<s>
In	O
mathematics	O
,	O
Manin	B-Algorithm
matrices	I-Algorithm
,	O
named	O
after	O
Yuri	O
Manin	O
who	O
introduced	O
them	O
around	O
1987	O
–	O
88	O
,	O
are	O
a	O
class	O
of	O
matrices	B-Architecture
with	O
elements	O
in	O
a	O
not-necessarily	O
commutative	O
ring	O
,	O
which	O
in	O
a	O
certain	O
sense	O
behave	O
like	O
matrices	B-Architecture
whose	O
elements	O
commute	O
.	O
</s>
<s>
In	O
particular	O
there	O
is	O
natural	O
definition	O
of	O
the	O
determinant	O
for	O
them	O
and	O
most	O
linear	B-Language
algebra	I-Language
theorems	O
like	O
Cramer	O
's	O
rule	O
,	O
Cayley	O
–	O
Hamilton	O
theorem	O
,	O
etc	O
.	O
</s>
<s>
Any	O
matrix	O
with	O
commuting	O
elements	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
These	O
matrices	B-Architecture
have	O
applications	O
in	O
representation	O
theory	O
in	O
particular	O
to	O
Capelli	O
's	O
identity	O
,	O
Yangian	O
and	O
quantum	O
integrable	O
systems	O
.	O
</s>
<s>
Manin	B-Algorithm
matrices	I-Algorithm
are	O
particular	O
examples	O
of	O
Manin	O
's	O
general	O
construction	O
of	O
"	O
non-commutative	O
symmetries	O
"	O
which	O
can	O
be	O
applied	O
to	O
any	O
algebra	O
.	O
</s>
<s>
Taking	O
(	O
q	O
)	O
-(super )	O
-commuting	O
variables	O
one	O
will	O
get	O
(	O
q	O
)	O
-(super )	O
-analogs	O
of	O
Manin	B-Algorithm
matrices	I-Algorithm
,	O
which	O
are	O
closely	O
related	O
to	O
quantum	O
groups	O
.	O
</s>
<s>
In	O
that	O
sense	O
it	O
should	O
be	O
stressed	O
that	O
(	O
q	O
)	O
-Manin	O
matrices	O
are	O
defined	O
only	O
by	O
half	O
of	O
the	O
relations	O
of	O
related	O
quantum	O
group	O
Funq(GL )	O
,	O
and	O
these	O
relations	O
are	O
enough	O
for	O
many	O
linear	B-Language
algebra	I-Language
theorems	O
.	O
</s>
<s>
Matrices	B-Architecture
with	O
generic	O
noncommutative	O
elements	O
do	O
not	O
admit	O
a	O
natural	O
construction	O
of	O
the	O
determinant	O
with	O
values	O
in	O
a	O
ground	O
ring	O
and	O
basic	O
theorems	O
of	O
the	O
linear	B-Language
algebra	I-Language
fail	O
to	O
hold	O
true	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
if	O
one	O
considers	O
certain	O
specific	O
classes	O
of	O
matrices	B-Architecture
with	O
non-commutative	O
elements	O
,	O
then	O
there	O
are	O
examples	O
where	O
one	O
can	O
define	O
the	O
determinant	O
and	O
prove	O
linear	B-Language
algebra	I-Language
theorems	O
which	O
are	O
very	O
similar	O
to	O
their	O
commutative	O
analogs	O
.	O
</s>
<s>
Examples	O
include	O
:	O
quantum	O
groups	O
and	O
q-determinant	O
;	O
Capelli	O
matrix	O
and	O
Capelli	O
determinant	O
;	O
super-matrices	O
and	O
Berezinian	O
.	O
</s>
<s>
Manin	B-Algorithm
matrices	I-Algorithm
is	O
a	O
general	O
and	O
natural	O
class	O
of	O
matrices	B-Architecture
with	O
not-necessarily	O
commutative	O
elements	O
which	O
admit	O
natural	O
definition	O
of	O
the	O
determinant	O
and	O
generalizations	O
of	O
the	O
linear	B-Language
algebra	I-Language
theorems	O
.	O
</s>
<s>
An	O
n	O
by	O
m	O
matrix	O
M	O
with	O
entries	O
Mij	O
over	O
a	O
ring	O
R	O
(	O
not	O
necessarily	O
commutative	O
)	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
if	O
all	O
elements	O
in	O
a	O
given	O
column	O
commute	O
and	O
if	O
for	O
all	O
i	O
,	O
j	O
,	O
k	O
,	O
l	O
it	O
holds	O
that	O
 [ Mij , Mkl ] 	O
