Witt Algebra Research Articles

Brackets with (τ,σ)-derivations and (p,q)-deformations of Witt and Virasoro algebras

Abstract The aim of this paper is to study some brackets defined on (τ,σ)-derivations satisfying quasi-Lie identities. Moreover, we provide examples of ( p , q ) ${(p,q)}$ -deformations of Witt and Virasoro algebras as well as 𝔰 ⁢ 𝔩 $\mathfrak{sl}$ (2) algebra. These constructions generalize the results obtained by Hartwig, Larsson and Silvestrov on σ-derivations, arising in connection with discretizations and deformations of algebras of vector fields.

Derivations and Automorphism Group of Original Deformative Schrödinger-Virasoro Algebra

In this paper, we determine the derivation algebra and the automorphism group of the original deformative Schrödinger-Virasoro algebra, which is the semi-direct product Lie algebra of the Witt algebra and its tensor density module Ig(a,b).

On nilpotent commuting varieties of r-tuples in the Witt algebra

Let g be the Witt algebra over an algebraically closed field k of characteristic p>3, and G=Aut(g) be the automorphism group of g with Lie(G)=g0. A result [12, Theorem 5.2] of A. Suslin, E. Friedlander and C. Bendel implies that the spectrum of the cohomology ring for the r-th Frobenius kernel of G is homeomorphic to the commuting variety of r-tuples of nilpotent elements in g0. As an analogue of Ngo's result [5, Theorem 1.2.1] in the case of the classical Lie algebra sl2 and a generalization of our previous work [15], in this paper we show that the varieties of r-tuples of nilpotent elements in g, as well as certain subalgebras, are reducible. Irreducible components and their dimensions are precisely presented. Moreover, these nilpotent commuting varieties of r-tuples are neither normal nor Cohen–Macaulay. These results are different from those in the case of the classical Lie algebra sl2.

Derivations of a parametric family of subalgebras of the Weyl algebra

An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra Ah generated by elements x,y, which satisfy yx−xy=h, where h∈F[x]. When h≠0, the algebra Ah is subalgebra of the Weyl algebra A1 and can be viewed as differential operators with polynomial coefficients. This paper determines the derivations of Ah and the Lie structure of the first Hochschild cohomology group HH1(Ah)=DerF(Ah)/InderF(Ah) of outer derivations over an arbitrary field. In characteristic 0, we show that HH1(Ah) has a unique maximal nilpotent ideal modulo which HH1(Ah) is 0 or a direct sum of simple Lie algebras that are field extensions of the one-variable Witt algebra. In positive characteristic, we obtain decomposition theorems for DerF(Ah) and HH1(Ah) and describe the structure of HH1(Ah) as a module over the center of Ah.

On the sub-KP hierarchy and its constraints, revisited

In this paper, we are interested in a series of sub-hierarchies of the KP hierarchy introduced by Date et al. in [4], which we call the ℬ𝒞r-KP hierarchy for r ∈ ℤ≥0. In a unified way, we construct additional symmetries of the ℬ𝒞r-KP hierarchy and show that all of them form a [Formula: see text]-algebra. Then, we introduce the constrained ℬ𝒞r-KP hierarchy, and construct its additional symmetries and show that all of them form a Witt algebra.

The universal enveloping algebra of the Witt algebra is not noetherian

This work is prompted by the long standing question of whether it is possible for the universal enveloping algebra of an infinite dimensional Lie algebra to be noetherian. To address this problem, we answer a 23-year-old question of Carolyn Dean and Lance Small; namely, we prove that the universal enveloping algebra of the Witt (or centerless Virasoro) algebra is not noetherian. To show this, we prove our main result: the universal enveloping algebra of the positive part of the Witt algebra is not noetherian. We employ algebro-geometric techniques from the first author's classification of (noncommutative) birationally commutative projective surfaces.As a consequence of our main result, we also show that the enveloping algebras of many other (infinite dimensional) Lie algebras are not noetherian. These Lie algebras include the Virasoro algebra and all infinite dimensional Z-graded simple Lie algebras of polynomial growth.

NILPOTENT ELEMENTS IN THE JACOBSON–WITT ALGEBRA OVER A FINITE FIELD

It is shown in this paper that the number of nilpotent elements in the Jacobson–Witt algebra W n over a finite field $$ {\mathbb F} $$ q is equal to the expected power of q.

ORBIT CLOSURES IN THE WITT ALGEBRA AND ITS DUAL SPACE

Working over an algebraically closed field of characteristic p > 3, we calculate the orbit closures in the Witt algebra W under the action of its automorphism group G. We also outline how the same techniques can be used to determine closures of orbits of all heights except p - 1 (in which case we only obtain a conditional statement) in the dual space W* under the induced action of G. As a corollary we prove that the algebra of invariants k[W*]G is trivial.

