Commutative Polynomial Algebra Research Articles

Bases of fixed point subalgebras on nilpotent Leibniz algebras

Let K be a field of characteristic zero, X={x_(1,) x_2,…,x_n} and R_m={r_(1,) ,…,r_m} be two sets of variables, F be the free left nitpotent Leibniz algebra generated by X, and K[R_m ] be the commutative polynomial algebra generated by R_m over the base field K. The fixed point subalgebra of an automorphism φ is the subalgebra of F consisting of elements that are invariant under the automorphism. In this work, we consider specific automorphisms of F and determine the fixed point subalgebras of these automorphisms. Then, we find bases of these fixed point subalgebras. In addition, we get generators of these subalgebras as a free K[R_m ] -module.

İZİ SIFIR OLAN JENERİK MATRİSLER CEBİRİ İÇİN NOWICKI SANISI

A locally nilpotent linear derivation δ of the commutative polynomial algebra K[Xd ]=K[x1,…,xd ] of rank d is called Weitzenböck. It is well known that the subalgebra K[Xd ]^δ of K[Xd ] consisting of polynomials which are sent to zero by δ is finitely generated. Let the Weitzenböck derivation δ act on K[Xd,Yd ] such that δ(yi )=xi, δ(xi )=0, i=1,…,d. The explicit form of generators of the algebra K[Xd,Yd ]^δ was conjectured by Nowicki in 1994. In this study, we consider the Nowicki conjecture in the algebra W generated by two traceless generic matrices with entries from commutative associative unitary polynomial algebra with six variables, and obtain the free generators of the algebra W^δ of constants in this algebra.

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 $$n\ge 2$$ 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 $$F_n^{S_n}$$ .

Constructive arithmetics in Ore localizations enjoying enough commutativity

This paper continues a research program on constructive investigations of non-commutative Ore localizations, initiated in our previous papers, and particularly touches the constructiveness of arithmetics within such localizations. Earlier we have introduced monoidal, geometric and rational types of localizations of domains as objects of our studies. Here we extend this classification to rings with zero divisors and consider Ore sets of the mentioned types which are commutative enough: such a set either belongs to a commutative algebra or it is central or its elements commute pairwise. By using the systematic approach we have developed before, we prove that arithmetic within the localization of a commutative polynomial algebra is constructive and give the necessary algorithms. We also address the important question of computing the local closure of ideals which is also known as the desingularization, and present an algorithm for the computation of the symbolic power of a given ideal in a commutative ring. We also provide algorithms to compute local closures for certain non-commutative rings with respect to Ore sets with enough commutativity.

The structure monoid and algebra of a non-degenerate set-theoretic solution of the Yang–Baxter equation

For a finite involutive non-degenerate solution ( X , r ) (X,r) of the Yang–Baxter equation it is known that the structure monoid M ( X , r ) M(X,r) is a monoid of I-type, and the structure algebra K [ M ( X , r ) ] K[M(X,r)] over a field K K shares many properties with commutative polynomial algebras; in particular, it is a Noetherian PI-domain that has finite Gelfand–Kirillov dimension. In this paper we deal with arbitrary finite (left) non-degenerate solutions. Although the structure of both the monoid M ( X , r ) M(X,r) and the algebra K [ M ( X , r ) ] K[M(X,r)] is much more complicated than in the involutive case, we provide some deep insights. In this general context, using a realization of Lebed and Vendramin of M ( X , r ) M(X,r) as a regular submonoid in the semidirect product A ( X , r ) ⋊ Sym ⁡ ( X ) A(X,r)\rtimes \operatorname {Sym} (X) , where A ( X , r ) A(X,r) is the structure monoid of the rack solution associated to ( X , r ) (X,r) , we prove that K [ M ( X , r ) ] K[M(X,r)] is a finite module over a central affine subalgebra. In particular, K [ M ( X , r ) ] K[M(X,r)] is a Noetherian PI-algebra of finite Gelfand–Kirillov dimension bounded by | X | |X| . We also characterize, in ring-theoretical terms of K [ M ( X , r ) ] K[M(X,r)] , when ( X , r ) (X,r) is an involutive solution. This characterization provides, in particular, a positive answer to the Gateva-Ivanova conjecture concerning cancellativity of M ( X , r ) M(X,r) . These results allow us to control the prime spectrum of the algebra K [ M ( X , r ) ] K[M(X,r)] and to describe the Jacobson radical and prime radical of K [ M ( X , r ) ] K[M(X,r)] . Finally, we give a matrix-type representation of the algebra K [ M ( X , r ) ] / P K[M(X,r)]/P for each prime ideal P P of K [ M ( X , r ) ] K[M(X,r)] . As a consequence, we show that if K [ M ( X , r ) ] K[M(X,r)] is semiprime, then there exist finitely many finitely generated abelian-by-finite groups, G 1 , … , G m G_1,\dotsc ,G_m , each being the group of quotients of a cancellative subsemigroup of M ( X , r ) M(X,r) such that the algebra K [ M ( X , r ) ] K[M(X,r)] embeds into M v 1 ⁡ ( K [ G 1 ] ) × ⋯ × M v m ⁡ ( K [ G m ] ) \operatorname {M}_{v_1}(K[G_1])\times \dotsb \times \operatorname {M}_{v_m}(K[G_m]) , a direct product of matrix algebras.

