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Register a new userTransformation Groups on Real Plane and their Differential Invariants
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Added by
Maryna Nesterenko ms.
Publication date
2005
fields
Physics
and research's language is
English
Authors
Maryna Nesterenko
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Complete sets of bases of differential invariants, operators of invariant differentiation and Lie determinants of continuous transformation groups acting on the real plane are constructed. As a necessary preliminary, realizations of finite-dimensional Lie algebras on the real plane are revisited.
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We find the complete equivalence group of a class of (1+1)-dimensional second-order evolution equations, which is infinite-dimensional. The equivariant moving frame methodology is invoked to construct, in the regular case of the normalization procedure, a moving frame for a group related to the equivalence group in the context of equivalence transformations among equations of the class under consideration. Using the moving frame constructed, we describe the algebra of differential invariants of the former group by obtaining a minimum generating set of differential invariants and a complete set of independent operators of invariant differentiation.
Recursive algebraic construction of two infinite families of polynomials in $n$ variables is proposed as a uniform method applicable to every semisimple Lie group of rank $n$. Its result recognizes Chebyshev polynomials of the first and second kind as the special case of the simple group of type $A_1$. The obtained not Laurent-type polynomials are proved to be equivalent to the partial cases of the Macdonald symmetric polynomials. Basic relation between the polynomials and their properties follow from the corresponding properties of the orbit functions, namely the orthogonality and discretization. Recurrence relations are shown for the Lie groups of types $A_1$, $A_2$, $A_3$, $C_2$, $C_3$, $G_2$, and $B_3$ together with lowest polynomials.
The role played by Deligne-Beilinson cohomology in establishing the relation between Chern-Simons theory and link invariants in dimensions higher than three is investigated. Deligne-Beilinson cohomology classes provide a natural abelian Chern-Simons action, non trivial only in dimensions $4l+3$, whose parameter $k$ is quantized. The generalized Wilson $(2l+1)$-loops are observables of the theory and their charges are quantized. The Chern-Simons action is then used to compute invariants for links of $(2l+1)$-loops, first on closed $(4l+3)$-manifolds through a novel geometric computation, then on $mathbb{R}^{4l+3}$ through an unconventional field theoretic computation.
Nonrelativistic conformal groups, indexed by l=N/2, are analyzed. Under the assumption that the mass parametrizing the central extension is nonvanishing the coadjoint orbits are classified and described in terms of convenient variables. It is shown that the corresponding dynamical system describes, within Ostrogradski framework, the nonrelativistic particle obeying (N+1)-th order equation of motion. As a special case, the Schroedinger group and the standard Newton equations are obtained for N=1 (l=1/2).
We show that non-linear Schwarzian differential equations emerging from covariance symmetry conditions imposed on linear differential operators with hypergeometric function solutions, can be generalized to arbitrary order linear differential operators with polynomial coefficients having selected differential Galois groups. For order three and order four linear differential operators we show that this pullback invariance up to conjugation eventually reduces to symmetric powers of an underlying order-two operator. We give, precisely, the conditions to have modular correspondences solutions for such Schwarzian differential equations, which was an open question in a previous paper. We analyze in detail a pullbacked hypergeometric example generalizing modular forms, that ushers a pullback invariance up to operator homomorphisms. We expect this new concept to be well-suited in physics and enumerative combinatorics. We finally consider the more general problem of the equivalence of two different order-four linear differential Calabi-Yau operators up to pullbacks and conjugation, and clarify the cases where they have the same Yukawa couplings.
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