Hopf-algebraic structure of combinatorial objects and differential operators Robert University
Grossman*
of Illinois
Richard University
G.
at Larson
of Illinois July,
Chicago t
at
Chicago
1989
Abstract In this space of
paper
which
this
structure
operators directions
to
Much
will
of the
material
many
describe
spanned
by
how
structure
rems,
this and
*Supported ?Supported
of trees,
combinatorics
the
how
a
of equivalence can
it can
in part in part
(NASA-C_,-I _3.h3(_0) L}F Ct_c, INAT_-:,IAL t,}PERATL)PS (Illinois
be
survey
3 we indicate in this
and
give
some
classes to give
applied
in the
and some
paper
of finite
calculus
differential
possible
future
is work-in-progress.
answers.
imposed
proofs
a vector
applications to
of trees
structure
be used
on
some
1,
structure
Hopf-algebraic
set
structure
and
ire_Section
discussed
questions
Hopf-algebraic binatorics
We
a t{opf-algebraic
a family
in _etie_2. In Section for this work.
We describe
1
we describe
has as basis
on
rooted
of classical
and
the trees,
vector and
combinatorial
of differential
operators.
by NASA Grant NAG 2-513. by NSF Grant DMS 870-1085.
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coin-
space indicate theoThe
coalgebra
structure
structure
defined
defined such
for individual as the
of rooted
this
numbers
objects,
rooted
similar
Other
to the
structure
rather
coalgebra
defined
than
there
for a class
applications
k will denote
or the complex
of finite
is very
coalgebra
of Hopf
was
of objects algebras
to
in [12].
paper,
By a tree we will mean classes
space
the
trees.
can be found
Throughout real
on this
combinatorial
family
combinatorics the
we define
in [9]. Itowever,
of characteristic
0 such
as
numbers.
a finite
trees,
a field
rooted
and
let
tree.
k{T}
Let T be the be the
set of equivalence
k-vector
space
which
has
T as a basis. We first root
has
define
tl ® t2, where range
over
subsets.
st,
the
The
trees.
Let
some
have
node
trees. tree
sl,
only
be shown
homomorphisms, The nodes.
so that
bialgebra Because
We summarize Theorem rooted An
tile above
a field
Hopf
algebra,
then
P(A)=
the
primitive
root
and
that
structure
elements A I A(_)
child
1 of of
(n + 1) _ the
trivial
product.
It
see [.3]. with
it is a Ilop[
7_ + 1
algebra.
theorem.
IIopf
classes
of finite
algebra.
of primitive
llop[
elements.
of A are defined = 1 ®
[email protected]
n+
are algebra
all trees
of cocommutative
space
is t2 E T
si the
with basis all equivalence
structure
tree
7"subtrees
for this
For details,
connected
tl,
sum of these
connected,
two
other
If t2 has the each
unit
following
of t into
that
oftl.
as basis
of t2
comultiplication
to attach
coalgebra
whose
2 r terms root
every
Suppose
left
has
in the
0 is the
{ae
and
is graded
graded
of the
that
to be the
k{T},_
k{T}
root
1 and
is associative,
defining
space k {T}
feature the
is defined
discussion
of characteristic
to
of the of tile
of the
t2 by making
is a bialgebra.
is a cocomm_ltative
important
over
tree
is a right
maps
bialgebra
of the
product
is graded:
1 The vector trees
tit2
k{T}
k{T} the
to the
this
the
tree
are (n + 1) _ ways
of tile root
that
children
on k{T}.
children
there
that
trivial
structure
product
be shown
the
It is immediate
s_ as roots
of t2. The
consisting
can also
s_ be the
root),
If t E T is a tree is tile sum
of tl and
the
coproduct.
...,
A(t)
of tile children
sends
algebra
sl, ...,
It can
root
the
the
on k{T}.
coproduct
of the
e which
define
structure the
partitions
for this
(counting
tl which
st,
children
map
We next nodes
....
all 2_ possible
to 0 is a counit cocommutative.
are
the coalgebra
children
by }.
algebras If A is a
It can be shown that Denote Two
the
important
connected
Hopf
is a Lie subalgebra
enveloping
theorems
Hopf
Theorem
P(A)
universal
algebras
concerning
are the
of the
Lie algebra
the structure
L by U(L).
of cocommutative
graded
following.
2 (Milnor-Moore)
algebra.
of A-.
algebra
Let A be a cocommutative
graded
connected
Then a _- U(P(A))
as Hopf
algebras.
Theorem dered
3 (Poincar_-Birkhoff-Witt)
basis xl,
...,
x,_, ....
Let
{x;: ... is a basis for See page
[10,
page
A natural
244] or
l i, < ... < i,; o
