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.

tIOPF-A LGc-_)SA I £- -STRUC TUR: _ O:;JECTS AND _IFFFR_NT[AL Univ.} 10 p C$CL 12A C,I_I6T

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

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.

tIOPF-A LGc-_)SA I £- -STRUC TUR: _ O:;JECTS AND _IFFFR_NT[AL Univ.} 10 p C$CL 12A C,I_I6T

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