1 Reminder of notation
| {\mathfrak{a}} | Lie algebra over a field K | elements x,y,\ldots |
| {\mathfrak{a}}' | dual vector space of {\mathfrak{a}} | elements x',y',\ldots |
| \Lambda({\mathfrak{a}}) | exterior algebra of {\mathfrak{a}} | elements \alpha,\beta,\ldots |
| \Lambda({\mathfrak{a}}') | exterior algebra of {\mathfrak{a}}' | elements \alpha',\beta',\ldots |
The operator \theta(x) is the infinitesimal transformation defined by x\in{\mathfrak{a}} on \Lambda({\mathfrak{a}}); from now on, we denote by \theta^*(x) (instead of \theta(x)) the operator that is the transpose of -\theta(x) acting on \Lambda({\mathfrak{a}}').
The operator e(x) is the (left) exterior product with x\in{\mathfrak{a}} on \Lambda({\mathfrak{a}}); the operator i(x) is the interior product with x\in{\mathfrak{a}} on \Lambda({\mathfrak{a}}'); i(x) and e(x) are the transpose of one another.
The operator \partial is the boundary operator on \Lambda({\mathfrak{a}}); the operator \delta is the coboundary operator on \Lambda({\mathfrak{a}}'); \delta is the transpose of -\partial.
These operators satisfy the following relations: \begin{aligned} \partial\partial &= 0, \\\delta\delta &= 0, \\\theta(x) &= e(x)\partial+\partial e(x), \\\theta^*(x) &= i(x)\delta+\delta i(x), \\\partial\theta(x) &= \theta(x)\partial, \\\delta\theta^*(x) &= \theta^*(x)\delta. \end{aligned} \tag{1}
The homology vector space \operatorname{H}({\mathfrak{a}}) is defined by \partial acting on \Lambda({\mathfrak{a}}); the cohomology algebra \operatorname{H}({\mathfrak{a}}') is defined by \delta acting on \Lambda({\mathfrak{a'}}). There is a multiplicative structure on \operatorname{H}({\mathfrak{a}}') because \delta is an antiderivation. The vector spaces \operatorname{H}({\mathfrak{a}}) and \operatorname{H}({\mathfrak{a}}') are canonically in duality.
2 Direct sum of two Lie algebras
We say that {\mathfrak{a}} is the direct sum of two sub-algebras {\mathfrak{b}},{\mathfrak{c}}\subseteq{\mathfrak{a}}, and we write {\mathfrak{a}}={\mathfrak{b}}\oplus{\mathfrak{c}}, if
- the vector space {\mathfrak{a}} is the direct sum of the subspaces {\mathfrak{b}} and {\mathfrak{c}} ; and
- [x,y]=0 for x\in{\mathfrak{b}} and y\in{\mathfrak{c}}.
If this is the case, then \Gamma({\mathfrak{a}}) is canonically isomorphic to the tensor product \Gamma({\mathfrak{b}})\otimes\Gamma({\mathfrak{c}}); if we transport the multiplicative structure of \Gamma({\mathfrak{a}}) to this tensor product, then we see that (\beta_1\otimes\gamma_1)\cdot(\beta_2\otimes\gamma_2) = (-1)^{pq}(\beta_1\wedge\beta_2)\otimes(\gamma_1\otimes\gamma_2) (where p=\deg\beta_2 and q=\deg\gamma_1), which describes the notion of the tensor product of graded algebras. Finally, the boundary operator \partial on \Gamma({\mathfrak{b}})\otimes\Gamma({\mathfrak{c}}) satisfies \partial(\beta\otimes\gamma) = (\partial\beta)\otimes\gamma + \overline{\beta}\otimes(\partial\gamma) (where \overline{\beta}=(-1)^p\beta for \beta of degree p). It thus follows that the vector space \operatorname{H}({\mathfrak{a}}) can be identified with the tensor product \operatorname{H}({\mathfrak{b}})\otimes\operatorname{H}({\mathfrak{c}}).
