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In-Depth Information
≤
≤
dual are isomorphic, we can also think if
t
˕
living in
U
W
. We define the rank
of a bilinear map
˕
to be the rank of the corresponding tensor
t
˕
.If
A
is an finite
dimensional associative algebra with unity, that is,
A
is a ring which is also a finite
dimensional vector space over some field
k
, then themultiplicationmap in
A
is a bilin-
ear mapping
A
V
of
A
is the rank of its multiplication map.
If we think in coordinates, we get the tensor that corresponds to
A
as follows.
Choose a basis
x
1
,...,
×
A
≥
A
. The rank
R
(
A
)
x
n
of
A
. The product of any two elements of
A
is again an
element of
A
and can be written as a linear combination of
x
1
,...,
x
n
. In particular
n
x
i
·
x
j
=
1
˃
i
,
j
,
k
x
k
.
k
=
The so-called
structural constants
˃
i
,
j
,
k
are the entries of the tensor (with respect to
the chosen basis). Since a change of basis is an isomorphim of vector spaces, we get
that the rank of this tensor is independent of the chosen basis.
The best lower bounds for the rank of an algebra and for any other tensor of order
three are of the form3 dim
A
−
(
)
o
dim
A
. Very recently, Landsberg proved this for the
algebra
k
n
×
n
of
n
×
n
-matrices. An earlier example with an easier proof is the algebra
[
X
1
,...,
X
n
]
/
k
I
d
where
I
d
is the ideal generated by all monomials of degree
d
,see
[
Blä01
]. Because the families of algebras have a “regular” structure, it is clear that the
corresponding tensors are explicit. We just have to compute the structural constants.
For instance, in the case of the algebra
k
n
×
n
with the standard basis, we have
⊡
e
i
,
j
if
i
=
j
,
e
i
,
i
e
j
,
j
=
0
otherwise.
(Note that we use double indices since dim
k
n
×
n
n
2
.) In the second case, if we
=
take all monomials of degree
d
as a basis, we get a similar expression.
It is a major open problem to find explicit tensors or explicit families of algebras
with a larger rank.
<
k
n
k
n
k
n
with
Open Problem 4
1. Is there an explicit family of tensors
t
n
∗
≤
≤
0.
2. Can we even achieve this for tensors corresponding to the multiplication in an
algebra, i.e., is there an explicit family of algebras
A
n
with
R
R
(
t
n
)
∧
(
3
+
∂)
n
for some
∂ >
(
A
n
)
∧
(
3
+
∂)
dim
A
n
for some
∂ >
0. Of course, dim
A
n
should go to infinity.
6.3.2 From Tensors of Order Three to Higher Order Tensors
We can use the lower bounds of the rank of tensors of order three to obtains bounds
for the rank of higher order tensors. Up to lower order terms, they match the current
best lower bounds (see the next section).
