estimate is exact iff the set $α$ is strongly free: so we obtain some new characterizations of such sets. As a consequence it is proved that over a field of zero characteristic for the class of finitely defined graded algebras there is no algorithm to answe…
Let $G$ be a group of order $k$. We consider the algebra $M_k(\mathbb{C})$ of $k$ by $k$ matrices over the complex numbers and view it as a crossed product with respect to $G$ by embedding $G$ in the symmetric group $S_k$ via the regular representation and embedding $S_k$ in $M_k(\mathbb{C})$ in the usual way. This induces a natural $G$-grading on $M_k(\mathbb{C})$ which we call a crossed-product grading. We study the graded $*$-identities for $M_k(\mathbb{C})$ equipped with such a crossed-product grading and the transpose involution. To each multilinear monomial in the free graded algebra wit…
A noncommutative Grassmannian NGr(m, n) is introduced by Efimov, Luntz, and Orlov in `Deformation theory of objects in homotopy and derived categories III: Abelian categories' as a noncommutative algebra associated to an exceptional collection of n-m+1 coherent sheaves on P^n. It is a graded Calabi--Yau Z-algebra of dimension n-m+1. We show that this algebra is coherent provided that the codimension d = n-m of the Grassmannian is two. According to op. cit., this gives a t-structure on the derived category of the coherent sheaves on the noncommutative Grassmannian. The proof is quite different …
We study symplectic structures on filiform Lie algebras -- nilpotent Lie algebras of the maximal length of the descending central sequence. In the present article we classify the Lie algebras with the structure relations of the following form: $$[e_i,e_j]=(j-i)e_{i+j}+\sum_{l{=}1}c_{ij}^l e_{i+j+l}, i+j \le n.$$ In even dimensions the subspace of symplectic Lie algebras has the codimension one.
We consider the algebra M_k(C) of k-by-k matrices over the complex numbers and view it as a crossed product with a group G of order k by embedding G in the symmetric group S_k via the regular representation and embedding S_k in M_k(C) in the usual way. This induces a natural G-grading on M_k(C) which we call a crossed product grading. This grading is the so called elementary grading defined by any k-tuple (g_1,g_2,..., g_k) of distinct elements g_i in G. We study the graded polynomial identities for M_k(C) equipped with a crossed product grading. To each multilinear monomial in the free graded…
We study identities of finite dimensional algebras over a field of characteristic zero, graded by an arbitrary groupoid $Γ$. First we prove that its graded colength has a polynomially bounded growth. For any graded simple algebra $A$ we prove the existence of the graded PI-exponent, provided that $Γ$ is a commutative semigroup. If $A$ is simple in a non-graded sense the existence of the graded PI-exponent is proved without any restrictions on $Γ$.
We initiate a study of the growth and matrix-valued Hilbert series of non-negatively graded twisted Calabi-Yau algebras that are homomorphic images of path algebras of weighted quivers, generalizing techniques previously used to investigate Artin-Schelter regular algebras and graded Calabi-Yau algebras. Several results are proved without imposing any assumptions on the degrees of generators or relations of the algebras. We give particular attention to twisted Calabi-Yau algebras of dimension d at most 3, giving precise descriptions of their matrix-valued Hilbert series and partial results desc…
In this note we draw a connection between noncommutative algebra and geometric group theory. Specifically, we ask whether it is possible to bound the sequence of codimensions for an associative PI-algebra using techniques from geometric group theory. The classic and best known bound on codimension growth was derived by finding a particularly nice spanning set for the multilinear polynomials of degree n inside the free algebra. This spanning set corresponds to permutations in the symmetric group which are so-called d-good, where d is the degree of an identity satisfied by the algebra. The motiv…
