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For a positively graded Lie algebra $L$ over $\mathbb{Q}$, finite-dimensional in each degree, its cochain algebra is free graded commutative on the graded dual of $sL$, with differential dual to the Lie bracket and its Koszul signs. The Jacobi identity gives $d^2=0$. A Lie element of degree $r$ contributes a cochain generator of degree $r+1$. Its cohomology computes ${\mathrm{Ext}}_{U(L)}(\mathbb{Q},\mathbb{Q})$ with total grading; the standard free enveloping-algebra resolution gives this identification. For a free graded Lie algebra, the tensor-algebra augmentation has a length-one free resolution, explaining the square-zero cohomology of a rational wedge sum of spheres.

For a positively graded vector space $V$, the free graded Lie algebra $\mathbb{L}(V)$ receives a linear map from $V$ and extends every graded linear map $V\to L$ uniquely to a Lie homomorphism. Over $\mathbb{Q}$, it is the Lie subalgebra generated by $V$ inside the tensor algebra $T(V)$, using $[u,v]=uv-(-1{)}^{|u||v|}vu$. Its universal enveloping algebra is $T(V)$: both have the same universal property for maps from $V$ into associative algebras. The primitives of the tensor Hopf algebra with primitive generators are exactly $\mathbb{L}(V)$.

A graded Lie algebra has a graded Lie bracket preserving total degree, with $[u,v]=-(-1{)}^{|u||v|}[v,u]$ and $[u,[v,w]]=[[u,v],w]+(-1{)}^{|u||v|}[v,[u,w]]$ for homogeneous vectors in the indicated degrees. A Gerstenhaber algebra uses this sign rule after shifting degrees by one.

The bracket on a graded Lie algebra is a bilinear operation satisfying graded antisymmetry and the graded Jacobi identity. For homogeneous elements of degrees $r,s$, $[u,v]=-(-1{)}^{rs}[v,u]$. A Gerstenhaber bracket uses this convention on the shifted degrees $r=|u|-1$ and $s=|v|-1$.

For a graded Lie algebra over characteristic zero, the associated graded algebra of its universal enveloping algebra under the word-length filtration is its graded symmetric algebra. Ordered homogeneous monomials form a basis, with exponent at most one on each odd Lie basis element. The defining graded commutation relations reorder every word, and the Jacobi identity resolves the triple-reordering ambiguities. In particular the primitive elements of a Hopf algebra of $U(L)$ are precisely $L$: in its associated graded algebra, a primitive polynomial of symmetric length greater than one would have a nonzero mixed term under the diagonal, which is impossible in characteristic zero.

Use shifted degreesIts graded Lie bracket is the sign-adjusted Whitehead bracket from Question 2, equivalently the Samelson product on loops. This regrading is essential: the geometric bracket has degree minus one in the unshifted sphere grading. A free graded Lie algebra on a graded vector space $V$ means $\mathbb{L}(V)$ with the universal extension property for linear maps from $V$ to any graded Lie algebra.

Here are the general results used. The Bott–Samelson theorem says that, for a connected based CW complex $X$, the inclusion $X\to \Omega \Sigma X$ induces an isomorphism of graded algebraswith the Pontryagin product. The Milnor–Moore theorem says that for simply connected $Y$, rational loop Hurewicz homomorphism identifies $L_Y$ with the primitives of that loop-homology Hopf algebra and induces $U(L_Y)\cong H_{\ast }(\Omega Y;\mathbb{Q})$. The graded Poincaré-Birkhoff-Witt theorem identifies the associated graded algebra of $U(L)$, for its word-length filtration, with the graded symmetric algebra on $L$; hence its primitive elements of a Hopf algebra are exactly $L$ over $\mathbb{Q}$. Finally the rational Whitehead theorem identifies rational homology equivalences of simply connected CW complexes with rational homotopy equivalences. Rationalization of a topological space induces the corresponding isomorphisms of loop-space homotopy and rational homology.