Similarly, \Gamma({\mathfrak{a}}') can be identified with the tensor product \Gamma({\mathfrak{b}}')\otimes\Gamma({\mathfrak{c}}') of graded algebras. The duality between \Gamma({\mathfrak{b}})\otimes\Gamma({\mathfrak{c}}) and \Gamma({\mathfrak{b}}')\otimes\Gamma({\mathfrak{c}}') is defined by \langle\beta\otimes\gamma,\beta'\otimes\gamma'\rangle = \langle\beta,\beta'\rangle \cdot \langle\gamma,\gamma'\rangle. The operator \delta on \Gamma({\mathfrak{b}}')\otimes\Gamma({\mathfrak{c}}') satisfies \delta(\beta'\otimes\gamma') = (\delta\beta')\otimes\gamma' + \overline{\beta'}\otimes(\delta\gamma'), and the cohomology algebra \operatorname{H}({\mathfrak{a}}') can be identified with the tensor product \operatorname{H}({\mathfrak{b}}')\otimes\operatorname{H}({\mathfrak{c}}') of graded algebras.
If {\mathfrak{b}} and {\mathfrak{c}} are the Lie algebras of compact connected Lie groups G_1 and G_2 (respectively), then {\mathfrak{a}}={\mathfrak{b}}\oplus{\mathfrak{c}} is the Lie algebra of the product group G_1\times G_2; we recover the following theorem: the cohomology algebra (with real coefficients) of G_1\times G_2 is canonically identified with the tensor product of the cohomology algebras of G_1 and G_2.
3 Semi-simple Lie algebras
Every time that we speak of semi-simplicity, the field K will be assumed to be of characteristic 0. We have the following equivalent characterisations of semi-simple Lie algebras
every square-zero ideal (i.e. giving rise to an invariant abelian subgroup) is the ideal \{0\};
the quadratic form \operatorname{tr}(\theta(x)\theta(y)) is regular;
{\mathfrak{a}} is the direct sum of simple algebras (i.e. algebras of dimension >1 with only trivial ideals, namely \{0\} and the whole algebra itself); such a decomposition is then unique. (In principal, determining \operatorname{H}({\mathfrak{a}}) and \operatorname{H}({\mathfrak{a}}') for semi-simple {\mathfrak{a}} thus reduces to determining the homology and cohomology of the simple algebras) ; and
every representation of {\mathfrak{a}} in the endomorphism algebra of a vector space E (of finite dimension over K) is completely reducible (i.e. every invariant subspace admits an invariant complement).
Every quotient algebra of a semi-simple algebra is semi-simple. We thus deduce: if {\mathfrak{a}} is semi-simple, then {\mathfrak{a}} is identical to the “derived” algebra {\mathfrak{a}}^2. In particular, a semi-simple algebra is unimodular (cf. §I).
For every Lie algebra {\mathfrak{a}}, the map x\mapsto\theta(x) is a representation of {\mathfrak{a}} in the endomorphism algebra of the vector space \Gamma({\mathfrak{a}}); similarly, x\mapsto\theta^*({\mathfrak{a}}) is a representation in the endomorphism algebra of \Gamma({\mathfrak{a}}'). These representations vanish on the centre {\mathfrak{c}} of {\mathfrak{a}}; we will be sure that they are completely reducible if the algebra {\mathfrak{a}}/{\mathfrak{c}} is semi-simple, or, in other words, if {\mathfrak{a}} is the direct sum of a semi-simple algebra and an abelian algebra. Such a Lie algebra is said to be reductive; we are particularly interested in the homology and cohomology of reductive Lie algebras. The Lie algebra of a compact group is always reductive.
4 Homology of reductive Lie algebras
In the representation x\mapsto\theta(x) of {\mathfrak{a}} in the endomorphism algebra of \Gamma({\mathfrak{a}}), by (1), the \theta(x) is a of a cycle is a boundary; similarly, the \theta^*(x) of a cocycle is a coboundary.
