Laurent Polynomial Algebra Research Articles

Transposed Poisson structures on Virasoro-type (super)algebras

We explore transposed Poisson structures on the Lie algebra: the deformed twisted Schrödinger-Virasoro algebra D(λ), as well as two Lie superalgebras: the super-BMS3 algebra and the twisted N=1 Schrödinger-Neveu-Schwarz algebra. Initially, we demonstrate the absence of non-trivial transposed Poisson structures on the Lie algebra D(λ) for λ≠1 and provide an example of a transposed Poisson algebra with associative and Lie parts isomorphic to the algebra of triadic extended Laurent polynomials and D(1). Subsequently, we establish that the super-BMS3 algebra possesses non-trivial 12-superderivations but lacks a non-trivial transposed Poisson structure. Finally, we prove that the twisted N=1 Schrödinger-Neveu-Schwarz algebra does not have non-trivial 12-superderivations and thus lacks non-trivial transposed Poisson structures.

Exel–Pardo algebras with a twist

Takeshi Katsura associated a C^{\ast} -algebra C^{\ast}_{A,B} to a pair of square matrices A\ge 0 and B of the same size with integral coefficients and gave sufficient conditions on (A,B) to be simple purely infinite (SPI). We call such a pair a KSPI pair. It follows from a result of Katsura that any separable C^{\ast} -algebra \mathfrak{A} which is a cone of a map \xi:C(S^{1})^{n}\rightarrow C(S^{1})^{n} in Kasparov’s triangulated category KK is KK -isomorphic to C^{\ast}_{A,B} for some KSPI pair (A,B) . In this article, we introduce, for the data of a commutative ring \ell , non-necessarily square matrices A , B and a matrix C of the same size with coefficients in the group \mathcal{U}(\ell) of invertible elements, an \ell -algebra \mathcal{O}_{A,B}^{C} , the twisted Katsura algebra of the triple (A,B,C) . When A and B are square and C is trivial, we recover the Katsura \ell -algebra first considered by Enrique Pardo and Ruy Exel. We show that if \ell is a field of characteristic 0 and (A,B) is KSPI, then \mathcal{O}_{A,B}^{C} is SPI, and that any \ell -algebra which is a cone of a map \xi:\ell[t,t^{-1}]^{n}\rightarrow \ell[t,t^{-1}]^{n} in the triangulated bivariant algebraic K -theory category kk is kk -isomorphic to \mathcal{O}_{A,B}^{C} for some (A,B,C) as above so that (A,B) is KSPI. Katsura \ell -algebras are examples of the Exel–Pardo algebras L(G,E,\phi) associated to a group G acting on a directed graph E and a 1 -cocycle \phi:G\times E^{1}\rightarrow G . Similarly, twisted Katsura algebras are examples of the twisted Exel–Pardo \ell -algebras L(G,E,\phi_{c}) we introduce in the current article; they are associated to data (G,E,\phi) as above twisted by a 1 -cocycle c:G\times E^{1}\rightarrow \mathcal{U}(\ell) . The algebra L(G,E,\phi_{c}) can be variously described by generators and relations, as a quotient of a twisted semigroup algebra, as a twisted Steinberg algebra, as a corner skew Laurent polynomial algebra, and as a universal localization of a tensor algebra. We use each of these guises of L(G,E,\phi_{c}) to study its K -theoretic, regularity, and (purely infinite) simplicity properties. For example, we show that if \ell is a field of characteristic 0 , G and E are countable, and E is regular, then L(G,E,\phi_{c}) is simple whenever the Exel–Pardo C^{\ast} -algebra C^{\ast}(G,E,\phi) is, and is SPI if in addition the Leavitt path algebra L(E) is SPI.

Simple Harish-Chandra modules over the superconformal current algebra

In this paper, we classify the simple Harish-Chandra modules over the superconformal current algebra gˆ, which is the semi-direct sum of the N=1 superconformal algebra with the affine Lie superalgebra g˙⊗A⊕CC1, where g˙ is a finite-dimensional simple Lie algebra, and A is the tensor product of the Laurent polynomial algebra and the Grassmann algebra. As an application, we can directly get the classification of the simple Harish-Chandra modules over the N=1 Heisenberg-Virasoro algebra.

