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Planar trees, free nonassociative algebras, invariants, and elliptic integrals
, 2007
"... We consider absolutely free nonassociative algebras and, more generally, absolutely free algebras with (maybe infinitely) many multilinear operations. Such algebras are described in terms of labeled reduced planar rooted trees. This allows to apply combinatorial techniques to study their Hilbert se ..."
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We consider absolutely free nonassociative algebras and, more generally, absolutely free algebras with (maybe infinitely) many multilinear operations. Such algebras are described in terms of labeled reduced planar rooted trees. This allows to apply combinatorial techniques to study their Hilbert series and the asymptotics of their coefficients. These algebras satisfy the NielsenSchreier property and their subalgebras are also free. Then, over a field of characteristic 0, we investigate the subalgebras of invariants under the action of a linear group, their sets of free generators and their Hilbert series. It has turned out that, except in the trivial cases, the algebra of invariants is never finitely generated. In important partial cases the Hilbert series of the algebras of invariants and the generating functions of their sets of free generators are expressed in terms of elliptic integrals.
Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras
 J. Algebra
"... Abstract. In commutative algebra, a Weitzenböck derivation is a nonzero triangular linear derivation of the polynomial algebra K[x1,..., xm] in several variables over a field K of characteristic 0. The classical theorem of Weitzenböck states that the algebra of constants is finitely generated. (This ..."
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Abstract. In commutative algebra, a Weitzenböck derivation is a nonzero triangular linear derivation of the polynomial algebra K[x1,..., xm] in several variables over a field K of characteristic 0. The classical theorem of Weitzenböck states that the algebra of constants is finitely generated. (This algebra coincides with the algebra of invariants of a single unipotent transformation.) In this paper we study the problem of finite generation of the algebras of constants of triangular linear derivations of finitely generated (not necessarily commutative or associative) algebras over K assuming that the algebras are free in some sense (in most of the cases relatively free algebras in varieties of associative or Lie algebras). In this case the algebra of constants also coincides with the algebra of invariants of some unipotent transformation. The main results are the following: 1. We show that the subalgebra of constants of a factor algebra can be lifted to the subalgebra of constants. 2. For all varieties of associative algebras which are not nilpotent in Lie sense the subalgebras of constants of the relatively free algebras of rank ≥ 2 are not finitely generated. 3. We describe the generators of the subalgebra of constants for all factor algebras K〈x, y〉/I modulo a GL2(K)invariant ideal I. 4. Applying known results from commutative algebra, we construct classes of automorphisms of the algebra generated by two generic 2 × 2 matrices. We obtain also some partial results on relatively free Lie algebras. 1.
INVARIANTS OF UNIPOTENT TRANSFORMATIONS ACTING ON NOETHERIAN RELATIVELY FREE ALGEBRAS
, 2004
"... Abstract. The classical theorem of Weitzenböck states that the algebra of invariants K[X] g of a single unipotent transformation g ∈ GLm(K) acting on the polynomial algebra K[X] = K[x1,..., xm] over a field K of characteristic 0 is finitely generated. This algebra coincides with the algebra of cons ..."
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Abstract. The classical theorem of Weitzenböck states that the algebra of invariants K[X] g of a single unipotent transformation g ∈ GLm(K) acting on the polynomial algebra K[X] = K[x1,..., xm] over a field K of characteristic 0 is finitely generated. This algebra coincides with the algebra of constants K[X] δ of a linear locally nilpotent derivation δ of K[X]. Recently the author and C. K. Gupta have started the study of the algebra of invariants Fm(V) g where Fm(V) is the relatively free algebra of rank m in a variety V of associative algebras. They have shown that Fm(V) g is not finitely generated if V contains the algebra UT2(K) of 2 ×2 upper triangular matrices. The main result of the present paper is that the algebra Fm(V) g is finitely generated if and only if the variety V does not contain the algebra UT2(K). As a byproduct of the proof we have established also the finite generation of the algebra of invariants T g nm where Tnm is the mixed trace algebra generated by m generic n × n matrices and the traces of their products.
COMPUTING WITH RATIONAL SYMMETRIC FUNCTIONS AND APPLICATIONS TO INVARIANT THEORY AND PIALGEBRAS
 SERDICA MATH. J. 38 (2012), 137–188
, 2012
"... Let K be a field of any characteristic. Let the formal power series f(x1,...,xd) = ∑ αnx n1 1 ···xnd d = ∑ m(λ)Sλ(x1,...,xd), αn,m(λ) ∈ K, be a symmetric function decomposed as a series of Schur functions. When f is a rational function whose denominator is a product of binomials of the ..."
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Let K be a field of any characteristic. Let the formal power series f(x1,...,xd) = ∑ αnx n1 1 ···xnd d = ∑ m(λ)Sλ(x1,...,xd), αn,m(λ) ∈ K, be a symmetric function decomposed as a series of Schur functions. When f is a rational function whose denominator is a product of binomials of the
Cocharacters of polynomial identities of block triangular matrices, arXiv: 1112.0792v1 [math.RA
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Central Polynomials, Weak Identities And Parallel Computers In Matrix Algebras
"... We present a new fast parallel divideandconquer type algorithm for computing symmetric polynomials of 5 5 matrices. These algorithms are directly extendable to higher order matrices and polynomials of any degree. This allows us to look for new weak polynomial identities in the algebra of 55 matri ..."
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We present a new fast parallel divideandconquer type algorithm for computing symmetric polynomials of 5 5 matrices. These algorithms are directly extendable to higher order matrices and polynomials of any degree. This allows us to look for new weak polynomial identities in the algebra of 55 matrices. Previous algorithms that were used to discover new weak polynomial identities for the 3 3 and 4 4 matrices would have required approximately T = 6:10 15 ops if applied to the 55 case. Our new algorithm required only about T 2=3 as many ops (appr.(3:1)10 10 ops). Our algorithm parallelizes pretty well. We used this algorithm to disprove a conjecture about the existence of weak identity of degree 15 in 11 indeterminates for 5 5 matrices. INTRODUCTION In the theory of the algebras with polynomial identities it is of particular interest to compute the dimension of the linear space F spanned by the following homogeneous polynomials of total degree n: f (x 1 ; :::; x p )...
Invariant and coinvariant spaces for the algebra of symmetric polynomials in . . .
, 2010
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2 COMPUTING WITH RATIONAL SYMMETRIC FUNCTIONS AND APPLICATIONS TO INVARIANT THEORY AND PIALGEBRAS
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