Let x\mapsto\tau(x) be a completely reducible representation of a Lie algebra {\mathfrak{a}} in the endomorphism algebra of a finite-dimensional vector space E endowed with an operator d such that dd=0; suppose that \tau(x)d=d\tau(x), and that \tau(x) sends cycles to boundaries. Let F be the subspace of “invariant” elements of E (i.e. elements \alpha such that \tau(x)\cdot\alpha=0 for all x\in{\mathfrak{a}}). Then F is stable under d, and the canonical homology from the homology group \operatorname{H}(F) to the homology group \operatorname{H}(E) is an isomorphism from the former to the latter.
We can apply this to the representation x\mapsto\theta(x) (resp. x\mapsto\theta^*(x)) of a reductive algebra {\mathfrak{a}}; then F becomes {\mathfrak{J}} (resp. {\mathfrak{J}}'), the sub-algebra of invariant chains (resp. invariant cochains). But \partial is zero for every element of {\mathfrak{J}}, and \delta is zero for every element of {\mathfrak{J}}', thanks to the formulas \begin{aligned} 2\delta &= \sum_k e(x'_k)\theta(x_k) \\2\partial &= \sum_k i(x'_k)^*\theta(x_k) \end{aligned} (where (x_k) and (x'_k) are dual bases; see §I). So \operatorname{H}({\mathfrak{J}}) can be identified with {\mathfrak{J}}, and \operatorname{H}({\mathfrak{J}}') with {\mathfrak{J}}', and the lemma shows that {\mathfrak{J}}\to\operatorname{H}({\mathfrak{a}}) and {\mathfrak{J}}'\to\operatorname{H}({\mathfrak{a}}) are onto isomorphisms (i.e there is one and only one invariant chain in each homology class; id. for cochains and cohomology). Furthermore, {\mathfrak{J}}'\to\operatorname{H}({\mathfrak{a}}') is an isomorphism for the algebra structure; we can not say the same of {\mathfrak{J}}\to\operatorname{H}({\mathfrak{a}}), since \operatorname{H}({\mathfrak{a}}) has no multiplicative structure, but the identification of \operatorname{H}({\mathfrak{a}}) with {\mathfrak{J}}, which does have an algebra structure, precisely defines a multiplicative structure on \operatorname{H}({\mathfrak{a}}), and allows us to speak about the homology algebra of a reductive Lie algebra. We will prove that, if {\mathfrak{a}} is the Lie algebra of a compact connected group, then the multiplicative structure of \operatorname{H}({\mathfrak{a}}) is precisely that which is defined by the “Pontrjagin product”.
On any reductive {\mathfrak{a}}, there exists at least one regular invariant quadratic form; it defines an isomorphism from the algebra \Gamma({\mathfrak{a}}) to the algebra \Gamma({\mathfrak{a}}), and from {\mathfrak{J}} to {\mathfrak{J}}'; thus \operatorname{H}({\mathfrak{a}}) and \operatorname{H}({\mathfrak{a}}') are (non-canonically) isomorphic algebras.
5 The first three Betti numbers of a semi-simple Lie algebra
A preliminary remark: for \alpha' to be an invariant cochain, it is necessary and sufficient that it be a cocycle, and that the i(x)\cdot\alpha' be cocycles for all x\in{\mathfrak{a}} (by (1)).
The equation \langle x\wedge y,\delta x'\rangle = -\langle[x,y],x'\rangle shows that, if x' is a cocycle, then x' is orthogonal to the derived algebra {\mathfrak{a}}^2, and is thus zero if {\mathfrak{a}} is semi-simple.
Every invariant cochain \alpha' of degree 2 is zero, since i(x)\cdot\alpha' is a cocycle of degree 1 for all x, and is thus zero.
