235,283 research outputs found
Locally parabolic subgroups in Coxeter groups of arbitrary ranks
Despite the significance of the notion of parabolic closures in Coxeter
groups of finite ranks, the parabolic closure is not guaranteed to exist as a
parabolic subgroup in a general case. In this paper, first we give a concrete
example to clarify that the parabolic closure of even an irreducible reflection
subgroup of countable rank does not necessarily exist as a parabolic subgroup.
Then we propose a generalized notion of "locally parabolic closure" by
introducing a notion of "locally parabolic subgroups", which involves parabolic
ones as a special case, and prove that the locally parabolic closure always
exists as a locally parabolic subgroup. It is a subgroup of parabolic closure,
and we give another example to show that the inclusion may be strict in
general. Our result suggests that locally parabolic closure has more natural
properties and provides more information than parabolic closure. We also give a
result on maximal locally finite, locally parabolic subgroups in Coxeter
groups, which generalizes a similar well-known fact on maximal finite parabolic
subgroups.Comment: 7 pages; (v2) 11 pages, examples added, main theorem slightly updated
(v3) references updated, minor changes performed, to appear in Journal of
Algebr
Parabolic subalgebras, parabolic buildings and parabolic projection
Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have
a rich geometry determined by their parabolic subgroups and subalgebras, which
carry the structure of a building in the sense of J. Tits. We present herein an
elementary approach to the geometry of parabolic subalgebras, over an arbitrary
field of characteristic zero, which does not rely upon the structure theory of
semisimple Lie algebras. Indeed we derive such structure theory, from root
systems to the Bruhat decomposition, from the properties of parabolic
subalgebras. As well as constructing the Tits building of a reductive Lie
algebra, we establish a "parabolic projection" process which sends parabolic
subalgebras of a reductive Lie algebra to parabolic subalgebras of a Levi
subquotient. We indicate how these ideas may be used to study geometric
configurations and their moduli.Comment: 26 pages, v2 minor clarification
Parabolic-like maps
In this paper we introduce the notion of parabolic-like mapping, which is an
object similar to a polynomial-like mapping, but with a parabolic external
class, i.e. an external map with a parabolic fixed point. We prove a
straightening theorem for parabolic-like maps, which states that any
parabolic-like map of degree 2 is hybrid conjugate to a member of the family
Per_1(1), and this member is unique (up to holomorphic conjugacy) if the filled
Julia set of the parabolic-like map is connected.Comment: 32 pages, 12 figure
Harnack's Inequality for Parabolic De Giorgi Classes in Metric Spaces
In this paper we study problems related to parabolic partial differential
equations in metric measure spaces equipped with a doubling measure and
supporting a Poincare' inequality. We give a definition of parabolic De Giorgi
classes and compare this notion with that of parabolic quasiminimizers. The
main result, after proving the local boundedness, is a scale and location
invariant Harnack inequality for functions belonging to parabolic De Giorgi
classes. In particular, the results hold true for parabolic quasiminimizers
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