- Source: Manin triple
In mathematics, a Manin triple
(
g
,
p
,
q
)
{\displaystyle ({\mathfrak {g}},{\mathfrak {p}},{\mathfrak {q}})}
consists of a Lie algebra
g
{\displaystyle {\mathfrak {g}}}
with a non-degenerate invariant symmetric bilinear form, together with two isotropic subalgebras
p
{\displaystyle {\mathfrak {p}}}
and
q
{\displaystyle {\mathfrak {q}}}
such that
g
{\displaystyle {\mathfrak {g}}}
is the direct sum of
p
{\displaystyle {\mathfrak {p}}}
and
q
{\displaystyle {\mathfrak {q}}}
as a vector space. A closely related concept is the (classical) Drinfeld double, which is an even dimensional Lie algebra which admits a Manin decomposition.
Manin triples were introduced by Vladimir Drinfeld in 1987, who named them after Yuri Manin.
In 2001 Delorme classified Manin triples where
g
{\displaystyle {\mathfrak {g}}}
is a complex reductive Lie algebra.
Manin triples and Lie bialgebras
There is an equivalence of categories between finite-dimensional Manin triples and finite-dimensional Lie bialgebras.
More precisely, if
(
g
,
p
,
q
)
{\displaystyle ({\mathfrak {g}},{\mathfrak {p}},{\mathfrak {q}})}
is a finite-dimensional Manin triple, then
p
{\displaystyle {\mathfrak {p}}}
can be made into a Lie bialgebra by letting the cocommutator map
p
→
p
⊗
p
{\displaystyle {\mathfrak {p}}\to {\mathfrak {p}}\otimes {\mathfrak {p}}}
be the dual of the Lie bracket
q
⊗
q
→
q
{\displaystyle {\mathfrak {q}}\otimes {\mathfrak {q}}\to {\mathfrak {q}}}
(using the fact that the symmetric bilinear form on
g
{\displaystyle {\mathfrak {g}}}
identifies
q
{\displaystyle {\mathfrak {q}}}
with the dual of
p
{\displaystyle {\mathfrak {p}}}
).
Conversely if
p
{\displaystyle {\mathfrak {p}}}
is a Lie bialgebra then one can construct a Manin triple
(
p
⊕
p
∗
,
p
,
p
∗
)
{\displaystyle ({\mathfrak {p}}\oplus {\mathfrak {p}}^{*},{\mathfrak {p}},{\mathfrak {p}}^{*})}
by letting
q
{\displaystyle {\mathfrak {q}}}
be the dual of
p
{\displaystyle {\mathfrak {p}}}
and defining the commutator of
p
{\displaystyle {\mathfrak {p}}}
and
q
{\displaystyle {\mathfrak {q}}}
to make the bilinear form on
g
=
p
⊕
q
{\displaystyle {\mathfrak {g}}={\mathfrak {p}}\oplus {\mathfrak {q}}}
invariant.
Examples
Suppose that
a
{\displaystyle {\mathfrak {a}}}
is a complex semisimple Lie algebra with invariant symmetric bilinear form
(
⋅
,
⋅
)
{\displaystyle (\cdot ,\cdot )}
. Then there is a Manin triple
(
g
,
p
,
q
)
{\displaystyle ({\mathfrak {g}},{\mathfrak {p}},{\mathfrak {q}})}
with
g
=
a
⊕
a
{\displaystyle {\mathfrak {g}}={\mathfrak {a}}\oplus {\mathfrak {a}}}
, with the scalar product on
g
{\displaystyle {\mathfrak {g}}}
given by
(
(
w
,
x
)
,
(
y
,
z
)
)
=
(
w
,
y
)
−
(
x
,
z
)
{\displaystyle ((w,x),(y,z))=(w,y)-(x,z)}
. The subalgebra
p
{\displaystyle {\mathfrak {p}}}
is the space of diagonal elements
(
x
,
x
)
{\displaystyle (x,x)}
, and the subalgebra
q
{\displaystyle {\mathfrak {q}}}
is the space of elements
(
x
,
y
)
{\displaystyle (x,y)}
with
x
{\displaystyle x}
in a fixed Borel subalgebra containing a Cartan subalgebra
h
{\displaystyle {\mathfrak {h}}}
,
y
{\displaystyle y}
in the opposite Borel subalgebra, and where
x
{\displaystyle x}
and
y
{\displaystyle y}
have the same component in
h
{\displaystyle {\mathfrak {h}}}
.