Hamiltonian Formalism in Quantum Mechanics

Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their properties are formulated in a series of Conjectures

On the lifting of the Nagata automorphism

It is proved that the Nagata automorphism (Nagata coordinates, respectively) of the polynomial algebra $F[x,y,z]$ over a field $F$ cannot be lifted to a $z$-automorphism ($z$-coordinate, respectively) of the free associative algebra $K$. The proof is based on the following two new results which have their own interests: degree estimate of ${Q*_FF}$ and tameness of the automorphism group ${\text{Aut}_Q(Q*_FF)}$.Comment: 15 page

Primitive ideals and automorphisms of quantum matrices.

Let K be a field and q be a nonzero element of K that is not a root of unity. We give a criterion for (0) to be a primitive ideal of the algebra O-q(M-m,M-n) of quantum matrices. Next, we describe all height one primes of these two problems are actually interlinked since it turns out that (0) is a primitive ideal of O-q(M-m,M-n) whenever O-q(M-m,M-n) has only finitely many height one primes. Finally, we compute the automorphism group of O-q(M-m,M-n) in the case where m not equal n. In order to do this, we first study the action of this group on the prime spectrum of O-q(M-m,M-n). Then, by using the preferred basis of O-q(M-m,M-n) and PBW bases, we prove that the automorphism group of O-q(M-m,M-n) is isomorphic to the torus (K*)(m+n=1) when m not equal n and (m, n) not equal (1, 3) (3, 1)

On Plane Cremona Transformations of fixed degree

We study the quasi-projective variety Bird of plane Cremona transformations defined by three polynomials of fixed degree d and its subvariety Bir◦d where the three polynomials have no common factor. We compute their dimension and the decomposition in irreducible components. We prove that Bird is connected for each d and Bir◦d is connected when d <7

On commuting polynomial automorphisms of Ck{\mathbb{C}}^{k} , k ≥ 3

We characterize the polynomial automorphisms of $C^3,$ which commute with a regular automorphism. We use their meromorphic extension to $P^3$ and consider their dynamics on the hyperplane at infinity. We conjecture the additional hypothesis under which the same characterization is true in all dimensions. We give a partial answer to a question of S. Smale that in our context can be formulated as follows: can any polynomial automorphism of $C^k$ be the uniform limit on compact sets of polynomial automorphisms with trivial centralizer