arXiv subjects
Publications and source records attributed to Vesselin Drensky.
We give a simple proof for the existence of weak central polynomials for the algebra of multiplications of a finite-dimensional simple (non-associative) algebra. As an example we present explicit weak central polynomials in the cases of the three-dimensional simple Lie algebra and the Jordan algebra of the two-dimensional vector space with non-degenerate symmetric bilinear form.
The variety of bicommutative algebras is the class of all nonassociative algebras satisfying the polynomial identities $(x_1x_2)x_3=(x_1x_3)x_2$ and $x_1(x_2x_3)=x_2(x_1x_3)$. In this paper we provide a complete description of varieties of bicommutative algebras over a field of characteristic zero that satisfy a polynomial identity of degree three. Furthermore, we establish a sufficient and necessary condition for a variety of bicommutative algebras to have a distributive lattice of subvarieties.
This is a survey on the finite basis problem for varieties of algebraic systems. Our exposition is in two directions: (i) We give numerous examples of varieties which are not finitely based. (ii) We give examples of important varieties with the property that they and their subvarieties are finitely based. A special attention is paid on the varieties of semigroups, groups, associative, Lie and other nonassociative algebras.
Let $K\langle X_d\rangle$ be the free associative algebra of rank $d \geq 2$ over a field $K$. Lane in 1976 and Kharchenko in 1978 proved that the algebra of invariants $K\langle X_d\rangle^G$ is free for any subgroup $G \leq \text{GL}_d(K)$ and any field $K$. Later, Kharchenko introduced an additional action of the symmetric group $\text{Sym}(n)$ on the homogeneous component of degree $n$ of $K\langle X_d\rangle$, given by permuting the positions of the variables. This equips $K\langle X_d\rangle$ with the structure of a $(K\langle X_d\rangle,\circ)$-$S$-algebra. Then Koryukin showed that the algebra of invariants $K\langle X_d\rangle^G$ is finitely generated for every reductive group $G$ with respect to this action. In our paper we study the algebra $K\langle x_1,\ldots,x_d\rangle^{C_d}$ of invariants of the cyclic group $C_d$, $d\geq 2$, where $K$ is an arbitrary field of characteristic 0. We compute the Hilbert series of $K\langle x_1,\ldots,x_d \rangle^{C_d}$. When $K=\mathbb C$ we find a vector space basis of ${\mathbb C}\langle x_1,\ldots,x_d \rangle^{C_d}$ and explicitly describe the generators of ${\mathbb C}\langle x_1,\ldots,x_d \rangle^{C_d}$ as a free algebra. Moreover, we describe a finite generating set for the $S$-algebra $({\mathbb C}\langle x_1,\ldots,x_d \rangle^{C_d},\circ)$. We also transfer the results for $K=\mathbb C$ to the case of an arbitrary field of characteristic 0 for the $S$-algebra $(K\langle x_1,x_2,x_3 \rangle^{C_3},\circ)$ and find a minimal generating set for it as an $S$-algebra.
Let $K\langle X_d\rangle$ denote the free associative algebra of rank $d \geq 2$ over a field $K$. By results of Lane (1976) and Kharchenko (1978), the algebra of invariants $K\langle X_d\rangle ^G$ is free for any subgroup $G \leq \GL_d(K)$ and any field $K$. Koryukin (1984) introduced an additional action of the symmetric group $Sym(n)$ on the homogeneous component of degree $n$ of $K\langle X_d\rangle$, given by permuting the positions of the variables. This endows $K\langle X_d\rangle $ with the structure of a $(K\langle X_d\rangle,\circ)$-$S$-algebra. With respect to this action, Koryukin proved that the invariant algebra $K\langle X_d\rangle ^G$ is finitely generated for every reductive group $G$. In this paper we study the algebra ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ of invariants under the action of the dihedral group D_{2n} $ on the free associative algebra ${\mathbb C} \langle u,v\rangle$ of rank $2$. We compute the Hilbert series of ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ and construct an explicit set of generators for ${\mathbb C}\langle u,v\rangle^{D_{2n}}$ as a free algebra. Furthermore, we describe a finite generating set for the $S$-algebra ${\mathbb C}\langle u,v\rangle^{D_{2n}}$.
Formanek made the conjecture that the minimal degree of the central polynomials for the $n\times n$ matrix algebra over a field of characteristic 0 is $(n^2+3n-2)/2$ and this is true for $n\leq 3$. For $n=4$ there are examples of central polynomials of degree $13=(4^2+3\cdot 4-2)/2$ and we do not know whether there are central polynomials of lower degree. In this paper we discuss methods for searching for central polynomials of low degree and prove that the algebra of $4\times 4$ matrices does not have central polynomials in two variables of degree $\leq 12$. As a byproduct of our computations we obtain that this algebra does not have also polynomial identities in two variables of degree $\leq 12$.
