Vitor O. Ferreira

FREE SYMMETRIC AND UNITARY PAIRS IN DIVISION RINGS WITH INVOLUTION

Let D be a division ring with an involution and characteristic different from 2. Then, up to a few exceptions, D contains a pair of symmetric elements freely generating a free subgroup of its multiplicative group provided that (a) it is finite-dimensional and the center has a finite sufficiently large transcendence degree over the prime field, or (b) the center is uncountable, but not algebraically closed in D. Under conditions (a), if the involution is of the first kind, it is also shown that the unitary subgroup of the multiplicative group of D contains a free subgroup, with one exception. The methods developed are also used to exhibit free subgroups in the multiplicative group of a finite-dimensional division ring provided the center has a sufficiently large transcendence degree over its prime field.

A Hopf-Galois correspondence for free algebras

Abstract A Galois correspondence is exhibited between right coideals subalgebras of a finite-dimensional pointed Hopf algebra acting homogeneously and faithfully on a free associative algebra and free subalgebras containing the invariants of this action.

Free symmetric algebras in division rings generated by enveloping algebras of Lie algebras

For any Lie algebra L over a field, its universal enveloping algebra U(L) can be embedded in a division ring 𝔇(L) constructed by Lichtman. If U(L) is an Ore domain, 𝔇(L) coincides with its ring of fractions. It is well known that the principal involution of L, x ↦ -x, can be extended to an involution of U(L), and Cimpric proved that this involution can be extended to one on 𝔇(L). For a large class of noncommutative Lie algebras L over a field of characteristic zero, we show that 𝔇(L) contains noncommutative free algebras generated by symmetric elements with respect to (the extension of) the principal involution. This class contains all noncommutative Lie algebras such that U(L) is an Ore domain.

Constants of derivations on free associative algebras

It is proved that the subalgebra of constants of a derivation on a free associative algebra in prime characteristic is free provided that some constraining conditions are satisfied. As a particular case, it follows that the constants of the partial derivatives on a free algebra form a free subalgebra. The main result is also applied in order to provide a simplified proof of a previous result by the author on extensions of tensor rings. Accepted for publication in Glasgow Mathematical Journal as of 8 September 2000.

COMMUTATIVE MONOID AMALGAMS WITH NATURAL CORE

We consider commutative monoid amalgams having as their cores a conical cancellative monoid. In this context we present sufficient conditions for the amalgam to be strongly embeddable and for some properties of the factors to be transmitted to the amalgamated free product.

Rationality of the Hilbert series of Hopf-invariants of free algebras

It is shown that the subalgebra of invariants of a free associative algebra of finite rank under a linear action of a semisimple Hopf algebra has a rational Hilbert series with respect to the usual degree function, whenever the ground field has zero characteristic.

Free Fields in Malcev–Neumann Series Rings

It is shown that the skew field of Malcev–Neumann series of an ordered group frequently contains a free field of countable rank, i.e. the universal field of fractions of a free associative algebra of countable rank. This is an application of a criterion on embeddability of free fields on skew fields which are complete with respect to a valuation function, following K. Chiba. Other applications to skew Laurent series rings are discussed. Finally, embeddability questions on free fields of uncountable rank in Malcev–Neumann series rings are also considered.