Connected (graded) Hopf algebras

We study algebraic and homological properties of two classes of infinite-dimensional Hopf algebras over an algebraically closed field k k of characteristic 0 0 . The first class consists of those Hopf k k -algebras that are connected graded as algebras, and the second class are those Hopf k k -algebras that are connected as coalgebras. For many but not all of the results presented here, the Hopf algebras are assumed to have finite Gel’fand–Kirillov dimension. It is shown that if the Hopf algebra H H is a connected graded Hopf algebra of finite Gel’fand–Kirillov dimension n n , then H H is a noetherian domain which is Cohen–Macaulay, Artin–Schelter regular, and Auslander regular of global dimension n n . It has S 2 = I d H S^2 = \mathrm {Id}_H , and it is Calabi–Yau. Detailed information is also provided about the Hilbert series of H H . Our results leave open the possibility that the first class of algebras is (properly) contained in the second. For this second class, the Hopf k k -algebras of finite Gel’fand–Kirillov dimension n n with connected coalgebra, the underlying coalgebra is shown to be Artin–Schelter regular of global dimension n n . Both these classes of Hopf algebras share many features in common with enveloping algebras of finite-dimensional Lie algebras. For example, an algebra in either of these classes satisfies a polynomial identity only if it is a commutative polynomial algebra. Nevertheless, we construct, as one of our main results, an example of a Hopf k k -algebra H H of Gel’fand–Kirillov dimension 5, which is connected graded as an algebra and connected as a coalgebra, but is not isomorphic as an algebra to U ( g ) U(\mathfrak {g}) for any Lie algebra g \mathfrak {g} .

Left semi-braces and solutions of the Yang–Baxter equation

Abstract Let r : X 2 → X 2 {r\colon X^{2}\rightarrow X^{2}} be a set-theoretic solution of the Yang–Baxter equation on a finite set X. It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive, then the algebra K ⁢ 〈 x ∈ X ∣ x ⁢ y = u ⁢ v ⁢ if ⁢ r ⁢ ( x , y ) = ( u , v ) 〉 {K\langle x\in X\mid xy=uv\text{ if }r(x,y)=(u,v)\rangle} shares many properties with commutative polynomial algebras in finitely many variables; in particular, this algebra is Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions r B {r_{B}} that are associated to a left semi-brace B; such solutions can be degenerate or can even be idempotent. In order to do so, we first describe such semi-braces and then prove some decompositions results extending those of Catino, Colazzo and Stefanelli.

PBW-Basis for Universal Enveloping Algebras of Differential Graded Poisson Algebras

For any differential graded (DG for short) Poisson algebra A given by generators and relations, we give a “formula” for computing the universal enveloping algebra $$A^e$$ of A. Moreover, we prove that $$A^e$$ has a Poincare–Birkhoff–Witt basis provided that A is a graded commutative polynomial algebra. As an application of the PBW-basis, we show that a DG symplectic ideal of a DG Poisson algebra A is the annihilator of a simple DG Poisson A-module, where A is the DG Poisson homomorphic image of a DG Poisson algebra R whose underlying algebra structure is a graded commutative polynomial algebra.