In particular, it is always \geqslant 1. An invariant quadratic form is a symmetric bilinear form f(x,y) such that f(\theta(z)\cdot x,y)+f(x,\theta(z)\cdot y)=0 for all z\in{\mathfrak{a}}. We define, as follows, an isomorphism from the vector space ({\mathfrak{J}}')^{(3)} of invariant cochains of degree 3 to the space of invariant quadratic forms: let \alpha' be invariant and of degree 3; for every x\in{\mathfrak{a}}, we know that i(x)\cdot\alpha' is a cocycle of degree 2, and thus cohomologous to 0; there thus exists a unique cochain x' of degree 1 such that \delta x'=i(x)\cdot\alpha'; the map x\mapsto x' from {\mathfrak{a}} to {\mathfrak{a}}' defines a bilinear form f(x,y)=\langle y,x'\rangle. We have f(x,[y,z]) = -\langle x\wedge y\wedge z,\alpha'\rangle, \tag{2} and f is symmetric and invariant. Conversely, if f is a symmetric invariant bilinear form, then there exists a unique cochain \alpha' such that (2) holds, and this cochain is invariant. QED.
This converse holds even without the semi-simplicity assumption. In particular, the invariant symmetric bilinear form \operatorname{tr}\theta(x)\theta(y) defines an invariant cochain of degree 3, called the Cartan cochain.
If {\mathfrak{a}} is a Lie algebra such that the uniqueness (up to a scalar multiple) of an invariant quadratic form is assured, then the third Betti number of {\mathfrak{a}} is equal to 1. This is notably the case for the algebra of a compact connected Lie group, or for a simple Lie algebra over an algebraically closed field.
6 Homomorphisms from one Lie algebra to another
Let {\mathfrak{b}} and {\mathfrak{a}} be arbitrary Lie algebras (over the same field K), and let \varphi be a homomorphism from {\mathfrak{b}} to {\mathfrak{a}}; then \varphi extends to a homomorphism, again denoted by \varphi, from the algebra \Gamma({\mathfrak{b}}) to the algebra \Gamma({\mathfrak{a}}). The transpose \varphi^* (a homomorphism from {\mathfrak{a}}' to {\mathfrak{b}}') can be extended to a homomorphism, again denoted by \varphi^*, from the algebra \Gamma({\mathfrak{a}}') to the algebra \Gamma({\mathfrak{b}}'); furthermore, the extensions of \varphi and \varphi^* are transpose to one another. We have \partial\varphi=\varphi\partial and \delta\varphi^*=\varphi^*\delta, whence a homomorphism \widetilde{\varphi} from \operatorname{H}({\mathfrak{b}}) to \operatorname{H}({\mathfrak{a}}), and a homomorphism \widetilde{\varphi}^* from \operatorname{H}^({\mathfrak{a}}') to \operatorname{H}({\mathfrak{b}}'), with \widetilde{\varphi}^* being the transpose of \widetilde{\varphi}. The homomorphism \widetilde{\varphi}^* is also a homomorphism for the multiplicative structures.
If the algebras {\mathfrak{a}} and {\mathfrak{b}} are reductive, then \widetilde{\varphi} is a homomorphism for the multiplicative structures of \operatorname{H}({\mathfrak{b}}) and \operatorname{H}({\mathfrak{a}}).
This can be proved by using:
Let {\mathfrak{a}} be an arbitrary Lie algebra; then, for any chains \alpha and \beta of \Gamma({\mathfrak{a}}), the chain \partial(\alpha\wedge\beta) + (\partial\alpha)\wedge\beta - \overline{\alpha}\wedge(\partial\beta) is orthogonal to the invariant cochains.
