- Source: Duflo isomorphism
In mathematics, the Duflo isomorphism is an isomorphism between the center of the universal enveloping algebra of a finite-dimensional Lie algebra and the invariants of its symmetric algebra. It was introduced by Michel Duflo (1977) and later generalized to arbitrary finite-dimensional Lie algebras by Kontsevich.
The Poincaré-Birkoff-Witt theorem gives for any Lie algebra
g
{\displaystyle {\mathfrak {g}}}
a vector space isomorphism from the polynomial algebra
S
(
g
)
{\displaystyle S({\mathfrak {g}})}
to the universal enveloping algebra
U
(
g
)
{\displaystyle U({\mathfrak {g}})}
. This map is not an algebra homomorphism. It is equivariant with respect to the natural representation of
g
{\displaystyle {\mathfrak {g}}}
on these spaces, so it restricts to a vector space isomorphism
F
:
S
(
g
)
g
→
U
(
g
)
g
{\displaystyle F\colon S({\mathfrak {g}})^{\mathfrak {g}}\to U({\mathfrak {g}})^{\mathfrak {g}}}
where the superscript indicates the subspace annihilated by the action of
g
{\displaystyle {\mathfrak {g}}}
. Both
S
(
g
)
g
{\displaystyle S({\mathfrak {g}})^{\mathfrak {g}}}
and
U
(
g
)
g
{\displaystyle U({\mathfrak {g}})^{\mathfrak {g}}}
are commutative subalgebras, indeed
U
(
g
)
g
{\displaystyle U({\mathfrak {g}})^{\mathfrak {g}}}
is the center of
U
(
g
)
{\displaystyle U({\mathfrak {g}})}
, but
F
{\displaystyle F}
is still not an algebra homomorphism. However, Duflo proved that in some cases we can compose
F
{\displaystyle F}
with a map
G
:
S
(
g
)
g
→
S
(
g
)
g
{\displaystyle G\colon S({\mathfrak {g}})^{\mathfrak {g}}\to S({\mathfrak {g}})^{\mathfrak {g}}}
to get an algebra isomorphism
F
∘
G
:
S
(
g
)
g
→
U
(
g
)
g
.
{\displaystyle F\circ G\colon S({\mathfrak {g}})^{\mathfrak {g}}\to U({\mathfrak {g}})^{\mathfrak {g}}.}
Later, using the Kontsevich formality theorem, Kontsevich showed that this works for all finite-dimensional Lie algebras.
Following Calaque and Rossi, the map
G
{\displaystyle G}
can be defined as follows. The adjoint action of
g
{\displaystyle {\mathfrak {g}}}
is the map
g
→
E
n
d
(
g
)
{\displaystyle {\mathfrak {g}}\to \mathrm {End} ({\mathfrak {g}})}
sending
x
∈
g
{\displaystyle x\in {\mathfrak {g}}}
to the operation
[
x
,
−
]
{\displaystyle [x,-]}
on
g
{\displaystyle {\mathfrak {g}}}
. We can treat map as an element of
g
∗
⊗
E
n
d
(
g
)
{\displaystyle {\mathfrak {g}}^{\ast }\otimes \mathrm {End} ({\mathfrak {g}})}
or, for that matter, an element of the larger space
S
(
g
∗
)
⊗
E
n
d
(
g
)
{\displaystyle S({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})}
, since
g
∗
⊂
S
(
g
∗
)
{\displaystyle {\mathfrak {g}}^{\ast }\subset S({\mathfrak {g}}^{\ast })}
. Call this element
a
d
∈
S
(
g
∗
)
⊗
E
n
d
(
g
)
{\displaystyle \mathrm {ad} \in S({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})}
Both
S
(
g
∗
)
{\displaystyle S({\mathfrak {g}}^{\ast })}
and
E
n
d
(
g
)
{\displaystyle \mathrm {End} ({\mathfrak {g}})}
are algebras so their tensor product is as well. Thus, we can take powers of
a
d
{\displaystyle \mathrm {ad} }
, say
a
d
k
∈
S
(
g
∗
)
⊗
E
n
d
(
g
)
.