ℤ2-GRADED POISSON ALGEBRAS, THEIR DEFORMATIONS AND COHOMOLOGY IN LOW DIMENSIONS

We study ℤ2-graded Poisson structures defined on ℤ2-graded commutative polynomial algebras. In small-dimensional cases, we obtain the associated Poisson ℤ2-graded cohomology and in some cases, deformations of these Poisson brackets and P∞-algebra structures. We highlight differences and analogies between this ℤ2-graded context and the non-graded context, by studying, for example, the links between Poisson cohomology and singularities.

Matrix representations of finitely generated Grassmann algebras and some consequences

We prove that the m-generated Grassmann algebra can be embedded into a 2m−1×2m−1 matrix algebra over a factor of a commutative polynomial algebra in m indeterminates. Cayley-Hamilton and standard identities for n × n matrices over the m-generated Grassmann algebra are derived from this embedding. Other related embedding results are also presented.

Introduction to Non Commutative Algebraic Geometry

Ordinary commutative algebraic geometry is based on commutative polynomial algebras over an algebraically closed eld k. Here we make a natural generalization to matrix polynomial k-algebras which are non-commutative coordinate rings of non-commutative varieties

Extended letterplace correspondence for nongraded noncommutative ideals and related algorithms

Let K〈xi〉 be the free associative algebra generated by a finite or a countable number of variables xi. The notion of "letterplace correspondence" introduced in [R. La Scala and V. Levandovskyy, Letterplace ideals and non-commutative Gröbner bases, J. Symbolic Comput. 44(10) (2009) 1374–1393; R. La Scala and V. Levandovskyy, Skew polynomial rings, Gröbner bases and the letterplace embedding of the free associative algebra, J. Symbolic Comput. 48 (2013) 110–131] for the graded (two-sided) ideals of K〈xi〉 is extended in this paper also to the nongraded case. This amounts to the possibility of modelizing nongraded noncommutative presented algebras by means of a class of graded commutative algebras that are invariant under the action of the monoid ℕ of natural numbers. For such purpose we develop the notion of saturation for the graded ideals of K〈xi,t〉, where t is an extra variable and for their letterplace analogues in the commutative polynomial algebra K[xij, tj], where j ranges in ℕ. In particular, one obtains an alternative algorithm for computing inhomogeneous noncommutative Gröbner bases using just homogeneous commutative polynomials. The feasibility of the proposed methods is shown by an experimental implementation developed in the computer algebra system Maple and by using standard routines for the Buchberger algorithm contained in Singular.

A deformation of commutative polynomial algebras in even numbers of variables

Abstract We introduce and study a deformation of commutative polynomial algebras in even numbers of variables. We also discuss some connections and applications of this deformation to the generalized Laguerre orthogonal polynomials and the interchanges of right and left total symbols of differential operators of polynomial algebras. Furthermore, a more conceptual re-formulation for the image conjecture [18] is also given in terms of the deformed algebras. Consequently, the well-known Jacobian conjecture [8] is reduced to an open problem on this deformation of polynomial algebras.

On Some Degenerate Deformations of Commutative Polynomial Algebras

This paper gives a classification of the prime ideals, primitive ideals, and irreducible representations of where K is an algebraically closed field, n ≥ 3, and β ij ∈ K. The classification of the prime ideals of S proves that, under the Zariski topology, the topological dimension of spec S is no greater than the Gelfand-Kirillov dimension of S. Included is an appendix by A. Berliner, in which it is shown that the finite dimensional irreducible representations are all one-dimensional.

Positive cones in enveloping algebras

The concept of strong ordering on enveloping algebras of finite-dimensional Lie algebras is introduced and studied as a generalization of the corresponding notion for the commutative polynomial algebra. A linear functional f on an enveloping algebra E ( G) is called strongly positive if f( x) ⩾ 0 for all x ϵ E ( G) which are mapped on positive operators for all G-integrable irreducible representations of E ( G). We prove that for each real connected Lie group G ≠ R 1 there are positive, not strongly positive, linear functionals on E ( G). A non-commutative problem of moments is defined. It has a solution iff the corresponding linear functional is strongly positive.