arXiv subjects

Vesselin Drensky

Publications and source records attributed to Vesselin Drensky.

At least 37 records · Page 2Linked to original sources

arXiv2019

Symmetric polynomials in the free metabelian Lie algebras

Let $K[X_n]$ be the commutative polynomial algebra in the variables $X_n=\{x_1,\ldots,x_n\}$ over a field $K$ of characteristic zero. A theorem from undergraduate course of algebra states that the algebra $K[X_n]^{S_n}$ of symmetric polynomials is generated by the elementary symmetric polynomials which are algebraically independent over $K$. In the present paper we study a noncommutative and nonassociative analogue of the algebra $K[X_n]^{S_n}$ replacing $K[X_n]$ with the free metabelian Lie algebra $F_n$ of rank $n\geq 2$ over $K$. It is known that the algebra $F_n^{S_n}$ is not finitely generated but its ideal $(F_n')^{S_n}$ consisting of the elements of $F_n^{S_n}$ in the commutator ideal $F_n'$ of $F_n$ is a finitely generated $K[X_n]^{S_n}$-module. In our main result we describe the generators of the $K[X_n]^{S_n}$-module $(F_n')^{S_n}$ which gives the complete description of the algebra $F_n^{S_n}$.

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arXiv2019

Locally Nilpotent Derivations of Free Algebra of Rank Two

In commutative algebra, if $δ$ is a locally nilpotent derivation of the polynomial algebra $K[x_1,\ldots,x_d]$ over a field $K$ of characteristic 0 and $w$ is a nonzero element of the kernel of $δ$, then $Δ=wδ$ is also a locally nilpotent derivation with the same kernel as $δ$. In this paper we prove that the locally nilpotent derivation $Δ$ of the free associative algebra $K\langle X,Y\rangle$ is determined up to a multiplicative constant by its kernel. We show also that the kernel of $Δ$ is a free associative algebra and give an explicit set of its free generators.

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arXiv2019

Generalized Nowicki conjecture

Let $B$ be an integral domain over a field $K$ of characteristic 0. The derivation $δ$ of $B[Y_d]=B[y_1,\ldots,y_d]$ is elementary if $δ(B)=0$ and $δ(y_i)\in B$, $i=1,\ldots,d$. Then the elements $u_{ij}=δ(y_i)y_j-δ(y_j)y_i$, $1\leq i 0$, $i=1,\ldots,d$, was handled by Khoury in the first proof of the Nowicki conjecture given by him in 2004. As a consequence of the proof of Kuroda in 2009 if $δ(y_i)=f_i(x_i)$, for any nonconstant polynomials $f_i(x_i)$, $i=1,\ldots,d$, then $B[Y_d]^δ=K[X_d,Y_d]^δ$ is generated by $X_d$ and $U_d=\{u_{ij}=f_i(x_i)y_j-y_if_j(x_j)\mid 1\leq i<j\leq d\}$. In the present paper we have found a presentation of the algebra \[ K[X_d,Y_d]^δ=K[X_d,U_d\mid R=S=0], \] \[ R=\{r(i,j,k,l)\mid 1\leq i<j<k<l\leq d\}, S=\{s(i,j,k)\mid 1\leq i<j<k\leq d\}, \] and a basis of $K[X_d,Y_d]^δ$ as a vector space. As a corollary we have shown that the defining relations $R\cup S$ form the reduced Gröbner basis of the ideal which they generate with respect to a specific admissible order. This is an analogue of the result of Makar-Limanov and the author in their proof of the Nowicki conjecture in 2009.

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arXiv2019

Another proof of the Nowicki conjecture

Let $K[X_d,Y_d]=K[x_1,\ldots,x_d,y_1,\ldots,y_d]$ be the polynomial algebra in $2d$ variables over a field $K$ of characteristic 0 and let $δ$ be the derivation of $K[X_d,Y_d]$ defined by $δ(y_i)=x_i$, $δ(x_i)=0$, $i=1,\ldots,d$. In 1994 Nowicki conjectured that the algebra $K[X_d,Y_d]^δ$ of constants of $δ$ is generated by $X_d$ and $x_iy_j-y_ix_j$ for all $1\leq i<j\leq d$. The affirmative answer was given by several authors using different ideas. In the present paper we give another proof of the conjecture based on representation theory of the general linear group $GL_2(K)$.