With this, we consider a reductive Lie algebra, and we will show that, if \alpha and\overline{\alpha}\wedge\partial\gamma are cycles, then \overline{\alpha}\wedge\partial\gamma is a boundary: indeed, by Lemma 2, \partial(\alpha\wedge\gamma)-\overline{\alpha}\wedge\partial\gamma is orthogonal to the invariant cochains, and, since it is a cycle, it is orthogonal to the coboundaries; it is thus orthogonal to all the cocycles, and thus is a boundary; so \overline{\alpha}\wedge\partial\gamma is indeed a boundary. Now let \alpha and \beta be invariant cycles, and \alpha_1 and \beta_1 cycles that are homologous to \alpha and \beta (respectively); we will show that, if \alpha_1\wedge\beta_1 is a cycle, then this cycle is homologous to \alpha\wedge\beta: we have \alpha\wedge\beta - \alpha_1\wedge\beta_1 = \alpha\wedge(\beta-\beta_1) + (\alpha-\alpha_1)\wedge\beta, and, by what we have just shown, both of the terms on the right-hand side are boundaries.
From this it follows that, in a reductive Lie algebra, if \alpha_1 and \alpha_2 are cycles such that \alpha_1\wedge\beta_1 is a cycle, then the homology class of \alpha_1\wedge\beta_1 is the product of the homology classes of \alpha_1 and \beta_1.
The proof of Theorem 1 goes as follows: \beta_1 and \beta_2 are invariant chains of \Gamma({\mathfrak{b}}), and so \varphi(\beta_1), \varphi(\beta_2), and \varphi(\beta_1\wedge\beta_2) are all cycles in \Gamma({\mathfrak{a}}), and so the class of \varphi(\beta_1\wedge\beta_2)=\varphi(\beta_1)\wedge\varphi(\beta_2) is the product of the classes of \varphi(\beta_1) and \varphi(\beta_2).
7 The Hopf–Samelson theorem
This theorem is proven for reductive Lie algebras, using Theorem 1 above. First, let {\mathfrak{a}} be an arbitrary Lie algebra, and \varphi the “diagonal map” from {\mathfrak{a}} to {\mathfrak{a}}\oplus{\mathfrak{a}}\subset\Gamma({\mathfrak{a}})\otimes\Gamma({\mathfrak{a}}): \varphi(x) = x\oplus x = x\otimes1 + 1\otimes x. From this, we obtain a homomorphism \varphi from \Gamma({\mathfrak{a}}) to \Gamma({\mathfrak{a}})\otimes\Gamma({\mathfrak{a}}), such that \varphi(\alpha) = \alpha\times1 + \ldots + 1\otimes\alpha. The transpose map \varphi^* from \Gamma({\mathfrak{a}}')\otimes\Gamma({\mathfrak{a}}') to \Gamma({\mathfrak{a}}') is an algebra homomorphism (§II.6); it easily follows that \varphi^*(\alpha'\otimes\beta') = \alpha'\wedge\beta'.
The homomorphism \widetilde{\varphi} from \operatorname{H}({\mathfrak{a}}) to \operatorname{H}({\mathfrak{a}})\otimes\operatorname{H}({\mathfrak{a}}) satisfies \widetilde{\varphi}(\widetilde{\alpha}) = \widetilde{\alpha}\otimes1 + \ldots + 1\otimes\widetilde{\alpha} where \widetilde{\alpha} denotes a homology class. The transpose homomorphism \widetilde{\varphi}^* is an algebra homomorphism, and satisfies \widetilde{\varphi}^*(\widetilde{\alpha}'\otimes\widetilde{\beta}') = \widetilde{\alpha}'\cdot\widetilde{\beta}' (where the product \cdot is in \operatorname{H}({\mathfrak{a}}')).
