Lie bialgebras, degreewise duality, and root-graded Manin triples

Let $k$ be a field of characteristic $0$. A Lie bialgebra is a Lie algebra $\mathfrak{a}$ over $k$ together with a linear map ${\delta }_{\mathfrak{a}}:\mathfrak{a}\to {\mathrm{\Lambda }}^2\mathfrak{a}$ such that

$\mathrm{Alt}({\delta }_{\mathfrak{a}}\otimes \mathrm{id}){\delta }_{\mathfrak{a}}=0$

and ${\delta }_{\mathfrak{a}}([x,y])=[x,{\delta }_{\mathfrak{a}}(y)]-[y,{\delta }_{\mathfrak{a}}(x)]$ for all $x,y\in \mathfrak{a}$. The bracket action on ${\mathrm{\Lambda }}^2\mathfrak{a}$ is $[x,u\wedge v]=[x,u]\wedge v+u\wedge [x,v]$.

Let $Q$ be an additive abelian group. Suppose $\mathfrak{a}={⨁}_{\alpha \in Q}{\mathfrak{a}}_{\alpha }$ and $\mathfrak{c}={⨁}_{\alpha \in Q}{\mathfrak{c}}_{\alpha }$ are $Q$-graded vector spaces with finite-dimensional graded pieces, and suppose that, within each space separately, only finitely many pairs of nonzero graded pieces have degrees summing to any fixed degree. A pairing between them is degreewise perfect if ${\mathfrak{a}}_{\alpha }$ pairs perfectly with ${\mathfrak{c}}_{-\alpha }$ and all other degree pairs are orthogonal. It then identifies $\mathfrak{c}$ with the restricted graded dual ${\mathfrak{a}}^{\mathrm{g}\mathrm{r}\mathrm{\prime }}:={⨁}_{\alpha \in Q}{\mathfrak{a}}_{\alpha }^{\ast }$. The induced pairing on exterior powers is the determinant pairing: $⟨x_1\wedge \cdots \wedge x_m,y_1\wedge \cdots \wedge y_m⟩=\det (⟨x_i,y_j⟩{)}_{i,j=1}^m$.

For the Lie bialgebra duality below, require both brackets to preserve degree: $[{\mathfrak{a}}_{\alpha },{\mathfrak{a}}_{\beta }]\subseteq {\mathfrak{a}}_{\alpha +\beta }$ and $[{\mathfrak{c}}_{\alpha },{\mathfrak{c}}_{\beta }]\subseteq {\mathfrak{c}}_{\alpha +\beta }$. Two such Lie bialgebras are dual when these pairings are degreewise perfect and their cobrackets are transposes of the opposite brackets: $⟨{\delta }_{\mathfrak{a}}(x),y\wedge z⟩=⟨x,[y,z]⟩$ and $⟨x\wedge x^{\mathrm{\prime }},{\delta }_{\mathfrak{c}}(y)⟩=⟨[x,x^{\mathrm{\prime }}],y⟩$. For $x\in {\mathfrak{a}}_{\gamma }$, degree preservation and orthogonality make $⟨x,[{\mathfrak{c}}_{\alpha },{\mathfrak{c}}_{\beta }]⟩$ vanish unless $\alpha +\beta =-\gamma$. There are only finitely many such nonzero pairs, and their finite-dimensional perfect pairings identify the transpose with a unique element of $({\mathrm{\Lambda }}^2\mathfrak{a}{)}_{\gamma }$ under the determinant pairing. The same argument applies with the two algebras exchanged, and linear extension handles arbitrary elements. Thus the transposes lie in the ordinary exterior squares rather than formal infinite sums.

A root-graded Manin triple is a $Q$-graded Lie algebra $\mathfrak{d}={⨁}_{\alpha \in Q}{\mathfrak{d}}_{\alpha }$ with finite-dimensional graded pieces, a symmetric invariant bilinear form $B$ pairing ${\mathfrak{d}}_{\alpha }$ perfectly with ${\mathfrak{d}}_{-\alpha }$ and satisfying $B({\mathfrak{d}}_{\alpha },{\mathfrak{d}}_{\beta })=0$ whenever $\alpha +\beta \mathrm{\ne }0$, and graded Lie subalgebras ${\mathfrak{d}}_{+}$ and ${\mathfrak{d}}_{-}$ such that $\mathfrak{d}={\mathfrak{d}}_{+}\oplus {\mathfrak{d}}_{-}$ as a vector space and $B({\mathfrak{d}}_{+},{\mathfrak{d}}_{+})=B({\mathfrak{d}}_{-},{\mathfrak{d}}_{-})=0$. The cross pairing is degreewise perfect: a vector in $({\mathfrak{d}}_{+}{)}_{\alpha }$ annihilating $({\mathfrak{d}}_{-}{)}_{-\alpha }$ also annihilates $({\mathfrak{d}}_{+}{)}_{-\alpha }$ by isotropy, so it is zero by perfectness on the double. The same argument applies with the two halves exchanged, and finite dimensionality gives perfectness of the cross pairing. All other degree pairs are orthogonal by the condition on $B$. Require finite degree decompositions within each of ${\mathfrak{d}}_{+}$ and ${\mathfrak{d}}_{-}$ separately, as above; a triple satisfying this requirement is called locally finite. No finite-decomposition condition is imposed on the whole double. This permits opposite Borels with infinitely many roots, since each Borel has support in one root cone. The finite-dimensional Manin-triple definition is the special case with finitely many nonzero graded pieces.