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arXiv2019

Noncommutative invariant theory of symplectic and orthogonal groups

We present a method for computing the Hilbert series of the algebra of invariants of the complex symplectic and orthogonal groups acting on graded noncommutative algebras with homogeneous components which are polynomial modules of the general linear group. We apply our method to compute the Hilbert series for different actions of the symplectic and orthogonal groups on the relatively free algebras of the varieties of associative algebras generated, respectively, by the Grassmann algebra and the algebra of $2\times 2$ upper triangular matrices. These two varieties are remarkable with the property that they are the only minimal varieties of exponent 2.

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arXiv2019

The Nowicki Conjecture for free metabelian Lie algebras

Let $K[X_d]=K[x_1,\ldots,x_d]$ be the polynomial algebra in $d$ variables over a field $K$ of characteristic 0. The classical theorem of Weitzenböck from 1932 states that for linear locally nilpotent derivations $δ$ (known as Weitzenböck derivations) the algebra of constants $K[X_{d}]^δ$ is finitely generated. When the Weitzenböck derivation $δ$ acts on the polynomial algebra $K[X_d,Y_d]$ in $2d$ variables by $δ(y_i)=x_i$, $δ(x_i)=0$, $i=1,\ldots,d$, Nowicki conjectured that $K[X_d,Y_d]^δ$ is generated by $X_d$ and $x_iy_j-y_ix_j$ for all $1\leq i<j\leq d$. There are several proofs based on different ideas confirming this conjecture. Considering arbitrary Weitzenböck derivations of the free $d$-generated metabelian Lie algebra $F_d$, with few trivial exceptions, the algebra $F_d^δ$ is not finitely generated. However, the vector subspace $(F_d')^δ$ of the commutator ideal $F_d'$ of $F_d$ is finitely generated as a $K[X_d]^δ$-module. In this paper we study an analogue of the Nowicki conjecture in the Lie algebra setting and give an explicit set of generators of the $K[X_d,Y_d]^δ$-module $(F_{2d}')^δ$.

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arXiv2019

Classical invariant theory for free metabelian Lie algebras

Let $KX_d$ be a vector space with basis $X_d=\{x_1,\ldots,x_d\}$ over a field $K$ of characteristic 0. One of the main topics of classical invariant theory is the study of the algebra of invariants $K[X_d]^{SL_2(K)}$, where $KX_d$ is a module of the special linear group $SL_2(K)$ isomorphic to a direct sum $V_{k_1}\oplus\cdots\oplus V_{k_r}$ and $V_k$ is the $SL_2(K)$-module of binary forms of degree $k$. Noncommutative invariant theory deals with the algebra of invariants $F_d({\mathfrak V})^G$ of the group $G<GL_d(K)$ acting on the relatively free algebra $F_d({\mathfrak V})$ of a variety of $K$-algebras $\mathfrak V$. In this paper we consider the free metabelian Lie algebra $F_d({\mathfrak A}^2)$ which is the relatively free algebra in the variety ${\mathfrak A}^2$ of metabelian (solvable of class 2) Lie algebras. We study the algebra $F_d({\mathfrak A}^2)^{SL_2(K)}$ of $SL_2(K)$-invariants of $F_d({\mathfrak A}^2)$. We describe the cases when this algebra is finitely generated. This happens if and only if $KX_d\cong V_1\oplus V_0\oplus\cdots\oplus V_0$ or $KX_d\cong V_2$ as an $SL_2(K)$-module (and in the trivial case $KX_d\cong V_0\oplus\cdots\oplus V_0$). For small $d$ we give a list of generators even when $F_d({\mathfrak A}^2)^{SL_2(K)}$ is not finitely generated. The methods for establishing that the algebra $F_d({\mathfrak A}^2)^{SL_2(K)}$ is not finitely generated work also for other relatively free algebras $F_d({\mathfrak V})$ and for other groups $G$.