If {\mathfrak{a}} is now a reductive algebra, then the homomorphism \widetilde{\varphi} is also a homomorphism for the multiplicative structures of \operatorname{H}({\mathfrak{a}}) and \operatorname{H}({\mathfrak{a}})\otimes\operatorname{H}({\mathfrak{a}}). We thus find ourselves in the following algebraic situation:
K is a field of characteristic 0; A is a graded algebra of finite rank over K whose multiplication law satisfies the usual commutativity and anticommutativity conditions for an exterior algebra; A' is a graded algebra that is (non-canonically) isomorphic to A, in canonical duality with A; the degree 0 elements of A (resp. of A') are given by multiples of the unit, and can be identified with the elements of K. The duality between A\otimes A and A'\otimes A' is defined by \langle\alpha\otimes\beta,\alpha'\otimes\beta'\rangle = \langle\alpha,\alpha'\rangle\cdot\langle\beta,\beta'\rangle, and the transpose of the canonical homomorphism \widetilde{\varphi}^* from A'\otimes A' to A' (such that \widetilde{\varphi}^*(\alpha'\otimes\beta')=\alpha'\cdot\beta') is \widetilde{\varphi}, which is a homomorphism from A to A\otimes A that respects the multiplicative structures. In such a situation, we say that the homogeneous elements of degree \geqslant 1 of A (resp. of A') that are orthogonal to the products \alpha'\cdot\beta' with \alpha' and \beta' of degree \geqslant 1 (resp. to the products \alpha\cdot\beta etc.) are primitive. We can then prove the following theorem of algebra:
The primitive elements of are odd degree; they generate a vector subspace P of A (resp. P' of A'); the canonical map P\to A (resp. P'\to A') can be extended to give an isomorphism from the exterior algebra \Gamma(P) onto A (resp. an isomorphism from \Gamma(P') onto A'); when restricted to P and P', the duality between A and A' defines a duality between P and P'; this duality can be extended in the usual way to give a duality between \Gamma(P) and \Gamma(P') that, by identifying \Gamma(P) with A, and \Gamma(P') with A', recovers the existing duality between A and A'.
The above statement thus applies if we take A to be the homology algebra \operatorname{H}({\mathfrak{a}}) of a reductive Lie algebra, and A' to then be the cohomology algebra \operatorname{H}({\mathfrak{a}}').
In the case where {\mathfrak{a}} is the Lie algebra of a compact connected group, the dimension of the space P of primitive elements is equal to the rank of the group, that is, to the maximal dimension of the abelian sub-algebras of {\mathfrak{a}}. We still do not have an algebraic proof of this fact, which would prove this statement for a reductive Lie algebra.
8 Homomorphisms from a reductive algebra to a reductive algebra
Let {\mathfrak{a}} and {\mathfrak{b}} be reductive algebras, and \varphi a homomorphism from {\mathfrak{b}} to {\mathfrak{a}}; we keep the notation form §II.6. Then every primitive element of \operatorname{H}({\mathfrak{b}}) is sent, by \widetilde{\varphi}, to a primitive element of \operatorname{H}({\mathfrak{a}}); every primitive element of \operatorname{H}({\mathfrak{a}}') is sent, by \widetilde{\varphi}^*, to a primitive element of \operatorname{H}({\mathfrak{b}}'). Taking Theorem 1 into account (from §II.6), we thus deduce:
(cf. Samelson). The ideal of zeros of \widetilde{\varphi} is the ideal of \operatorname{H}({\mathfrak{b}}) generated by the primitive elements of \operatorname{H}({\mathfrak{b}}) whose image under \widetilde{\varphi} is zero; the sub-algebra given by the image of \widetilde{\varphi} is the sub-algebra of \operatorname{H}({\mathfrak{a}}) generated by the unit and by the primitive elements of \operatorname{H}({\mathfrak{a}}) that belong to the image of \widetilde{\varphi}. The analogous properties hold for cohomology algebras.
Let n denote the dimension of {\mathfrak{b}}. For \widetilde{\varphi} to be bijective, it suffices that \widetilde{\varphi}(\omega)\neq0, where \omega is the generating element of \operatorname{H}_n({\mathfrak{b}}).
The next talk will be dedicated to the study of relative homology and cohomology: given a sub-algebra {\mathfrak{b}} of {\mathfrak{a}}, we define the homology (resp. cohomology) of the “homogeneous space” {\mathfrak{a}}/{\mathfrak{b}}, and we study the relations between the homology (resp. cohomology) of {\mathfrak{a}}, of {\mathfrak{b}}, and of {\mathfrak{a}}/{\mathfrak{b}}.