Note on 2d binary operadic harmonic oscillator
It is explained how the time evolution of the operadic variables may be introduced. As an example, a 2-dimensional binary operadic Lax representation of the harmonic oscillator is found.
math-ph↗
arXiv subjects
Eugen Paal
Publications and source records attributed to Eugen Paal.
Moufang loops and generalized Lie-Cartan theorem
Generalized Lie-Cartan theorem for linear birepresentations of an analytic Moufang loop is considered. The commutation relations of the generators of the birepresentation were found. In particular, the Lie algebra of the multiplication group of the birepresentation is explicitly given.
math.RT↗
Note on operadic harmonic oscillator
It is explained how the time evolution of the operadic variables may be introduced. As an example, an operadic Lax representation of the harmonic oscillator is considered.
math-ph↗
How to compose Lagrangian?
A method for constructing Lagrangians for the Lie transformation groups is explained. As examples, the Lagrangians for real plane rotations and affine transformations of the real line are constructed.
math-ph↗
Invitation to operadic dynamics
Based on the Gerstenhaber Theory, clarification is made of how operadic dynamics may be introduced. Operadic observables satisfy the Gerstenhaber algebra identities and their time evolution is governed by operadic evolution equation. The notion of an operadic Lax pair is also introduced. As an example, an operadic (representation of) harmonic oscillator is proposed.
math-ph↗
Moufang transformations and Noether currents
The Noether currents generated by continuous Moufang tranformations are constructed and their equal-time commutators are found. The corresponding charge algebra turns out to be a birepresentation of the tangent Mal'ltsev algebra of an analytic Moufang loop.
math-ph↗
Note on Moufang-Noether currents
The derivative Noether currents generated by continuous Moufang tranformations are constructed and their equal-time commutators are found. The corresponding charge algebra turns out to be a birepresentation of the tangent Mal'ltsev algebra of an analytic Moufang loop.
math-ph↗
Plane rotations and Hamilton-Dirac mechanics
Canonical formalism for SO(2) is developed. This group can be seen as a toy model of the Hamilton-Dirac mechanics with constraints. The Lagrangian and Hamiltonian are explicitly constructed and their physical interpretation are given. The Euler-Lagrange and Hamiltonian canonical equations coincide with the Lie equations. It is shown that the constraints satisfy CCR. Consistency of the constraints is checked.
math-ph↗
Toy operads
Didactic operadic entertainment for pedestrians. The following visual toys are included: operad of little squares, operad of planar rooted trees, and an operad algebra example.
math.QA↗
Moufang loops and Lie algebras
It is explicitly shown how the Lie algebras can be associated with the analytic Moufang loops. The resulting Lie algebra commutation relations are well known from the theory of alternative algebras and can be seen as a preliminary step to quantum Moufang loops.
math-ph↗
Operadic deformations as a tool for cogravity
The deformation equation and its integrability condition (Bianchi identity) of a non-(co)associative deformation in operad algebra are found. Based on physical analogies, cogravity equations are proposed.
math.QA↗
Coalgebras, Cocompositions and Cohomology
The (co)homology theory of n-ary (co)compositions is a functor associating to $n$-ary (co)composition a complex. We present unified approach to the cohomology theory of coassociative and Lie coalgebras and for $2n$-ary cocompositions. This approach points to a possible generalization.
hep-th↗