Formula omitted]-graded identities of the Lie algebras U1

Let K be an infinite field of characteristic different from two and let U1 be the Lie algebra of the derivations of the algebra of Laurent polynomials K[t,t−1]. The algebra U1 admits a natural Z-grading. We provide a basis for the graded identities of U1 and prove that they do not admit any finite basis. Moreover, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension ≤1, if a basis of the multilinear graded identities is known. As a consequence of this latter result we are able to provide a basis of the graded identities of the Lie algebra W1 of the derivations of the polynomial ring K[t]. The Z-graded identities for W1, in characteristic 0, were described in [8]. As a consequence of our results, we give an alternative proof of the main result, Theorem 1, in [8], and generalize it to positive characteristic. We also describe a basis of the graded identities for the special linear Lie algebra slq(K) with the Pauli gradings where q is a prime number.

Realization of representations of the Hom-Lie algebra 𝔰𝔩 q (2, ℂ) of Jackson

The weight modules of the Lie algebra are well known. In the first part of this paper we deal with a realization of weight modules of in the space , where is the algebra of Laurent polynomials. In the second part, we consider the Hom-Lie algebra of Jackson where . The q-analogue of the above realization in the space is considered. We obtain two kinds of q-modules. The regular q-modules which have limits the modules obtained in the classical realization when q goes to 1 . The other q-modules have no limits when q goes to 1 and they are called singular modules.

Classifications of Prime Ideals and Simple Modules of the Weyl Algebra $A_1$ in Prime Characteristic

Let $K$ be an arbitrary field of characteristic $p>0$. Classifications of prime ideals and simple modules are obtained for the Weyl algebra $A_1=K\langle x,\partial \, : \, \partial x-x\partial =1\rangle$, the skew polynomial algebra $\mathbb{A} = K[h][x;\sigma ]$ and the skew Laurent polynomial algebra $\cal{A} := K[h][x^{\pm 1};\sigma ]$ where $\sigma (h) = h-1$. In particular, classifications of prime, completely prime, maximal and primitive ideals are obtained for the above algebras. The quotient ring (of fractions) of each prime factor algebra of $A_1$, $\mathbb{A}$ and $\cal{A}$ is described. It is either a matrix algebra over a field or else a cyclic algebra. These descriptions are a key fact in the classification of completely prime ideals and simple modules for the algebras above.

Growth of generalized Weyl algebras over polynomial algebras and Laurent polynomial algebras

We study the growth and Gelfand-Kirillov dimension (GK-dimension) of the generalized Weyl algebra (GWA) A = D(σ, a), where D is a polynomial algebra or a Laurent polynomial algebra. Several necessary and sufficient conditions for GKdim(A) = GKdim(D) + 1 are given. In particular, we prove a dichotomy of the GK-dimension of GWAs over the polynomial algebra in two indeterminates, namely, GKdim(A) is either 3 or ∞ in this case. Our results generalize several existing results in the literature and can be applied to determine the growth, GK-dimension, simplicity and cancellation properties of some GWAs.

-graded identities of the Lie algebras in characteristic 2

AbstractLetKbe any field of characteristic two and let$U_1$and$W_1$be the Lie algebras of the derivations of the algebra of Laurent polynomials$K[t,t^{-1}]$and of the polynomial ringK[t], respectively. The algebras$U_1$and$W_1$are equipped with natural$\mathbb{Z}$-gradings. In this paper, we provide bases for the graded identities of$U_1$and$W_1$, and we prove that they do not admit any finite basis.

On $p$-adic entropy of some solenoid dynamical systems

To a dynamical system is attached a non-negative real number called entropy. In 1990, Lind, Schmidt and Ward proved that the entropy for the dynamical system induced by the Laurent polynomial algebra over the ring of the rational integers is described by the Mahler measure. In 2009, Deninger introduced the $p$-adic entropy and obtained a $p$-adic analogue of Lind-Schmidt-Ward's theorem by using the $p$-adic Mahler measures. In this paper, we prove the existence and the explicit formula about $p$-adic entropies for two dynamical systems; one is induced by the Laurent polynomial algebra over the ring of the integers of a number field $K$, and the other is defined by the solenoid.