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arXiv2019

Varieties of bicommutative algebras

Bicommutative algebras are nonassociative algebras satisfying the polynomial identities of right- and left-commutativity (xy)z=(xz)y and x(yz)=y(xz). We study subvarieties of the variety of all bicommutative algebras over a field of characteristic 0. We prove that if the subvariety satisfies a polynomial identity of degree k, then its codimension sequence is bounded by a polynomial of degree k-1. Since the exponent of the variety of all bicommutative algebras is equal to 2, this gives that it is minimal with respect to the codimension growth. Up to isomorphism there are three one-generated two-dimensional bicommutative algebras with one-dimensional square which are nonassociative. We present bases of their polynomial identities and show that one of these algebras generates the whole variety of bicommutative algebras.

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arXiv2018

Additive primitive length in relatively free algebras

The additive primitive length of an element $f$ of a relatively free algebra $F_d$ in a variety of algebras is equal to the minimal number $\ell$ such that $f$ can be presented as a sum of $\ell$ primitive elements. We give an upper bound for the additive primitive length of the elements in the $d$-generated polynomial algebra over a field of characteristic 0, $d>1$. The bound depends on $d$ and on the degree of the element. We show that if the field has more than two elements, then the additive primitive length in free $d$-generated nilpotent-by-abelian Lie algebras is bounded by 5 for $d=3$ and by 6 for $d>3$. If the field has two elements only, then our bound are 6 for $d=3$ and 7 for $d>3$. This generalizes a recent result of Ela Aydın for two-generated free metabelian Lie algebras. In all cases considered in the paper the presentation of the elements as sums of primitive can be found effectively in polynomial time.

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arXiv2017

Noetherianity and Specht problem for varieties of bicommutative algebras

Nonassociative algebras satisfying the polynomial identities x(yz)=y(xz) and (xy)z=(xz)y are called bicommutative. We prove the following results: (i) Finitely generated bicommutative algebras are weakly noetherian, i.e., satisfy the ascending chain condition for two-sided ideals. (ii) We give the positive solution to the Specht problem (or the finite basis problem) for varieties of bicommutative algebras over an arbitrary field of any characteristic.

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arXiv2016

Invariant theory of relatively free right-symmetric and Novikov algebras

Algebras with the polynomial identity (x,y,z)=(x,z,y), where (x,y,z)=x(yz)-(xy)z is the associator, are called right-symmetric. Novikov algebras are right-symmetric algebras satisfying additionally the polynomial identity x(yz)=y(xz). We consider the d-generated free right-symmetric algebra F(R) and the free Novikov algebra F(N) over a filed K of characteristic 0. The general linear group GL(K,d) with its canonical action on the d-dimensional vector space with basis the generators of F(R) and F(N) acts on F(R) and F(N) as a group of linear automorphisms. For a subgroup G of GL(K,d) we study the algebras of G-invariants in F(R) and F(N). For a large class of groups G we show that these algebras of invariants are never finitely generated. The same result holds for any subvariety of the variety R of right-symmetric algebras which contains the subvariety L of left-nilpotent of class 3 algebras in R.

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arXiv2016

Cocharacters of polynomial identities of block triangular matrices

We give an algorithm which calculates the generating function of the cocharacter sequence of the polynomial identities of the algebra of upper block triangular (p+2q) x (p+2q) matrices over a field of characteristic zero with diagonal consisting of p copies of 1 x 1 and q copies of 2 x 2 matrices. We have found the explicit form of the multiplicities and their asymptotic behaviour for small values of p and q.

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arXiv2016

GK-dimension of the Lie algebra of generic $2\times 2$ matrices

Recently Machado and Koshlukov have computed the Gelfand-Kirillov dimension of the relatively free algebra $F_m=F_m(\text{var}(sl_2(K)))$ of rank $m$ in the variety of algebras generated by the three-dimensional simple Lie algebra $sl_2(K)$ over an infinite field $K$ of characteristic different from 2. They have shown that $\text{GKdim}(F_m)=3(m-1)$. The algebra $F_m$ is isomorphic to the Lie algebra generated by $m$ generic $2\times 2$ matrices. Now we give a new proof for $\text{GKdim}(F_m)$ using classical results of Procesi and Razmyslov combined with the observation that the commutator ideal of $F_m$ is a module of the center of the associative algebra generated by $m$ generic traceless $2\times 2$ matrices.

