Tensor calculus: unlearning vector calculus

Tensor calculus is critical in the study of the vector calculus of the surface of a body. Indeed, tensor calculus is a natural step-up for vector calculus. This paper presents some pitfalls of a traditional course in vector calculus in transitioning to tensor calculus. We show how a deeper emphasis on traditional topics such as the Jacobian can serve as a bridge for vector calculus into tensor calculus.     

Cohomology of Sheaves of Frechet Algebras and Spectral Theory

We propose a holomorphic functional calculus for a noncommutative operator family generating a supernilpotent Lie subalgebra. This calculus extends Taylor's holomorphic functional calculus.     

Sub-Riemannian calculus on hypersurfaces in Carnot groups

We develop a sub-Riemannian calculus for hypersurfaces in graded nilpotent Lie groups. We introduce an appropriate geometric framework, such as horizontal Levi-Civita connection, second fundamental form, and horizontal Laplace-Beltrami operator. We analyze the relevant minimal surfaces and prove some basic integration by parts formulas. Using the latter we establish general first and second variation formulas for the horizontal perimeter in the Heisenberg group. Such formulas play a fundamental role in the sub-Riemannian Bernstein problem.     

Algebraic Ricci solitons on metric Lie groups with zero Schouten–Weyl tensor

Algebraic Ricci solitons on Lie groups with left-invariant (pseudo)Riemannian metric and zero Schouten–Weyl tensor are studied. The absence of nontrivial algebraic Ricci solitons on metric Lie groups with zero Schouten–Weyl tensor and diagonalizable Ricci operator is proved.     

Formal power series, operator calculus, and duality on Lie algebras

This paper presents an operator calculus approach to computing with non-commutative variables. First, we recall the product formulation of formal exponential series. Then we show how to formulate canonical boson calculus on formal series. This calculus is used to represent the action of a Lie algebra on its universal enveloping algebra. As applications, Hamilton's equations for a general Hamiltonian, given as a formal series, are found using a double-dual representation, and a formulation of the exponential of the adjoint representation is given. With these techniques one can represent the Volterra product acting on the enveloping algebra. We illustrate with a three-step nilpotent Lie algebra.     

Exterior Difference System on Hypercubic Lattice

Based on the exterior differential calculus, we define the exterior difference system on the hypercubic lattice. The pairing formula, bracket, contract operator and Lie derivative operator for this system are also constructed. The discrete H. Cartan formulas are obtained too. After those preparations, we prove the discrete Frobenius Theorem, discrete retraction theorem and discrete Cartan–Kähler theorem.     

Tensor structures on a differentiable manifold

Summary A tensor structure is a class of equivalent G-structures and it is defined by a special tensor field. Such fields are characterized by the existence of a linear connection relative to which they have covariant derivative zero. Two tensor structures may admit a common subordinate structure. Exaples of such subordinate stuctures are given and some cases, when one stucture is a Riemannian metric, are considered. To Enrico Bompiani on his scientific Jubilee     

Some Results on Contact Metric Manifolds

If the sectional curvatures of plane sections containing the characteristic vector field of a contact metric manifold M are non-vanishing, then we prove that a second order parallel tensor on M is a constant multiple of the associated metric tensor. Next, we prove for a contact metric manifold of dimension greater than 3 and whose Ricci operator commutes with the fundamental collineation that, if its Weyl conformal tensor is harmonic, then it is Einstein. We also prove that, if the Lie derivative of the fundamental collineation along the characteristic vector field on a contact metric 3-manifold M satisfies a cyclic condition, then M is either Sasakian or locally isometric to certain canonical Lie-groups with a left invariant metric. Next, we prove that if a three-dimensional Sasakian manifold admits a non-Killing projective vector field, it is of constant curvature 1. Finally, we prove that a conformally recurrent Sasakian manifold is locally isometric to a unit sphere.     

Generation of modes on the three-dimensional sphere

We use the method of Lie generation of tensor fields, which works for fields of different tensor structures, to construct the complete system of scalar, vector, symmetric tensor, and spinor fields on the three-dimensional sphere. We construct the Pauli operator explicitly. We demonstrate the role of spin in forming the mode series. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 417–440, March, 2007.     

A new form and a ⋆-algebraic structure of quantum stochastic integrals in Fock space

An algebraic definition of the basic quantum process for the noncommutative stochastic calculus is given in terms of the Fock representation of a Lie ⋆-algebra of matrices in a pseudo-Euclidean space. An operator definition of the quantum stochastic integral is given and its continuity is proved in a projective limit uniform operator topology. A new form of quantum stochastic equations, revealing the ⋆-algebraic structure of quantum Ito's formula, is given. (Conferenza tenuta il 21 settembre 1988)     

Double exponential map on symmetric spaces

We establish an asymptotic formula for the double exponential map operator on affine symmetric spaces. This operator plays an important role in the geometric calculus of symbols of (pseudo)differential operators on manifolds with connection, whose foundations were laid by Sharafutdinov. To obtain this result, we essentially use the structural theory of symmetric spaces and techniques of the Lie group theory. One of the key moments is an application of the Campbell-Hausdorff series in Dynkin form.     

