ON THE USE OF GRAPHS FOR COMPUTING A BASIS, GROWTH AND HILBERT SERIES OF ASSOCIATIVE ALGEBRAS

1991 ◽  
Vol 68 (2) ◽  
pp. 417-428 ◽  
Author(s):  
V A Ufnarovskiĭ
Keyword(s):  
Hilbert Series ◽  
Associative Algebras

Weitzenböck derivations of free metabelian associative algebras

2017 ◽  
Vol 16 (03) ◽  
pp. 1750041 ◽  
Author(s):  
Rumen Dangovski ◽  
Vesselin Drensky ◽  
Şehmus Fındık
Keyword(s):  
Hilbert Series ◽  
Polynomial Algebra ◽  
Finitely Generated ◽  
Associative Algebras ◽  
Triangular Matrices ◽  
Commutator Ideal ◽  
Several Variables ◽  
Locally Nilpotent ◽  
Derivation Formula ◽  
Infinite Set

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

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

scholarly journals The Hilbert Series of a Linear Symplectic Circle Quotient

2014 ◽  
Vol 23 (1) ◽  
pp. 46-65 ◽  
Author(s):  
Hans-Christian Herbig ◽  
Christopher Seaton
Keyword(s):  
Hilbert Series

scholarly journals 2, 12, 117, 1959, 45171, 1170086, …: a Hilbert series for the QCD chiral Lagrangian

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Lukáš Gráf ◽  
Brian Henning ◽  
Xiaochuan Lu ◽  
Tom Melia ◽  
Hitoshi Murayama
Keyword(s):  
Hilbert Series ◽  
Charge Conjugation ◽  
External Fields ◽  
Dynkin Diagrams ◽  
Non Linear ◽  
Very High ◽  
Chiral Lagrangian

Abstract We apply Hilbert series techniques to the enumeration of operators in the mesonic QCD chiral Lagrangian. Existing Hilbert series technologies for non-linear realizations are extended to incorporate the external fields. The action of charge conjugation is addressed by folding the $$ \mathfrak{su}(n) $$ su n Dynkin diagrams, which we detail in an appendix that can be read separately as it has potential broader applications. New results include the enumeration of anomalous operators appearing in the chiral Lagrangian at order p8, as well as enumeration of CP-even, CP-odd, C-odd, and P-odd terms beginning from order p6. The method is extendable to very high orders, and we present results up to order p16.(The title sequence is the number of independent C-even and P-even operators in the mesonic QCD chiral Lagrangian with three light flavors of quarks, at chiral dimensions p2, p4, p6, …)

scholarly journals Total Differentiability and Monogenicity for Functions in Algebras of Order 4

2021 ◽  
Vol 15 (2) ◽  
Author(s):  
I. Sabadini ◽  
D. C. Struppa
Keyword(s):  
Analytic Functions ◽  
Associative Algebras ◽  
Basic Tool ◽  
Algebraic Analysis ◽  
Italian School ◽  
Relevant Role

AbstractIn this paper we discuss some notions of analyticity in associative algebras with unit. We also recall some basic tool in algebraic analysis and we use them to study the properties of analytic functions in two algebras of dimension four that played a relevant role in some work of the Italian school, but that have never been fully investigated.

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