=	O
 [ Mkj , Mil ] 	O
.	O
</s>
<s>
A	O
rectangular	B-Architecture
matrix	I-Architecture
M	O
is	O
called	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
if	O
for	O
any	O
2×2	O
submatrix	O
,	O
consisting	O
of	O
rows	O
i	O
and	O
k	O
,	O
and	O
columns	O
j	O
and	O
l	O
:	O
</s>
<s>
Below	O
are	O
presented	O
some	O
examples	O
of	O
the	O
appearance	O
of	O
the	O
Manin	O
property	O
in	O
various	O
very	O
simple	O
and	O
natural	O
questions	O
concerning	O
2×2	O
matrices	B-Architecture
.	O
</s>
<s>
The	O
general	O
idea	O
is	O
the	O
following	O
:	O
consider	O
well-known	O
facts	O
of	O
linear	B-Language
algebra	I-Language
and	O
look	O
how	O
to	O
relax	O
the	O
commutativity	O
assumption	O
for	O
matrix	O
elements	O
such	O
that	O
the	O
results	O
will	O
be	O
preserved	O
to	O
be	O
true	O
.	O
</s>
<s>
The	O
answer	O
is	O
:	O
if	O
and	O
only	O
if	O
M	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Then	O
y1	O
,	O
y2	O
commute	O
among	O
themselves	O
if	O
and	O
only	O
if	O
M	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
anticommute	O
among	O
themselves	O
and	O
φi2	O
=	O
0	O
)	O
if	O
and	O
only	O
if	O
M	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Observations	O
1	O
,	O
2	O
holds	O
true	O
for	O
general	O
n	O
×	O
m	O
Manin	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
They	O
demonstrate	O
original	O
Manin	O
's	O
approach	O
as	O
described	O
below	O
(	O
one	O
should	O
thought	O
of	O
usual	O
matrices	B-Architecture
as	O
homomorphisms	O
of	O
polynomial	O
rings	O
,	O
while	O
Manin	B-Algorithm
matrices	I-Algorithm
are	O
more	O
general	O
"	O
non-commutative	O
homomorphisms	O
"	O
)	O
.	O
</s>
<s>
Pay	O
attention	O
that	O
polynomial	O
algebra	O
generators	O
are	O
presented	O
as	O
column	O
vectors	O
,	O
while	O
Grassmann	O
algebra	O
as	O
row-vectors	O
,	O
the	O
same	O
can	O
be	O
generalized	O
to	O
arbitrary	O
pair	O
of	O
Koszul	O
dual	O
algebras	O
and	O
associated	O
general	O
Manin	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
if	O
and	O
only	O
if	O
M	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
holds	O
if	O
and	O
only	O
if	O
M	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
detcolumn(MN )	O
=	O
detcolumn(M )	O
det(N )	O
holds	O
true	O
for	O
all	O
complex-valued	O
matrices	B-Architecture
N	O
if	O
and	O
only	O
if	O
M	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Manin	B-Algorithm
matrices	I-Algorithm
considered	O
here	O
are	O
examples	O
of	O
this	O
general	O
construction	O
applied	O
to	O
polynomial	O
algebras	O
C[ x1	O
,	O
xn	O
...	O
]	O
.	O
</s>
<s>
Somewhat	O
surprising	O
fact	O
is	O
that	O
this	O
construction	O
applied	O
to	O
the	O
polynomial	O
algebra	O
C[ x1	O
,	O
...	O
,	O
xn ]	O
will	O
give	O
not	O
the	O
usual	O
algebra	O
of	O
matrices	B-Architecture
Matn	O
(	O
more	O
precisely	O
algebra	O
of	O
function	O
on	O
it	O
)	O
,	O
but	O
much	O
bigger	O
non-commutative	O
algebra	O
of	O
Manin	B-Algorithm
matrices	I-Algorithm
(	O
more	O
precisely	O
algebra	O
generated	O
by	O
elements	O
Mij	O
.	O
</s>
<s>
The	O
elements	O
commute	O
among	O
themselves	O
if	O
and	O
only	O
if	O
M	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
two	O
propositions	O
above	O
imply	O
that	O
the	O
algebra	O