The nilpotent commuting variety of the Witt algebra

Let g be the Witt algebra over an algebraically closed field k of characteristic p>3. Let N={x∈g|x[p]=0} be the nilpotent variety of g, and C(N):={(x,y)∈N×N|[x,y]=0} the nilpotent commuting variety of g. As an analogue of Premet's result in the case of classical Lie algebras (Premet (2003) [6]), we show that the variety C(N) is reducible and equidimensional. Irreducible components of C(N) and their dimension are precisely given. Furthermore, the nilpotent commuting varieties of Borel subalgebras are also determined.

Entropy from near-horizon geometries of Killing horizons

We derive black hole entropy based on the near-horizon symmetries of black hole space-times. To derive these symmetries we make use of an $(R,T)$-plane close to a Killing horizon. We identify a set of vector fields that preserves this plane and forms a Witt algebra. The corresponding algebra of Hamiltonians is shown to have a non-trivial central extension. Using the Cardy formula and the central charge we obtain the Bekenstein-Hawking entropy.

String worldsheet theory in hamiltonian framework and background independence

We analyze exact conformal invariance of string worldsheet theory in non-trivial backgrounds using hamiltonian framework. In the first part of this talk we consider the example of type IIB superstrings in Ramond-Ramond pp-wave background. In particular, we discuss the quantum definition of energy-momentum (EM) tensor and two methods of computing Virasoro algebra. One of the methods uses dynamical supersymmetries and indirectly establishes (partially) conformal invariance when the background is on-shell. We discuss the problem of operator ordering involved in the other method which attempts to compute the algebra directly. This method is supposed to work for off-shell backgrounds and therefore is more useful. In order to understand this method better we attempt a background independent formulation of the problem which is discussed in the second half of the talk. For a bosonic string moving in an arbitrary metric-background such a framework is obtained by following DeWitt's work (Phys.Rev.85:653–661,1952) in the context of particle quantum mechanics. In particular, we construct certain background independent analogue of quantum Virasoro generators and show that in spin-zero representation they satisfy the Witt algebra with additional anomalous terms that vanish for Ricci-flat backgrounds. We also report on a new result which states that the same algebra holds true in arbitrary tensor representations as well.

Witt型与Virasoro型Lie双代数的对偶

Structures of dual Lie bialgebras on the one sided Witt algebra, the Witt algebra and the Virasoro algebra are investigated. As a result, we obtain some infinite-dimensional Lie algebras.

On $${\mathbb{F}_q}$$ F q -Rational Structure of Nilpotent Orbits in the Witt Algebra

Let \({\mathfrak{g}=W_1}\) be the p-dimensional Witt algebra over an algebraically closed field \({k=\overline{\mathbb{F}}_q}\), where p > 3 is a prime and q is a power of p. Let G be the automorphism group of \({\mathfrak{g}}\). The Frobenius morphism FG (resp. \({F_\mathfrak{g}}\)) can be defined naturally on G (resp. \({\mathfrak{g}}\)). In this paper, we determine the \({F_\mathfrak{g}}\) -stable G-orbits in \({\mathfrak{g}}\). Furthermore, the number of \({\mathbb{F}_q}\) -rational points in each \({F_\mathfrak{g}}\) -stable orbit is precisely given. Consequently, we obtain the number of \({\mathbb{F}_q}\) -rational points in the nilpotent variety.

The string equation and the τ-function Witt constraints for the discrete Kadomtsev-Petviashvili hierarchy

Using the tool of pseudo-difference operators, the string equation for the discrete Kadomtsev-Petviashvili (KP) hierarchy is introduced and a general expression for the string equation is formulated. Furthermore, the operators of the constraints that the string equation imposes on the τ-function of the p-reduced dKP hierarchy are presented. The algebra, which part of the constraints spans, and eigenvalues of part of the constraints corresponding to the τ-function are calculated. It is also shown that this algebra is exactly a Witt algebra not only for p = 2, but also for a general \documentclass[12pt]{minimal}\begin{document}$p\in \mathbb {N}$\end{document}p∈N, and that the τ-function of the p-reduced dKP hierarchy constrained by the string equation is a vacuum vector of a Witt algebra.