Virtual and arrow Temperley–Lieb algebras, Markov traces, and virtual link invariants

Let [Formula: see text] be the algebra of Laurent polynomials in the variable [Formula: see text] and let [Formula: see text] be the algebra of Laurent polynomials in the variable [Formula: see text] and standard polynomials in the variables [Formula: see text] For [Formula: see text] we denote by [Formula: see text] the virtual braid group on [Formula: see text] strands. We define two towers of algebras [Formula: see text] and [Formula: see text] in terms of diagrams. For each [Formula: see text] we determine presentations for both, [Formula: see text] and [Formula: see text]. We determine sequences of homomorphisms [Formula: see text] and [Formula: see text], we determine Markov traces [Formula: see text] and [Formula: see text], and we show that the invariants for virtual links obtained from these Markov traces are the [Formula: see text]-polynomial for the first trace and the arrow polynomial for the second trace. We show that, for each [Formula: see text] the standard Temperley–Lieb algebra [Formula: see text] embeds into both, [Formula: see text] and [Formula: see text], and that the restrictions to [Formula: see text] of the two Markov traces coincide.

Simple weight modules with finite-dimensional weight spaces over Witt superalgebras

Let Am,n be the tensor product of the Laurent polynomial algebra in m even variables and the exterior algebra in n odd variables over the complex field C, and the Witt superalgebra Wm,n be the Lie superalgebra of superderivations of Am,n. In this paper, we classify the simple weight Wm,n modules with finite-dimensional weight spaces with respect to the standard Cartan subalgebra of Wm,0. Every such module is either a simple quotient of a tensor module or a module of highest weight type.

Compactness property of Lie polynomials in the creation and annihilation operators of the q-oscillator

Given a real number q such that $$0<q<1$$ , the natural setting for the mathematics of a q-oscillator is an infinite-dimensional, separable Hilbert space that is said to provide an interpolation between the Bargmann–Segal space of holomorphic functions and the Hardy–Lebesgue space of analytic functions. The traditional basis states are interrelated by the creation and annihilation operators. Since the commutation relation is q-deformed, the commutator algebra for the creation and annihilation operators is not a low-dimensional Lie algebra like that for the canonical commutation relation. In this study, a characterization of the elements of the said commutator algebra is obtained using spectral properties of the creation and annihilation operators as these faithfully represent the generators of a q-deformed Heisenberg algebra. The derived algebra of the commutator algebra is precisely the set of all compact operators, and the resulting Calkin algebra is algebraically isomorphic to the complex algebra of Laurent polynomials in one indeterminate. As for any operator that is not in the commutator algebra, the action of such an operator on an arbitrary basis state can be approximated by a Lie series of elements from the commutator algebra.

Noncommutative Catalan Numbers

The goal of this paper is to introduce and study noncommutative Catalan numbers$$C_n$$ which belong to the free Laurent polynomial algebra $$\mathcal {L}_n$$ in n generators. Our noncommutative numbers admit interesting (commutative and noncommutative) specializations, one of them related to Garsia–Haiman (q, t)-versions, another—to solving noncommutative quadratic equations. We also establish total positivity of the corresponding (noncommutative) Hankel matrices $$H_n$$ and introduce accompanying noncommutative binomial coefficients.

Harish-Chandra modules for divergence zero vector fields on a torus

Let $A=\mathbb{C}[t_1^{\pm1},t_2^{\pm1}]$ be the algebra of Laurent polynomials in two variables and $B$ be the set of skew derivations of $A$. Let $L$ be the universal central extension of the derived Lie subalgebra of the Lie algebra $A\rtimes B$. Set $\widetilde{L}=L\oplus\mathbb{C} d_1\oplus\mathbb{C} d_2$, where $d_1$, $d_2$ are two degree derivations. A Harish-Chandra module is defined as an irreducible weight module with finite dimensional weight spaces. In this paper, we prove that a Harish-Chandra module of the Lie algebra $\widetilde{L}$ is a uniformly bounded module or a generalized highest weight (GHW for short) module. Furthermore, we prove that the nonzero level Harish-Chandra modules of $\widetilde{L}$ are GHW modules. Finally, we classify all the GHW Harish-Chandra modules of $\widetilde{L}$.

