Witt Type Research Articles

Superbiderivations of Simple Modular Lie Superalgebras of Witt Type and Special Type

Let [Formula: see text] be a simple modular Lie superealgebra of Witt type or special type over an algebraically closed field of characteristic [Formula: see text]. In this paper, all the superbiderivations of [Formula: see text] are studied. By means of weight space decompositions with respect to a suitable torus and the standard [Formula: see text]-grading structures of [Formula: see text], we show that the supersymmetric superbiderivations of [Formula: see text] are zero. Generalizing a result on the skewsymmetric biderivation of Lie algebras to the super case, we find that all the superbiderivations of [Formula: see text] are inner. As applications, the linear supercommuting maps and the supercommutative post-Lie superalgebra structures on [Formula: see text] are described.

Transposed Poisson structures on Witt type algebras

We describe 12-derivations, and hence transposed Poisson algebra structures, on Witt type Lie algebras V(f), where f:Γ→C is non-trivial and f(0)=0. More precisely, if |f(Γ)|≥4, then all the transposed Poisson algebra structures on V(f) are mutations of the group algebra structure (V(f),⋅) on V(f). If |f(Γ)|=3, then we obtain the direct sum of 3 subspaces of V(f), corresponding to cosets of Γ0 in Γ, with multiplications being different mutations of ⋅. The case |f(Γ)|=2 is more complicated, but also deals with certain mutations of ⋅. As a consequence, new Lie algebras that admit non-trivial Hom-Lie algebra structures are found.

Generalized quadratic commutator algebras of PBW-type

In recent years, various nonlinear algebraic structures have been obtained in the context of quantum systems as symmetry algebras, Painlevé transcendent models, and missing label problems. In this paper, we treat all these algebras as instances of the class of quadratic (and higher degree) commutator bracket algebras of Poincaré–Birkhoff–Witt type. We provide a general approach for simplifying the constraints arising from the diamond lemma and apply this in particular to give a comprehensive analysis of the quadratic case. We present new examples of quadratic algebras, which admit a cubic Casimir invariant. The connection with other approaches, such as Gröbner bases, is developed, and we suggest how our explicit and computational techniques can be relevant in other contexts.

Enriched pre-Lie operads and freeness theorems

In this paper, we study the \mathcal{C} -enriched pre-Lie operad defined by Calaque and Willwacher for any Hopf cooperad \mathcal{C} to produce conceptual constructions of the operads acting on various deformation complexes. Maps between Hopf cooperads lead to maps between the corresponding enriched pre-Lie operads; we prove criteria for the module action of the domain on the codomain to be free, on the left and on the right. In particular, this implies a new functorial Poincaré–Birkhoff–Witt type theorem for universal enveloping brace algebras of pre-Lie algebras.

On the orbital stability of periodic trajectories of a class of discontinuous systems

AbstractThis paper focuses on the stability analysis of periodic trajectories for non‐autonomous systems whose right‐hand sides are time‐periodic discontinuous functions. Such systems arise, in particular, in optimal control problems for nonlinear control systems with periodic bang‐bang inputs. We obtain sufficient orbital stability conditions of the Andronov–Witt type for this class of systems. The proposed stability conditions are applied to the analysis of periodic extremal trajectories of the controlled nonlinear chemical reaction model.

Quantization of the Lie superalgebra of Witt type

In this paper, we use the quantization technique by a Drinfel’d twist to quantize the Lie superalgebra of Witt type, whose Lie super-bialgebra structures were recently determined by Yue and Zhu, and we obtain a new class of non-commutative and non-cocommutative Hopf superalgebras.

Lie super-bialgebra structures on the Lie superalgebra of Witt type

In this article, Lie super-bialgebra structures on the Lie superalgebra of Witt type are investigated. They are shown to be triangular coboundary. As a byproduct, we obtain that the first cohomology group is trivial.

Cartan Invariants for Witt Lie Superalgebras with p-Characters of Height at Most One

We investigate projective modules of the reduced enveloping superalgebra $ U_{\chi }(\frak {g}) $ of the Lie superalgebra $ \frak {g}$ of Witt type with p-character χ of height at most one. We give a formula for the Cartan invariants of $ U_{\chi }(\frak {g}) $ which implies that $ U_{\chi }(\frak {g}) $ has only one block.