math.RA↗

arXiv2015

Weitzenboeck derivations of free metabelian associative algebras

By the classical theorem of Weitzenboeck the algebra of constants (i.e., the kernel) of a nonzero locally nilpotent linear derivation of the polynomial algebra K[X] in d variables over a field K of characteristic 0 is finitely generated. As a noncommutative generalization one considers the algebra of constants of a locally nilpotent linear derivation of a d-generated relatively free algebra F(V) in a variety V of unitary associative algebras over K. It is known that the algebra of constants of F(V) is finitely generated if and only if V satisfies a polynomial identity which does not hold for the algebra of 2 x 2 upper triangular matrices. Hence the free metabelian associative algebra F(M) is a crucial object to study. We show that the vector space of the constants in the commutator ideal F'(M) is a finitely generated module of the algebra of constants of the polynomial algebra K[U,V] in 2d variables, where the derivation acts on U and V in the same way as on X. For small d, we calculate the Hilbert series of the constants in F'(M) and find the generators of the related module. This gives also an (infinite) set of generators of the algebra of constants in F(M).

math.RA↗

arXiv2015

The Bulgarian solitaire and the mathematics around it

The Bulgarian solitaire is a mathematical card game played by one person. A pack of n cards is divided into several decks (or "piles"). Each move consists of the removing of one card from each deck and collecting the removed cards to form a new deck. The game ends when the same position occurs twice. It has turned out that when n=k(k+1)/2 is a triangular number, the game reaches the same stable configuration with size of the piles 1,2,...,k. The purpose of the paper is to tell the (quite amusing) story of the game and to discuss mathematical problems related with the Bulgarian solitaire. The paper is dedicated to the memory of Borislav Bojanov (1944-2009), a great mathematician, person, and friend, and one of the main protagonists in the story of the Bulgarian solitaire.

math.CO↗

arXiv2013

Weitzenboeck derivations of free metabelian Lie algebras

A nonzero locally nilpotent linear derivation of the polynomial algebra K[X] in d variables over a field K of characteristic 0 is called a Weitzenboeck derivation. The classical theorem of Weitzenboeck states that the algebra of constants (which coincides with the algebra of invariants of a single unipotent transformation) is finitely generated. Similarly one may consider the algebra of constants of a locally nilpotent linear derivation of a finitely generated (not necessarily commutative or associative) algebra which is relatively free in a variety of algebras over K. Now the algebra of constants is usually not finitely generated. Except for some trivial cases this holds for the algebra of constants of the free metabelian Lie algebra L/L" with d generators. We show that the vector space of the constants in the commutator ideal L'/L" is a finitely generated module over the algebra of constants in K[X]. For small d, we calculate the Hilbert series of the algebra of constants in L/L" and find the generators of the module of the constants in L'/L". This gives also an (infinite) set of generators of the Lie algebra of constants in L/L".

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arXiv2012

Central polynomials for matrices over finite fields

Let $c(x_1,...,x_d)$ be a multihomogeneous central polynomial for the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of positive characteristic $p$. We show that there exists a multihomogeneous polynomial $c_0(x_1,...,x_d)$ of the same degree and with coefficients in the prime field $F_p$ which is central for the algebra $M_n(F)$ for any (possibly finite) field $F$ of characteristic $p$. The proof is elementary and uses standard combinatorial techniques only.

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arXiv2012

Inner Automorphisms of Lie Algebras Related with Generic 2 x 2 Matrices

Let F_m=F_m(var(sl(2,K))) be the relatively free algebra of rank m in the variety of Lie algebras generated by the algebra sl(2,K) over a field K of characteristic 0. Translating an old result of Baker from 1901 we present a multiplication rule for the inner automorphisms of the completion of F_m with respect to the formal power series topology. Our results are more precise for m=2 when F_2 is isomorphic to the Lie algebra L generated by two generic traceless 2 x 2 matrices. We give a complete description of the group of inner automorphisms of the completion of L. As a consequence we obtain similar results for the automorphisms of the relatively free algebra F_m/F_m^{c+1} in the subvariety of var(sl(2,K)) consisting of all nilpotent algebras of class at most c in var(sl(2,K)).

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