We consider the 2-generated free metabelian associative and Lie algebras over the complex field and the invariants of the dihedral groups of finite order acting on these algebras. In the associative case we find a finite set of generators of the algebra of invariants. In the Lie case, when the algebra of invariants is not finitely generated, we give a minimal system of generators of the invariants in the commutator ideal as a module of the algebra of the invariants in the polynomial algebra in two variables. In both associative and Lie cases we compute the Hilbert series of the algebras of invariants.
We introduce the variety ${\mathfrak B}_{\textrm{sup}}$ of bicommutative superalgebras over an arbitrary field of characteristic different from 2. The variety consists of all nonassociative ${\mathbb Z}_2$-graded algebras satisfying the polynomial super-identities of super- left- and right-commutativity \[ x(yz)= (-1)^{\overline{x}\,\overline{y}} y(xz)\text{ and } (xy)z=(-1)^{\overline{y}\,\overline{z}} (xz)y, \] where $\overline{u}\in\{0,1\}$ is the parity of the homogeneous element $u$. We present an explicit construction of the free bicommutative superalgebras, find their bases as vector spaces and show that they share many properties typical for ordinary bicommutative algebras and super-commutative associative superalgebras. In particular, in the case of free algebras of finite rank we compute the Hilbert series and find explicitly its coefficients. As a consequence we give a formula for the codimension sequence. We establish an analogue of the classical Hilbert Basissatz for two-sided ideals. We see that the Gröbner-Shirshov bases of these ideals are finite, the Gelfand-Kirillov dimensions of finitely generated bicommutative superalgebras are nonnegative integers and the Hilbert series of finitely generated graded bicommutative superalgebras are rational functions. Concerning problems studied in the theory of varieties of algebraic systems, we prove that the variety of bicommutative superalgebras satisfies the Specht property. In the case of characteristic 0 we compute the sequence of cocharacters.
The purpose of this survey paper is to bring to a large mathematical audience (containing also non-algebraists) some topics of invariant theory both in the classical commutative and the recent noncommutative case. We have included only several topics from the classical invariant theory -- the finite generating (the Endlichkeitssatz) and the finite presenting (the Basissatz) of the algebra of invariants, the Molien formula for its Hilbert series and the Shephard-Todd-Chevalley theorem for the invariants of a finite group generated by pseudo-reflections. Then we give analogues of these results for free and relatively free associative and Lie algebras. Finally we deal with the algebra of generic matrices and the invariant theory related with it.
Let $F$ be a field of characteristic $0$ and let $E$ be the infinite dimensional Grassmann algebra over $F$. In the first part of this paper we give an algorithm calculating the generating function of the cocharacter sequence of the $n\times n$ upper triangular matrix algebra $UT_n(E)$ with entries in $E$, lying in a strip of a fixed size. In the second part we compute the double Hilbert series $H(E;\mathrm{T}_k,\mathrm{Y}_l)$ of $E$, then we define the $(k,l)$-multiplicity series of any PI-algebra. As an application, we derive from $H(E;\mathrm{T}_k,\mathrm{Y}_l)$ an easy algorithm determining the $(k,l)$-multiplicity series of $UT_n(E)$.
The variety of bicommutative algebras consists of all nonassociative algebras satisfying the polynomial identities of right- and left-commutativity $(x_1x_2)x_3=(x_1x_3)x_2$ and $x_1(x_2x_3)=x_2(x_1x_3)$. Let $F_d$ be the free $d$-generated bicommutative algebra over a field $K$ of characteristic 0. We study the algebra $F_d^G$ of invariants of a subgroup $G$ of the general linear group $GL_d(K)$. When $G$ is finite we search for analogies of classical results of invariant theory of finite groups acting on polynomial algebras: the Endlichkeitssatz of Emmy Noether, the Molien formula and the Chevalley-Shephard-Todd theorem and show the similarities and the differences in the case of bicommutative algebras. We also describe the symmetric polynomials in $F_d$.
Let $R$ be an associative algebra over a field $K$ generated by a vector subspace $V$. The polynomial $f(x_1,\ldots,x_n)$ of the free associative algebra $K\langle x_1,x_2,\ldots\rangle$ is a weak polynomial identity for the pair $(R,V)$ if it vanishes in $R$ when evaluated on $V$. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.