Functorial PBW theorems for post-Lie algebras

Using the categorical approach to Poincaré–Birkhoff–Witt type theorems from our previous work with Tamaroff, we prove three such theorems: for universal enveloping Rota–Baxter algebras of tridendriform algebras, for universal enveloping Rota–Baxter Lie algebras of post-Lie algebras, and for universal enveloping tridendriform algebras of post-Lie algebras. Similar results, though without functoriality of the PBW isomorphisms, were recently obtained by Gubarev. Our methods are completely different and mainly rely on methods of rewriting theory for shuffle operads.Communicated by Pavel Kolesnikov

Classification of Simple Cuspidal Modules Over a Lattice Lie Algebra of Witt Type

AbstractLet $W_{\unicode[STIX]{x1D70B}}$ be the lattice Lie algebra of Witt type associated with an additive inclusion $\unicode[STIX]{x1D70B}:\mathbb{Z}^{N}{\hookrightarrow}\mathbb{C}^{2}$ with $N>1$. In this article, the classification of simple $\mathbb{Z}^{N}$-graded $W_{\unicode[STIX]{x1D70B}}$-modules, whose multiplicities are uniformly bounded, is given.

Character Formulas for a Class of Simple Restricted Modules over the Simple Lie Superalgebras of Witt Type

Let F be an algebraically closed field of prime characteristic, and W(m, n, 1) be the simple restricted Lie superalgebra of Witt type over F, which is the Lie superalgebra of superderivations of the superalgebra $$\mathfrak{A}(m;1)\otimes\wedge(n)$$, where $$\mathfrak{A}(m;1)$$ is the truncated polynomial algebra with m indeterminants and ∧(n) is the Grassmann algebra with n indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of W(m, n, 1) with atypical weights of type I.

A note on type 2 q-Bernoulli and type 2 q-Euler polynomials

As is well known, power sums of consecutive nonnegative integers can be expressed in terms of Bernoulli polynomials. Also, it is well known that alternating power sums of consecutive nonnegative integers can be represented by Euler polynomials. In this paper, we show that power sums of consecutive positive odd q-integers can be expressed by means of type 2 q-Bernoulli polynomials. Also, we show that alternating power sums of consecutive positive odd q-integers can be represented by virtue of type 2 q-Euler polynomials. The type 2 q-Bernoulli polynomials and type 2 q-Euler polynomials are introduced respectively as the bosonic p-adic q-integrals on mathbb{Z}_{p} and the fermionic p-adic q-integrals on mathbb{Z}_{p}. Along the way, we will obtain Witt type formulas and explicit expressions for those two newly introduced polynomials.

Some identities of special numbers and polynomials arising from p-adic integrals on mathbb{Z}_{p}

In recent years, studying degenerate versions of various special polynomials and numbers has attracted many mathematicians. Here we introduce degenerate type 2 Bernoulli polynomials, fully degenerate type 2 Bernoulli polynomials, and degenerate type 2 Euler polynomials, and their corresponding numbers, as degenerate and type 2 versions of Bernoulli and Euler numbers. Regarding those polynomials and numbers, we derive some identities, distribution relations, Witt type formulas, and analogues for the Bernoulli interpretation of powers of the first m positive integers in terms of Bernoulli polynomials. The present study was done by using the bosonic and fermionic p-adic integrals on mathbb{Z}_{p}.

Free Gelfand–Dorfman–Novikov superalgebras and a Poincaré–Birkhoff–Witt type theorem

We construct linear bases of free Gelfand–Dorfman–Novikov (GDN) superalgebras. As applications, we prove a Poincaré–Birkhoff–Witt (PBW) type theorem, that is, every GDN superalgebra can be embedded into its universal enveloping associative differential supercommuative algebra. An Engel theorem is given.

Affine periplectic Brauer algebras

We formulate Nazarov–Wenzl type algebras Pˆd− for the representation theory of the periplectic Lie superalgebras p(n). We establish an Arakawa–Suzuki type functor to provide a connection between p(n)-representations and Pˆd−-representations. We also consider various tensor product representations for Pˆd−. The periplectic Brauer algebra Ad developed by Moon is a quotient of Pˆd−. In particular, actions induced by Jucys–Murphy elements can also be recovered under the tensor product representation of Pˆd−. Moreover, a Poincare–Birkhoff–Witt type basis for Pˆd− is obtained. A diagram realization of Pˆd− is also obtained.