{\displaystyle \mathrm {ad} ^{k}\in S({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}}).}
Going further, we can apply any formal power series to
a
d
{\displaystyle \mathrm {ad} }
and obtain an element of
S
¯
(
g
∗
)
⊗
E
n
d
(
g
)
{\displaystyle {\overline {S}}({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})}
, where
S
¯
(
g
∗
)
{\displaystyle {\overline {S}}({\mathfrak {g}}^{\ast })}
denotes the algebra of formal power series on
g
∗
{\displaystyle {\mathfrak {g}}^{\ast }}
. Working with formal power series, we thus obtain an element
e
a
d
−
e
−
a
d
a
d
∈
S
¯
(
g
∗
)
⊗
E
n
d
(
g
)
{\displaystyle {\sqrt {\frac {e^{\mathrm {ad} }-e^{-\mathrm {ad} }}{\mathrm {ad} }}}\in {\overline {S}}({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})}
Since the dimension of
g
{\displaystyle {\mathfrak {g}}}
is finite, one can think of
E
n
d
(
g
)
{\displaystyle \mathrm {End} ({\mathfrak {g}})}
as
M
n
(
R
)
{\displaystyle \mathrm {M} _{n}(\mathbb {R} )}
, hence
S
¯
(
g
∗
)
⊗
E
n
d
(
g
)
{\displaystyle {\overline {S}}({\mathfrak {g}}^{\ast })\otimes \mathrm {End} ({\mathfrak {g}})}
is
M
n
(
S
¯
(
g
∗
)
)
{\displaystyle \mathrm {M} _{n}({\overline {S}}({\mathfrak {g}}^{\ast }))}
and by applying the determinant map, we obtain an element
J
~
1
/
2
:=
d
e
t
e
a
d
−
e
−
a
d
a
d
∈
S
¯
(
g
∗
)
{\displaystyle {\tilde {J}}^{1/2}:=\mathrm {det} {\sqrt {\frac {e^{\mathrm {ad} }-e^{-\mathrm {ad} }}{\mathrm {ad} }}}\in {\overline {S}}({\mathfrak {g}}^{\ast })}
which is related to the Todd class in algebraic topology.
Now,
g
∗
{\displaystyle {\mathfrak {g}}^{\ast }}
acts as derivations on
S
(
g
)
{\displaystyle S({\mathfrak {g}})}
since any element of
g
∗
{\displaystyle {\mathfrak {g}}^{\ast }}
gives a translation-invariant vector field on
g
{\displaystyle {\mathfrak {g}}}
. As a result, the algebra
S
(
g
∗
)
{\displaystyle S({\mathfrak {g}}^{\ast })}
acts on
as differential operators on
S
(
g
)
{\displaystyle S({\mathfrak {g}})}
, and this extends to an action of
S
¯
(
g
∗
)
{\displaystyle {\overline {S}}({\mathfrak {g}}^{\ast })}
on
S
(
g
)
{\displaystyle S({\mathfrak {g}})}
. We can thus define a linear map
G
:
S
(
g
)
→
S
(
g
)
{\displaystyle G\colon S({\mathfrak {g}})\to S({\mathfrak {g}})}
by
G
(
ψ
)
=
J
~
1
/
2
ψ
{\displaystyle G(\psi )={\tilde {J}}^{1/2}\psi }
and since the whole construction was invariant,
G
{\displaystyle G}
restricts to the desired linear map
G
:
S
(
g
)
g
→
S
(
g
)
g
.
{\displaystyle G\colon S({\mathfrak {g}})^{\mathfrak {g}}\to S({\mathfrak {g}})^{\mathfrak {g}}.}
Properties
For a nilpotent Lie algebra the Duflo isomorphism coincides with the symmetrization map from symmetric algebra to universal enveloping algebra. For a semisimple Lie algebra the Duflo isomorphism is compatible in a natural way with the Harish-Chandra isomorphism.
References
Duflo, Michel (1977), "Opérateurs différentiels bi-invariants sur un groupe de Lie", Annales Scientifiques de l'École Normale Supérieure, Série 4, 10 (2): 265–288, doi:10.24033/asens.1327, ISSN 0012-9593, MR 0444841
Calaque, Damien; Rossi, Carlo A. (2011), Lectures on Duflo isomorphisms in Lie algebra and complex geometry, EMS Series of Lectures in Mathematics, Zürich: European Mathematical Society, doi:10.4171/096, hdl:21.11116/0000-0004-2127-B, ISBN 978-3-03719-096-8, MR 2816610