57,768 research outputs found
Cocommutative Calabi-Yau Hopf algebras and deformations
The Calabi-Yau property of cocommutative Hopf algebras is discussed by using
the homological integral, a recently introduced tool for studying infinite
dimensional AS-Gorenstein Hopf algebras. It is shown that the skew-group
algebra of a universal enveloping algebra of a finite dimensional Lie algebra
\g with a finite subgroup of automorphisms of \g is Calabi-Yau if and
only if the universal enveloping algebra itself is Calabi-Yau and is a
subgroup of the special linear group SL(\g). The Noetherian cocommutative
Calabi-Yau Hopf algebras of dimension not larger than 3 are described. The
Calabi-Yau property of Sridharan enveloping algebras of finite dimensional Lie
algebras is also discussed. We obtain some equivalent conditions for a
Sridharan enveloping algebra to be Calabi-Yau, and then partly answer a
question proposed by Berger. We list all the nonisomorphic 3-dimensional
Calabi-Yau Sridharan enveloping algebras
A note on the restricted universal enveloping algebra of a restricted Lie-Rinehart Algebra
Lie-Rinehart algebras, also known as Lie algebroids, give rise to Hopf
algebroids by a universal enveloping algebra construction, much as the
universal enveloping algebra of an ordinary Lie algebra gives a Hopf algebra,
of infinite dimension. In finite characteristic, the universal enveloping
algebra of a restricted Lie algebra admits a quotient Hopf algebra which is
finite-dimensional if the Lie algebra is. Rumynin has shown that suitably
defined restricted Lie algebroids allow to define restricted universal
enveloping algebras that are finitely generated projective if the Lie algebroid
is. This note presents an alternative proof and possibly fills a gap that
might, however, only be a gap in the author's understanding
Algebraic families of subfields in division rings
We describe relations between maximal subfields in a division ring and in its
rational extensions. More precisely, we prove that properties such as being
Galois or purely inseparable over the centre generically carry over from one to
another. We provide an application to enveloping skewfields in positive
characteristics. Namely, there always exist two maximal subfields of the
enveloping skewfield of a solvable Lie algebra, such that one is Galois and the
second purely inseparable of exponent 1 over the centre. This extends results
of Schue in the restricted case. Along the way we provide a description of the
enveloping algebra of the p-envelope of a Lie algebra as a polynomial extension
of the smaller enveloping algebra.Comment: 9 pages, revised according to referee comments, new titl
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