Invitation to composition

In 1963 [Ann. of Math. 78, 267-288], Gerstenhaber invented a comp(osition) calculus in the Hochschild complex of an associative algebra. In this paper, the first steps of the Gerstenhaber theory are exposed in an abstract (comp system) setting. In particular, as in the Hochschild complex, a graded Lie algebra and a pre-coboundary operator can be associated to every comp system. A derivation deviation of the pre-coboundary operator over the total composition is calculated in two ways, (the long) one of which is essentially new and can be seen as an example and elaboration of the auxiliary variables method proposed by Gerstenhaber in the early days of the comp calculus.     

The Lichnerowicz Laplacian on compact symmetric spaces

We show that, on a compact symmetric space, the Lichnerowicz Laplacian acting on the space of covariant tensor fields coincides with the Casimir operator and we deduce that, on a compact semisimple Lie group, the Lichnerowicz Laplacian is the mean of the left invariant Casimir operator and the right invariant Casimir operator. To cite this article: M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

扭的Heisenberg-Virasoro顶点算子代数的一个特征化（英文）

The twisted Heisenberg-Virasoro algebra is the universal central extension of the Lie algebra of differential operators on a circle of order at most one. In this paper, we first study the variety of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra, which is a finite set consisting of two nontrivial elements. Based on this property,we also show that the twisted Heisenberg-Virasoro vertex operator algebra is a tensor product of two vertex operator algebras. Moreover, associating to properties of semi-conformal vectors of the twisted Heisenberg-Virasoro vertex operator algebra, we charaterized twisted Heisenberg-Virasoro vertex operator algebras. This will be used to understand the classification problems of vertex operator algebras whose varieties of semi-conformal vectors are finite sets.     

The multiplicative Weyl functional calculus

The Weyl calculus discussed in the author's previous papers starts with a fixed set of n noncommuting self-adjoint operators and associates an operator to a real function of n variables. The calculus is not multiplicative with respect to point-wise multiplication of functions. However, if the n self-adjoint operators generate a unitary Lie group representation, a “skew product” of functions can be defined which yields multiplicativity. This skew product depends only on the Lie group, not on the particular representation. In the case of the Heisenberg group, this skew product makes it possible to write the Schrödinger equation as an integro-differential equation on the phase plane. Strong convergence of the dynamical group, as Planck's constant goes to zero, to the classical Hamiltonian flow is proved under various conditions on the Hamiltonian.     

A Family of Irreducible Representations of the Witt Lie Algebra with Infinite-Dimensional Weight Spaces

We define a 4-parameter family of generically irreducible and inequivalent representations of the Witt Lie algebra on which the infinitesimal rotation operator acts semisimply with infinite-dimensional eigenspaces. They are deformations of the (generically indecomposable) representations on spaces of polynomial differential operators between two spaces of tensor densities on S ¹, which are constructed by composing each such differential operator with the action of a rotation by a fixed angle.     

A non-abelian tensor product of precrossed modules in lie algebras

Abstract

In this paper, we introduce the non-abelian tensor square of precrossed modules in Lie algebras and investigate some of its properties. In particular, for an arbitrary Lie algebra L, we study the relation of the second homology of a precrossed L-module and the non-abelian exterior square. Also, we show how this non-abelian tensor product is related to the universal central extensions (with respect to the subcategory of crossed modules) of a precrossed module.     

Hahn Difference Operator and Associated Jackson–Nörlund Integrals

This paper is devoted for a rigorous investigation of Hahn’s difference operator and the associated calculus. Hahn’s difference operator generalizes both the difference operator and Jackson’s q-difference operator. Unlike these two operators, the calculus associated with Hahn’s difference operator receives no attention. In particular, its right inverse has not been constructed before. We aim to establish a calculus of differences based on Hahn’s difference operator. We construct a right inverse of Hahn’s operator and study some of its properties. This inverse also generalizes both Nörlund sums and the Jackson q-integrals. We also define families of corresponding exponential and trigonometric functions which satisfy first and second order difference equations, respectively.     

Spin characters of generalized symmetric groups

In 1911 Schur computed the spin character values of the symmetric group using two important ingredients: the first one later became famously known as the Schur Q-functions and the second one was certain creative construction of the projective characters on Clifford algebras. In the context of the McKay correspondence and affine Lie algebras, the first part was generalized to all wreath products by the vertex operator calculus in Frenkel et al. (Duke Math J 111:51–96, 2002) where a large part of the character table was produced. The current paper generalizes the second part and provides the missing projective character values for the wreath product of the symmetric group with a finite abelian group. Our approach relies on Mackey–Wigner’s little groups to construct irreducible modules. In particular, projective modules and spin character values of all classical Weyl groups are obtained.