Suppose first $Y$ has the rational homotopy type of $\Sigma X$. A reduced suspension is simply connected when $X$ is connected; a disconnected $X$ has $H_1(\Sigma X;\mathbb{Q})={\tilde{H}}_0(X;\mathbb{Q})\ne 0$, so it cannot have the rational homology of the simply connected $Y$. Put $V={\tilde{H}}_{\ast }(X;\mathbb{Q})$, in strictly positive degrees. Bott–Samelson theorem gives $H_{\ast }(\Omega Y;\mathbb{Q})\cong T(V)$ as an algebra. To deduce freeness of the primitive Lie algebra, its coproduct must also be considered: the initial Bott–Samelson theorem generators can have reduced-diagonal terms when $X$ has nontrivial cup products.

By Milnor–Moore theorem write this Hopf algebra as $U(L_Y)$. Its algebra indecomposables satisfyThe first identity follows from the enveloping relation: after all positive products are killed, every bracket is killed and no additional linear relation is introduced. Hence each homogeneous basis vector of $V$ has a primitive representative in $H_{\ast }(\Omega Y;\mathbb{Q})$. Choose such representatives $p_v$. Sending $v$ to $p_v$ defines an algebra map from $T(V)$ to itself whose effect on indecomposables is the identity. It is an isomorphism: $p_v=v+$ decomposable terms, and every factor in those terms has smaller positive degree than $v$. Degree induction solves recursively for each original generator and constructs the inverse. This works even with arbitrarily many generators in a degree.

Equip the source tensor algebra with primitive generators. The map is now a Hopf-algebra isomorphism, because the chosen images are primitive. The primitives in this tensor Hopf algebra are exactly $\mathbb{L}(V)$: $U(\mathbb{L}(V))=T(V)$ by the two universal properties, and the graded Graded Poincaré–Birkhoff–Witt theorem gives its primitive subspace. Consequently $L_Y\cong \mathbb{L}(V)$, proving the forward direction without assuming a primitive coproduct for the original generators.

Conversely suppose $L_Y\cong \mathbb{L}(W)$ for a positively graded $W$. Choose a homogeneous basis $w_{\lambda }$ of $W$. Represent its corresponding homotopy class by a based sphere map $S^{|w_{\lambda }|+1}\to Y_{\mathbb{Q}}$, and combine these maps intoThe wedge sum is a reduced suspension, namely $Z=\Sigma ({\bigvee }_{\lambda }{S}^{|w_{\lambda }|})$. The preceding direction, applied to this wedge sum, gives its free rational homotopy Lie algebra on the sphere inclusions. Indeed these particular tensor generators are primitive because the reduced diagonal of each sphere is zero, and loop Hurewicz homomorphism sends the adjoint sphere inclusion to that generator. Naturality of the bracket shows that $F_{\ast }$ sends the free generating basis to the prescribed free generating basis of $L_Y$. It is therefore an isomorphism of free graded Lie algebras, hence an isomorphism in every rational homotopy degree. The rational Whitehead theorem makes $F$ a rational homotopy equivalence. This proves the reverse implication and also shows that a wedge sum of spheres can be chosen as the reduced suspension.

For every chosen connected $X$ with $Y{\simeq }_{\mathbb{Q}}\Sigma X$, the required algebra statement is precisely the Bott–Samelson theorem isomorphism, transported along rational homotopy equivalence:Multiplication is concatenation of tensors and the degree-zero tensor is the unit. All freeness assertions here are graded, with the Koszul signs described above.

A derivation of an algebra is a $k$-linear map $D:A\to A$ satisfying $D(ab)=D(a)b+aD(b)$. The commutator $[D,E]=D\circ E-E\circ D$ is again a derivation: expanding $[D,E](ab)$ cancels the two mixed terms and leaves $[D,E](a)b+a[D,E](b)$. The commutator on endomorphisms is bilinear, antisymmetric, and satisfies the Jacobi identity by cancellation of its twelve triple-composition terms. Therefore ${\mathrm{Der}}_k(A)$ is a Lie algebra.