Let $\text{GL}(n) = \text{GL}(n, {\mathbb C})$ denote the complex general linear group and let $G \subset \text{GL}(n)$ be one of the classical complex subgroups $\text{O}(n)$, $\text{SO}(n)$, and $\text{Sp}(2k)$ (in the case $n = 2k$). We take a polynomial $\text{GL}(n)$-module $W$ and consider the symmetric algebra $S(W)$. Extending previous results for $G=\text{SL}(n)$, we develop a method for determining the Hilbert series $H(S(W)^G, t)$ of the algebra of invariants $S(W)^G$. Then we give explicit examples for computing $H(S(W)^G, t)$. As a further application, we extend our method to compute also the Hilbert series of the algebras of invariants $Λ(S^2 V)^G$ and $Λ(Λ^2 V)^G$, where $V = {\mathbb C}^n$ denotes the standard $GL(n)$-module.
Let $A_1,\ldots,A_s$ be unitary commutative rings which do not have non-trivial idempotents and let $A=A_1\oplus\cdots\oplus A_s$ be their direct sum. We describe all idempotents in the $2\times 2$ matrix ring $M_2(A[[X]])$ over the ring $A[[X]]$ of formal power series with coefficients in $A$ and in arbitrary set of variables $X$. We apply this result to the matrix ring $M_2({\mathbb Z}_n[[X]])$ over the ring ${\mathbb Z}_n[[X]]$ for an arbitrary positive integer $n$ greater than 1. Our proof is elementary and uses only the Cayley-Hamilton theorem (for $2\times 2$ matrices only) and, in the special case $A={\mathbb Z}_n$, the Chinese reminder theorem and the Euler-Fermat theorem.
A variety of associative algebras over a field of characteristic 0 is called minimal if its codimension sequence grows much faster than the codimension sequence of any of its proper subvarieties. By the results of Giambruno and Zaicev it follows that the number $b_n$ of minimal varieties of given exponent $n$ is finite. Using methods of the theory of colored (or weighted) compositions of integers, we show that the limit $β=\lim_{n\to\infty}\sqrt[n]{b_n}$ exists and can be expressed as the positive solution of an equation $a(t)=0$ where $a(t)$ is an explicitly given power series. Similar results are obtained for the number of minimal varieties with a given Gelfand-Kirillov dimension of their relatively free algebras of rank $d$. It follows from classical results on lacunary power series that the generating function of the sequence $b_n$, $n=1,2,\ldots$, is transcendental. With the same approach we construct examples of free graded semigroups $\langle Y\rangle$ with the following property. If $d_n$ is the number of elements of degree $n$ of $\langle Y\rangle$, then the limit $δ=\lim_{n\to\infty}\sqrt[n]{d_n}$ exists and is transcendental.
This article surveys results on graded algebras and their Hilbert series. We give simple constructions of finitely generated graded associative algebras $R$ with Hilbert series $H(R,t)$ very close to an arbitrary power series $a(t)$ with exponentially bounded nonnegative integer coefficients. Then we summarize some related facts on algebras with polynomial identity. Further we discuss the problem how to find series $a(t)$ which are rational/algebraic/transcendental over ${\mathbb Q}(t)$. Applying a classical result of Fatou we conclude that if a finitely generated graded algebra has a finite Gelfand-Kirillov dimension, then its Hilbert series is either rational or transcendental. In particular the same dichotomy holds for the Hilbert series of a finitely generated algebra with polynomial identity. We show how to use planar rooted trees to produce algebraic power series. Finally we survey some results on noncommutative invariant theory which show that we can obtain as Hilbert series various algebraic functions and even elliptic integrals.
Motivated by a recent Diophantine transport problem about how to transport profitably a group of persons or objects, we survey classical facts about solving systems of linear Diophantine equations and inequalities in nonnegative integers. We emphasize on the method of Elliott from 1903 and its further developed by MacMahon in his ``$Ω$-Calculus'' or Partition Analysis. As an illustration we obtain the solution of the considered transport problem in terms of a formal power series in several variables which is an expansion of a rational function of a special form.
For any power series $a(t)$ with exponentially bounded nonnegative integer coefficients we suggest a simple construction of a finitely generated monomial associative algebra $R$ with Hilbert series $H(R,t)$ very close to $a(t)$. If $a(t)$ is rational/algebraic/transcendental, then the same is $H(R,t)$. If the growth of the coefficients of $a(t)$ is polynomial, in the same way we construct a graded algebra $R$ preserving the polynomial growth of the coefficients of its Hilbert series $H(R,t)$. Applying a classical result of Fatou from 1906 we obtain that if a finitely generated graded algebra $R$ has a finite Gelfand-Kirillov dimension, then its Hilbert series is either rational or transcendental. In particular the same dichotomy holds for the Hilbert series of finitely generated algebras $R$ with polynomial identity.