Let
be the
free associative algebra
on
over the integers. Give
a
Hopf algebra
structure by means of the following co-multiplication, augmentation, and
antipode:
where
where the sum is over all strings
,
,
such that
.
This makes
a Hopf algebra, called the
Leibniz–Hopf algebra.
This Hopf algebra is important, e.g., in the theory of curves of non-commutative
formal groups (see
Formal group)
[a1],
[a2],
[a5].
Its commutative quotient
,
with the same co-multiplication, is the underlying Hopf algebra of the (big)
Witt vector functor
(see
Witt vector)
and it plays an important role in the classification theory of unipotent commutative
algebraic groups and in the theory of commutative formal groups (amongst other things)
[a3].
The Leibniz–Hopf algebra
is free as a
-module
and graded. Its graded dual is also a Hopf algebra, whose underlying algebra is the
overlapping shuffle algebra
.
As a
-module,
is free with basis
,
the free monoid (see
Free semi-group)
of all words in the alphabet
with the duality pairing
given by
The
overlapping shuffle product
of two such words
,
is equal to
where the sum is over all
and pairs of order-preserving injective mappings
,
such that
,
and where
with
if
,
and similarly for
.
For example,
The terms of maximal length of the overlapping shuffle product form the shuffle product,
see
Shuffle algebra.
A word
,
,
is
elementary
if the greatest common divisor of
is
.
With this terminology, the
Ditters–Scholtens theorem
[a4],
[a5]
says that, as an algebra over
,
the overlapping shuffle algebra
is the free commutative polynomial algebra with as generators the elementary
concatenation powers
of elementary Lyndon words (see
Lyndon word).
(E.g., the third concatenation power of
is
.)
In contrast with the case of the
shuffle algebra,
this theorem already holds over
(not just over
).