Cohomology of Hom-Lie superalgebras and q-deformed Witt superalgebra

Hom-Lie algebra (superalgebra) structure appeared naturally in q-deformations, based on σ-derivations of Witt and Virasoro algebras (superalgebras). They are a twisted version of Lie algebras (superalgebras), obtained by deforming the Jacobi identity by a homomorphism. In this paper, we discuss the concept of α k -derivation, a representation theory, and provide a cohomology complex of Hom-Lie superalgebras. Moreover, we study central extensions. As application, we compute derivations and the second cohomology group of a twisted osp(1, 2) superalgebra and q-deformed Witt superalgebra.

Representations of the Witt Algebra and Gl(n)-Opers

We exhibit a new link between certain representations of the Witt algebra and some Gl(n)-opers on the punctured disc. As applications, we discuss the connection with the KdV hierarchy and Virasoro constraints and how the Virasoro constraints of the so-called topological recursion fit in our approach.

On a Coordinate-Independent Description of String Worldsheet Theory

We rewrite the bosonic worldsheet theory in curved background in a language where it describes a single particle moving in an infinite-dimensional curved spacetime. This language is developed at a formal level without regularizing the infinite-dimensional traces. Then, we adopt DeWitt’s (Phys Rev 85:653, 1952) coordinate-independent formulation of quantum mechanics in the present context. This procedure enables us to define coordinate invariant quantum analogue of classical Virasoro generators, which we call DeWitt–Virasoro generators. This framework also enables us to calculate the invariant matrix elements of an arbitrary operator constructed out of the DeWitt–Virasoro generators between two arbitrary scalar states. Using these tools, we further calculate the DeWitt–Virasoro algebra in spin-zero representation. The result is given by the Witt algebra with additional anomalous terms that vanish for Ricci-flat backgrounds. Further analysis need to be performed to precisely relate this with the beta function computation of Friedan and others. Finally, we explain how this analysis improves the understanding of showing conformal invariance for certain pp-wave that has been recently discussed using hamiltonian framework.

Lie superalgebras of Krichever–Novikov type and their central extensions

Classically important examples of Lie superalgebras have been constructed starting from the Witt and Virasoro algebra. In this article we consider Lie superalgebras of Krichever–Novikov type. These algebras are multi-point and higher genus equivalents. The grading in the classical case is replaced by an almost-grading. The almost-grading is determined by a splitting of the set of points were poles are allowed into two disjoint subsets. With respect to a fixed splitting, or equivalently with respect to an almost-grading, it is shown that there is up to rescaling and equivalence a unique non-trivial central extension. It is given explicitly. Furthermore, a complete classification of bounded cocycles (with respect to the almost-grading) is given.

Counterexamples to a rank analog of the Shepherd-Leedham-Green-Mckay theorem on finite p-groups of maximal nilpotency class

By the Shepherd-Leedham-Green-McKay theorem on finite p-groups of maximal nilpotency class, if a finite p-group of order pn has nilpotency class n−1, then f has a subgroup of nilpotency class at most 2 with index bounded in terms of p. Some counterexamples to a rank analog of this theorem are constructed that give a negative solution to Problem 16.103 in The Kourovka Notebook. Moreover, it is shown that there are no functions r(p) and l(p) such that any finite 2-generator p-group whose all factors of the lower central series, starting from the second, are cyclic would necessarily have a normal subgroup of derived length at most l(p) with quotient of rank at most r(p). The required examples of finite p-groups are constructed as quotients of torsion-free nilpotent groups which are abstract 2-generator subgroups of torsion-free divisible nilpotent groups that are in the Mal’cev correspondence with “truncated” Witt algebras.

The Extended Symmetry Lie Algebra and the Asymptotic Expansion of the Transversal Correlation Function for the Isotropic Turbulence

The extended symmetry of the functional of length determined in an affine spaceK3of the correlation vectors for homogeneous isotropic turbulence is studied. The two-point velocity-correlation tensor field (parametrized by the time variablet) of the velocity fluctuations is used to equip this space by a family of the pseudo-Riemannian metricsdl2(t)(Grebenev and Oberlack (2011)). First, we observe the results obtained by Grebenev and Oberlack (2011) and Grebenev et al. (2012) about a geometry of the correlation spaceK3and expose the Lie algebra associated with the equivalence transformation of the above-mentioned functional for the quadratic formdlD22(t)generated bydl2(t)which is similar to the Lie algebra constructed by Grebenev et al. (2012). Then, using the properties of this Lie algebra, we show that there exists a nontrivial central extension wherein the central charge is defined by the same bilinear skew-symmetric formcas for the Witt algebra which measures the number of internal degrees of freedom of the system. For the applications in turbulence, as the main result, we establish the asymptotic expansion of the transversal correlation function for large correlation distances in the frame ofdlD22(t).