Jet modules for the centerless Virasoro-like algebra

In this paper, we studied the jet modules for the centerless Virasoro-like algebra which is the Lie algebra of the Lie group of the area-preserving diffeomorphisms of a [Formula: see text]-torus. The jet modules are certain natural modules over the Lie algebra of semi-direct product of the centerless Virasoro-like algebra and the Laurent polynomial algebra in two variables. We reduce the irreducible jet modules to the finite-dimensional irreducible modules over some infinite-dimensional Lie algebra and then characterize the irreducible jet modules with irreducible finite dimensional modules over [Formula: see text]. To determine the indecomposable jet modules, we use the technique of polynomial modules in the sense of [Irreducible representations for toroidal Lie algebras, J. Algebras 221 (1999) 188–231; Weight modules over exp-polynomial Lie algebras, J. Pure Appl. Algebra 191 (2004) 23–42]. Consequently, indecomposable jet modules are described using modules over the algebra [Formula: see text], which is the “positive part” of a Block type algebra studied first by [Some infinite-dimensional simple Lie algebras in characteristic [Formula: see text] related to those of Block, J. Pure Appl. Algebra 127(2) (1998) 153–165] and recently by [A [Formula: see text]-graded generalization of the Witt-algebra, preprint; Classification of simple Lie algebras on a lattice, Proc. London Math. Soc. 106(3) (2013) 508–564]).

Counting the Ideals of Given Codimension of the Algebra of Laurent Polynomials in Two Variables

We establish an explicit formula for the number Cn(q) of ideals of codimension (colength) n of the algebra Fq[x,y,x−1,y−1] of Laurent polynomials in two variables over a finite field Fq of cardinality q. This number is a palindromic polynomial of degree 2n in q. Moreover, Cn(q)=(q−1)2Pn(q), where Pn(q) is another palindromic polynomial; the latter is a q-analogue of the sum of divisors of n, which happens to be the number of subgroups of Z2 of index n.

Biderivations of the higher rank Witt algebra without anti-symmetric condition

Abstract The Witt algebra 𝔚 d of rank d(≥ 1) is the derivation algebra of Laurent polynomial algebras in d commuting variables. In this paper, all biderivations of 𝔚 d without anti-symmetric condition are determined. As an applications, commutative post-Lie algebra structures on 𝔚 d are obtained. Our conclusions recover and generalize results in the related papers on low rank or anti-symmetric cases.

$\mathbb A^1$-homotopy invariants of corner skew Laurent polynomial algebras

In this note we prove some structural properties of all the \mathbb A^1 -homotopy invariants of corner skew Laurent polynomial algebras. As an application, we compute the mod- l algebraic K -theory of Leavitt path algebras using solely the kernel/cokernel of the incidence matrix. This leads naturally to some vanishing and divisibility properties of the K -theory of these algebras.

GRADED CHAIN CONDITIONS AND LEAVITT PATH ALGEBRAS OF NO-EXIT GRAPHS

We obtain a complete structural characterization of Cohn–Leavitt algebras over no-exit objects as graded involutive algebras. Corollaries of this result include graph-theoretic conditions characterizing when a Leavitt path algebra is a directed union of (graded) matricial algebras over the underlying field and over the algebra of Laurent polynomials and when the monoid of isomorphism classes of finitely generated projective modules is atomic and cancelative. We introduce the nonunital generalizations of graded analogs of noetherian and artinian rings, graded locally noetherian and graded locally artinian rings, and characterize graded locally noetherian and graded locally artinian Leavitt path algebras without any restriction on the cardinality of the graph. As a consequence, we relax the assumptions of the Abrams–Aranda–Perera–Siles characterization of locally noetherian and locally artinian Leavitt path algebras.

Length two extensions of modules for the Witt algebra

ABSTRACTIn this paper, we establish an explicit classification of length two extensions of tensor modules for the Witt algebra using the cohomology of the Witt algebra with coefficients in the module of the space of homomorphisms between the two modules of interest. To do this we extended our module to a module that has a compatible action of the commutative algebra of Laurent polynomials in one variable. In this setting, we are be able to directly compute all possible 1-cocycles.