Filtered Hopf algebras and counting geodesic chords

We prove lower bounds on the growth of certain filtered Hopf algebras by means of a Poincare–Birkhoff–Witt type theorem for ordered products of primitive elements. When applied to the loop space homology algebra endowed with a natural length-filtration, these bounds lead to lower bounds for the number of geodesic paths between two points. Specifically, given a closed manifold \(M\) whose universal covering space is not homotopy equivalent to a finite complex and whose fundamental group has polynomial growth, for any Riemannian metric on \(M\), any pair of non-conjugate points \(p,q \in M\), and every component \({\mathcal C}\) of the space of paths from \(p\) to \(q\), the number of geodesics in \({\mathcal C}\) of length at most \(T\) grows at least like \(e^{\sqrt{T}}\). Using Floer homology, we extend this lower bound to Reeb chords on the spherisation of \(M\), and give a lower bound for the volume growth of the Reeb flow.

Monomial algebras defined by Lyndon words

Assume that X={x1,…,xg} is a finite alphabet and K is a field. We study monomial algebras A=K〈X〉/(W), where W is an antichain of Lyndon words in X of arbitrary cardinality. We find a Poincaré–Birkhoff–Witt type basis of A in terms of its Lyndon atoms N, but, in general, N may be infinite. We prove that if A has polynomial growth of degree d then A has global dimension d and is standard finitely presented, with d−1⩽|W|⩽d(d−1)/2. Furthermore, A has polynomial growth iff the set of Lyndon atoms N is finite. In this case A has a K-basis N={l1α1l2α2⋯ldαd|αi⩾0,1⩽i⩽d}, where N={l1,…,ld}. We give an extremal class of monomial algebras, the Fibonacci–Lyndon algebras, Fn, with global dimension n and polynomial growth, and show that the algebra F6 of global dimension 6 cannot be deformed, keeping the multigrading, to an Artin–Schelter regular algebra.

Representations of the Witt superalgebra [formula omitted

In this paper, we study representations of the Lie superalgebra W(2) of Witt type over an algebraically closed field of characteristic p>2. All irreducible modules and their dimensions are determined, up to isomorphisms. Thus, all irreducible modules and their dimensions of osp(2|2) are determined in prime characteristic, up to isomorphisms.

On p-adic interpolating function associated with modified Dirichlet's type of twisted q-Euler numbers and polynomials with weight α

The q-calculus theory is a novel theory that is based on finite difference re-scaling. The rapid development of q-calculus has led to the discovery of new generalizations of q-Euler polynomials involving q-integers. The present paper deals with the modified Dirichlet's type of twisted q-Euler polynomials with weight alpha. We apply the method of generating function and p-adic q-integral representation on Zp, which are exploited to derive further classes of q-Euler numbers and polynomials. To be more precise we summarize our results as follows, we obtain some combinatorial relations between modified Dirichlet's type of twisted q-Euler numbers and polynomials with weight alpha. Furthermore we derive witt's type formula and Distribution formula (Multiplication theorem) for modified Dirichlet's type of twisted q-Euler numbers and polynomials with weight alpha. In section three, by applying Mellin transformation we define q-analogue of modified twisted q-l-functions of Dirichlet's type and also we deduce that it can be written as modified Dirichlet's type of twisted q-Euler polynomials with weight alpha. Moreover we will find a link between modified twisted Hurwitz-zeta function and q-analogue of modified twisted q-l-functions of Dirichlet's type which yields a deeper insight into the effectiveness of this type of generalizations. In addition we consider q-analogue of partial zeta function and we derive behavior of the modified q-Euler L-function at s = 0. In final section, we construct p-adic twisted Euler q-L function with weight alpha and interpolate Dirichlet's type of twisted q-Euler polynomials with weight alpha at negative integers. Our new generating function possess a number of interesting properties which we state in this paper.

UNIVERSAL ENVELOPING ALGEBRAS OF PBW TYPE

AbstractWe continue our investigation of the general notion of universal enveloping algebra introduced in [A. Ardizzoni, A Milnor–Moore type theorem for primitively generated braided Bialgebras, J. Algebra 327(1) (2011), 337–365]. Namely, we study a universal enveloping algebra when it is of Poincaré–Birkhoff–Witt (PBW) type, meaning that a suitable PBW-type theorem holds. We discuss the problem of finding a basis for a universal enveloping algebra of PBW type: as an application, we recover the PBW basis both of an ordinary universal enveloping algebra and of a restricted enveloping algebra. We prove that a universal enveloping algebra is of PBW type if and only if it is cosymmetric. We characterise braided bialgebra liftings of Nichols algebras as universal enveloping algebras of PBW type.