generated	O
by	O
elements	O
of	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
is	O
a	O
bialgebra	O
coacting	O
on	O
the	O
polynomial	O
algebra	O
.	O
</s>
<s>
Any	O
matrix	O
with	O
commuting	O
elements	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Any	O
matrix	O
whose	O
elements	O
from	O
different	O
rows	O
commute	O
among	O
themselves	O
(	O
such	O
matrices	B-Architecture
sometimes	O
called	O
Cartier-Foata	O
matrices	B-Architecture
)	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Any	O
submatrix	O
of	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
One	O
can	O
interchange	O
rows	O
and	O
columns	O
in	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
the	O
result	O
will	O
also	O
be	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
One	O
can	O
add	O
row	O
or	O
column	O
multiplied	O
by	O
the	O
central	O
element	O
to	O
another	O
row	O
or	O
column	O
and	O
results	O
will	O
be	O
Manin	B-Algorithm
matrix	I-Algorithm
again	O
.	O
</s>
<s>
Consider	O
two	O
Manin	B-Algorithm
matrices	I-Algorithm
M	O
,	O
N	O
such	O
that	O
their	O
all	O
elements	O
commute	O
,	O
then	O
the	O
sum	O
M+N	O
and	O
the	O
product	O
MN	O
will	O
also	O
be	O
Manin	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
If	O
matrix	O
M	O
and	O
simultaneously	O
transpose	O
matrix	O
Mt	O
are	O
Manin	B-Algorithm
matrices	I-Algorithm
,	O
then	O
all	O
elements	O
of	O
M	O
commute	O
with	O
each	O
other	O
.	O
</s>
<s>
No-go	O
facts	O
:	O
Mk	O
is	O
not	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
in	O
general	O
(	O
except	O
k	O
=	O
-1	O
discussed	O
below	O
)	O
;	O
neither	O
det(M )	O
,	O
nor	O
Tr(M )	O
are	O
central	O
in	O
the	O
algebra	O
generated	O
by	O
Mij	O
in	O
general	O
(	O
in	O
that	O
respect	O
Manin	B-Algorithm
matrices	I-Algorithm
differs	O
from	O
quantum	O
groups	O
)	O
;	O
det(eM )	O
≠	O
eTr(M )	O
;	O
log(det(M )	O
)	O
≠	O
Tr(log(M )	O
)	O
.	O
</s>
<s>
xij	O
,	O
form	O
matrices	B-Architecture
X	O
,	O
D	O
with	O
the	O
corresponding	O
elements	O
.	O
</s>
<s>
The	O
determinant	O
of	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
can	O
be	O
defined	O
by	O
the	O
standard	O
formula	O
,	O
with	O
the	O
prescription	O
that	O
elements	O
from	O
the	O
first	O
columns	O
comes	O
first	O
in	O
the	O
product	O
.	O
</s>
<s>
Many	O
linear	B-Language
algebra	I-Language
statements	O
hold	O
for	O
Manin	B-Algorithm
matrices	I-Algorithm
even	O
when	O
R	O
is	O
not	O
commutative	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
determinant	O
can	O
be	O
defined	O
in	O
the	O
standard	O
way	O
using	O
permutations	B-Algorithm
and	O
it	O
satisfies	O
a	O
Cramer	O
's	O
rule	O
.	O
</s>
<s>
MacMahon	O
Master	O
theorem	O
holds	O
true	O
for	O
Manin	B-Algorithm
matrices	I-Algorithm
and	O
actually	O
for	O
their	O
generalizations	O
(	O
super	O
)	O
,	O
(	O
q	O
)	O
,	O
etc	O
.	O
</s>
<s>
The	O
inverse	O
to	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
M	O
can	O
be	O
defined	O
by	O
the	O
standard	O
formula	O
:	O
</s>
<s>
Assume	O
a	O
two-sided	O
inverse	O
to	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
M	O
exists	O
,	O
then	O
it	O
will	O
also	O
be	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
are	O
Manin	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