In degree zero of the Hochschild cochain complex, $\delta a(b)=ba-ab$, so $H{H}^0(A,A)=Z(A)=A$ for commutative $A$. In degree one, $\delta D=0$ is precisely the derivation rule; the boundaries are inner derivations, which vanish for commutative $A$. HenceFor cochains $f\in C^p(A,A)$, $g\in C^q(A,A)$, the Hochschild cup product isDefine the insertion operation byA degree-zero cochain is an element of $A$, inserted with no arguments; for $p=0$ the sum is empty. For the Gerstenhaber bracket we use the left graded Leibniz rule convention, compatible with the unsigned Hochschild cup product just displayed:Both degree-zero inputs have bracket zero. Another common insertion convention writes $f\circ g-(-1{)}^{(p-1)(q-1)}g\circ f$; the two brackets differ by $(-1{)}^{(p-1)(q-1)}$. With an unsigned Hochschild cup product, that convention uses the corresponding right graded Leibniz rule. The distinction matters for a degree-two cochain bracketed with a function. Either consistent convention gives the same degree-one Lie bracket and the same derivation action on functions.

If $m(a,b)=ab$, this convention gives $\delta f=[m,f]$. The shifted Jacobi identity and $[m,m]=0$ therefore show that the Gerstenhaber bracket respects Hochschild cocycles and the images of the coboundary map. The Hochschild cup product and Gerstenhaber bracket induce operations on Hochschild cohomology. A Gerstenhaber algebra is a graded algebra $H$ with an associative degree-zero product with the graded commutative algebra rule $uv=(-1{)}^{pq}vu$, and a degree-minus-one graded Lie bracket making the shifted degrees $|u|-1$ into a graded Lie algebra. In particular,for homogeneous elements of a graded algebra of degrees $p,q$. The shifted Jacobi identity isThe Hochschild cup product does not make the cochains a graded commutative algebra in general, but does make their cohomology a graded commutative algebra; the insertion operation supplies the homotopy for this assertion and for the graded Leibniz rule. Thus these axioms describe the induced Gerstenhaber algebra, not a claim of a graded commutative algebra structure on the cochain multiplication itself.

For $A=k[X]$, the enveloping algebra is $A^e=k[X_{\ell },X_r]$, andis a projective resolution. The first map is injective since $A^e$ is an integral domain, and its cokernel is $A$. Applying ${\mathrm{Hom}}_{A^e}(-,A)$ gives a zero coboundary map. ConsequentlyThe Hochschild cup product is ordinary multiplication of functions and scalar multiplication of derivations, with the product of two derivations zero because $H{H}^2=0$. Every derivation is $f{\partial }_X$, since it is determined by its value on $X$. The Gerstenhaber bracket iswith all other orders fixed by graded antisymmetry. These formulas fully determine the Gerstenhaber algebra.

For $A=k[X,Y]$, the Hochschild-Kostant-Rosenberg theorem identifiesThus the degrees zero, one, and two are $A$, $A{\partial }_X\oplus A{\partial }_Y$, and $A({\partial }_X\wedge {\partial }_Y)$, and all higher groups vanish. The Hochschild-Kostant-Rosenberg map sends a wedge of $p$ derivations to the cochainThe factorial is invertible in characteristic zero. Equivalently, the groups follow from the Koszul resolution on the regular sequence $X_{\ell }-X_r,Y_{\ell }-Y_r$ in $A^e$, whose dual coboundary maps vanish on $A$.

The Hochschild cup product becomes the exterior product, and our Gerstenhaber bracket becomes the left Schouten-Nijenhuis bracket. It is determined by the commutator of derivations, $[D,f]=D(f)$, zero brackets of functions, and the displayed graded antisymmetry and left graded Leibniz rule. For explicit signs, put $\Pi ={\partial }_X\wedge {\partial }_Y$ and $D=F{\partial }_X+G{\partial }_Y$. ThenThe last bracket has degree three, whose exterior power is zero. For two derivations, the coefficient functions of their commutator give the remaining formula. This specifies the entire Gerstenhaber algebra; under the alternate insertion convention mentioned above, the first displayed bracket changes sign, together with the Leibniz convention. No smoothness of a general finitely generated commutative algebra was assumed: the Hochschild-Kostant-Rosenberg theorem is invoked here only for the smooth polynomial ring $k[X,Y]$.

An element $x$ of positive degree is primitive when $\Delta x=x\otimes 1+1\otimes x$. Graded commutators of primitives are primitive, so they form a graded Lie algebra. In a universal enveloping algebra over $\mathbb{Q}$, the primitive elements of a Hopf algebra are exactly its given Lie algebra: the associated graded algebra of the word-length filtration is symmetric, where no nonzero homogeneous polynomial of length greater than one is additive under the diagonal.