Tensors, tensor densities

The classical theory of continuous distributions of dislocations has traditionally focused on the Burgers’ vectors and the dislocation density tensor as descriptions of defectiveness. We prove that, generally, there is an infinite number of tensor densities with similarly descriptive properties, and that there is a functional basis for this list which strictly includes the Burgers’ vectors and dislocation density. Moreover the changes of state which preserve these densities turn out to represent slip in certain surfaces associated with crystal geometry, so that the basic mechanism of plasticity emerges naturally from abstract ideas which neither anticipate nor involve the kinematics of particular types of crystal defects.

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ON MIXED TENSOR DENSITIES AS THE ALGEBRAIC CONCOMITANTS OF THE CURVATURE TENSOR IN THE TWO-DIMENSIONAL SPACE L²

Demonstratio Mathematica ◽

10.1515/dema-1975-0203 ◽

1975 ◽

Vol 8 (2) ◽

Author(s):

Leon Bieszk

Keyword(s):

Curvature Tensor ◽

Dimensional Space ◽

Two Dimensional ◽

Tensor Densities

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EXTENDED BOTT–VIRASORO ALGEBRA, SEMIDIRECT PRODUCT, ⋆-LIE ALGEBRA OF DIFFEOMORPHISM AND NONCOMMUTATIVE INTEGRABLE SYSTEMS

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887809003679 ◽

2009 ◽

Vol 06 (04) ◽

pp. 555-572

Author(s):

PARTHA GUHA

Keyword(s):

Lie Algebra ◽

Semidirect Product ◽

Virasoro Algebra ◽

Lie Derivative ◽

Coadjoint Representation ◽

Kdv Equations ◽

Two Component ◽

Universal Extension ◽

Product Algebra ◽

Tensor Densities

We study noncommutative theory of a coadjoint representation of a universal extension of Vect (S1) ⋉ C∞(S1) algebra using the action of ⋆-deformed matrix Hill's operators Δ⋆ on the space of ⋆-deformed tensor densities. The centrally extended semidirect product algebra [Formula: see text] is a Lie algebra of extended semidirect product of the Bott–Virasoro group [Formula: see text]. The study of deformed diffeomorphisms, deformed semidirect product algebra and deformed Lie derivative action of Δ⋆ on ⋆ deformed tensor-densities on S1 allow us to construct noncommutative two component Korteweg–de Vries (KdV) equations, in particular, we derive the noncommutative Ito equation. This leads to a geometric formulation of ⋆-deformed quantization of the centrally extended semidirect product algebra [Formula: see text] and two component noncommutative KdV equations.

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