The	O
Capelli	O
identity	O
from	O
19th	O
century	O
gives	O
one	O
of	O
the	O
first	O
examples	O
of	O
determinants	O
for	O
matrices	B-Architecture
with	O
non-commuting	O
elements	O
.	O
</s>
<s>
Manin	B-Algorithm
matrices	I-Algorithm
give	O
a	O
new	O
look	O
on	O
this	O
classical	O
subject	O
.	O
</s>
<s>
Take	O
Eij	O
be	O
matrices	B-Architecture
with	O
1	O
at	O
position	O
(	O
i	O
,	O
j	O
)	O
and	O
zeros	O
everywhere	O
else	O
.	O
</s>
<s>
It	O
is	O
matrix	O
with	O
elements	O
in	O
ring	O
of	O
matrices	B-Architecture
Matn	O
.	O
</s>
<s>
It	O
is	O
not	O
Manin	B-Algorithm
matrix	I-Algorithm
however	O
there	O
are	O
modifications	O
which	O
transform	O
it	O
to	O
Manin	B-Algorithm
matrix	I-Algorithm
as	O
described	O
below	O
.	O
</s>
<s>
The	O
matrix	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
matrix	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
On	O
the	O
right	O
hand	O
side	O
of	O
this	O
equality	O
one	O
recognizes	O
the	O
Capelli	O
determinant	O
(	O
or	O
more	O
precisely	O
the	O
Capelli	O
characteristic	O
polynomial	O
)	O
,	O
while	O
on	O
the	O
left	O
hand	O
side	O
one	O
has	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
with	O
its	O
natural	O
determinant	O
.	O
</s>
<s>
So	O
Manin	B-Algorithm
matrices	I-Algorithm
gives	O
new	O
look	O
on	O
Capelli	O
's	O
determinant	O
.	O
</s>
<s>
Moreover	O
,	O
Capelli	O
identity	O
and	O
its	O
generalization	O
can	O
be	O
derived	O
by	O
techniques	O
of	O
Manin	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
So	O
the	O
only	O
property	O
used	O
here	O
is	O
that	O
which	O
is	O
true	O
for	O
any	O
Manin	B-Algorithm
matrix	I-Algorithm
M	O
and	O
any	O
matrix	O
g	O
with	O
central	O
(	O
e.g.	O
</s>
<s>
Then	O
the	O
matrix	O
exp( 	O
-d/dz	O
)	O
T(z )	O
is	O
a	O
Manin	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
So	O
the	O
determinant	O
in	O
Yangian	O
theory	O
has	O
natural	O
interpretation	O
via	O
Manin	B-Algorithm
matrices	I-Algorithm
.	O
</s>
<s>
of	O
these	O
expressions	O
has	O
been	O
first	O
proposed	O
by	O
the	O
use	O
of	O
Newton	O
identities	O
for	O
Manin	B-Algorithm
matrices	I-Algorithm
:	O
</s>
<s>
the	O
particular	O
case	O
which	O
is	O
called	O
Manin	B-Algorithm
matrices	I-Algorithm
is	O
discussed	O
in	O
,	O
where	O
some	O
basic	O
properties	O
were	O
outlined	O
.	O
</s>
<s>
Quantum	O
matrices	B-Architecture
Funq(GLn )	O
can	O
be	O
defined	O
as	O
such	O
matrices	B-Architecture
that	O
T	O
and	O
simultaneously	O
Tt	O
are	O
q-Manin	O
matrices	B-Architecture
(	O
i.e.	O
</s>
<s>
After	O
original	O
Manin	O
's	O
works	O
there	O
were	O
only	O
a	O
few	O
papers	O
on	O
Manin	B-Algorithm
matrices	I-Algorithm
until	O
2003	O
.	O
</s>
<s>
But	O
around	O
and	O
some	O
after	O
this	O
date	O
Manin	B-Algorithm
matrices	I-Algorithm
appeared	O
in	O
several	O
not	O
quite	O
related	O
areas	O
:	O
obtained	O
certain	O
noncommutative	O
generalization	O
of	O
the	O
MacMahon	O
master	O
identity	O
,	O
which	O
was	O
used	O
in	O
knot	O
theory	O
;	O
applications	O
to	O
quantum	O
integrable	O
systems	O
,	O
Lie	O
algebras	O
has	O
been	O
found	O
in	O
;	O
generalizations	O
of	O
the	O
Capelli	O
identity	O
involving	O
Manin	B-Algorithm
matrices	I-Algorithm
appeared	O
in	O
.	O
</s>
