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differential graded Hopf algebra

Differential graded Hopf algebras

A \mathbb{Z}-graded Hopf algebra (pre-gha) is a \mathbb{Z}-graded vector space, which, for that grading, is both a \mathbb{Z}-graded algebra, (A,μ)(A,\mu), with unity, η:KA\eta : K \to A, and a \mathbb{Z}-graded coalgebra (A,Δ,ε)(A, \Delta, \varepsilon) such that:

  • η:KA\eta : K \to A is a morphism of \mathbb{Z}-graded coalgebras;
  • ε:AK\varepsilon : A \to K is a morphism of \mathbb{Z}-graded algebras;
  • μ:AAA\mu : A \otimes A \to A is a morphism of \mathbb{Z}-graded coalgebras

Remark

We can replace the third condition by:

  • Δ:AAA \Delta : A \to A \otimes A is a morphism of \mathbb{Z}-graded algebras.

Of course, wherever possible, we will abbreviate (A,Δ,μ,ϵ,η)(A,\Delta,\mu,\epsilon,\eta) to AA.

A morphism of pre-ghas is a linear map of degree zero compatible with both the algebra and coalgebra structures. We may write preGHApre GHA for the resulting category.

If AA and AA' are two pre-ghas, AAA\otimes A' is a pre-gha for the algebra and coalgebra structures already defined.

Derivations of Hopf algebras

Let AA be a pre-gha. A Hopf algebra derivation of AA of degree pp\in \mathbb{Z} is a linear mapping θHom p(A,A)\theta \in Hom_p(A,A), defining both an algebra and a coalgebra derivation.

A differential \partial of pre-ghas is a Hopf algebra derivation of degree -1 such that =0\partial\circ \partial = 0. The pair (A,)(A,\partial) is called a differential \mathbb{Z}-graded Hopf algebra (pre-dgha). Its homology H(A,)H(A,\partial) is also a pre-gha. A morphism of pre-dghas is a morphism, at the same time, of pre-ghas and pre-dgvs. This gives a category preDGHApre DGHA.

A pre-gha (A,Δ,μ,ϵ,η)(A,\Delta,\mu,\epsilon,\eta) is commutative if (A,μ)(A,\mu) is commutative and is cocommutative if (A,Δ,ε)(A,\Delta,\varepsilon) is cocommutative.

This gives categories preCDGHApre CDGHA and preCoDGHApre CoDGHA respectively.

A cocommutative (resp. commutative) dgha} is an object of preCoDGHApre CoDGHA (resp. preCDGHApre CDGHA, which has a lower (resp. upper) grading.

A cocommutative (resp. commutative) dgha AA is nn-connected if A¯ p=0\bar{A}_p = 0 (resp A¯ p=0\bar{A}^p = 0) for pnp\leq n.

Shuffle product on T(V)T(V)

Let VV be a pre-gvs. The gvs T(V)T(V) is a pre-cga for the shuffle product defined by

(v 1v p)(v p+1v n)= σε(σ)v σ 1(1)v σ 1(n),(v_1\otimes \ldots \otimes v_p)\star (v_{p+1}\otimes\ldots \otimes v_n) = \sum_\sigma \varepsilon(\sigma)v_{\sigma^{-1}(1)}\otimes\ldots \otimes v_{\sigma^{-1}(n)},

where the sum is over all (p,np)(p,n-p) shuffles, ε(σ)\varepsilon(\sigma) is the Koszul sign of σ\sigma and the elements v iv_i of VV are all homogeneous.

The commutative graded Hopf algebra structure on T(V)T(V)

The underlying algebra structure is T(V)T(V) with the shuffle product. The reduced diagonal is given by

Δ¯(v 1v n)= p=1 n1(v 1v p)(v p+1v n).\bar{\Delta}(v_1\otimes \ldots \otimes v_n) = \sum_{p=1}^{n-1} (v_1\otimes \ldots \otimes v_p)\otimes(v_{p+1}\otimes \ldots \otimes v_n).

The cocommutative graded Hopf algebra structure on T(V)T(V)

The underlying algebra structure this time is T(V)T(V) with the usual product

(v 1v p)(v p+1v n)=v 1v pv p+1v n,(v_1\otimes \ldots \otimes v_p)\cdot(v_{p+1}\otimes \ldots \otimes v_n) = v_1\otimes \ldots \otimes v_p\otimes v_{p+1}\otimes \ldots \otimes v_n,

but with the reduced diagonal given by

Δ¯(v 1v n)= p=1 n1 σε(σ)(v σ(1)v σ(p))(v σ(p+1)v σ(n)),\bar{\Delta}(v_1\otimes \ldots \otimes v_n) = \sum_{p=1}^{n-1}\sum_\sigma \varepsilon(\sigma) (v_{\sigma(1)}\otimes \ldots \otimes v_{\sigma(p)})\otimes(v_{{\sigma(p+1)}}\otimes \ldots \otimes v_{\sigma(n)}),

where the sum is over all pp and all (p,np)(p,n-p)-shuffles and, as usual, ε(σ)\varepsilon(\sigma) is the Koszul sign.

The diagonal Δ\Delta is thus defined by the conditions

  • Δv=v1+1v \Delta v = v\otimes 1 + 1\otimes v if vVv \in V;

  • Δ\Delta is a morphism of \mathbb{Z}-graded algebras.

A commutative and cocommutative \mathbb{Z}-graded Hopf algebra structure on V\bigwedge V is obtained by using the algebra and coalgebra structures defined in differential graded algebra and differential graded coalgebra. respectively.

The enveloping algebra of a Lie algebra, U(L)U(L).

Let LL be a pre-gla, U(L)U(L), is the quotient algebra of the tensor algebra T(L)T(L) by the two sided ideal generated by the elements

xy(1) |y||x|yx[x,y],x,y,L.x\otimes y - (-1)^{|y||x|}y\otimes x - [x,y], \quad x,y,\in L.

The diagonal Δ:LL×L\Delta : L \to L\times L, with Δ(x)=(x,x)\Delta(x) = (x,x) defines a homomorphism of pre-gas,

U(Δ):U(L)U(L×L)U(L)U(L),U(\Delta) : U(L)\to U(L\times L) \cong U(L)\otimes U(L),

which makes U(L)U(L) a pre-gha which is cocommutative and conilpotent.

If LL is a free Lie algebra on VV, then the enveloping algebra is the tensor algebra: U𝕃(V)T(V)U\mathbb{L}(V) \cong T(V).

Let (L,)(L,\partial) be a pre-dgla, the differential \partial extends to an algebra differential on T(L)T(L). With the quotient differential, U(L)U(L) becomes a cocommutative pre-dgha, which will be denoted U(L,)U(L,\partial).

The differential \partial determines a differential, also denoted \partial, on the cocommutative pre-gca L\bigwedge' L, (for which gca see differential graded coalgebra). It satisfies:

H(L,)H(L,).\bigwedge' H(L,\partial) \cong H(\bigwedge' L,\partial).

Let i:LU(L)i : L \to U(L) be the linear mapping LT(L)U(L)L\to T(L) \to U(L), then define e:LU(L)e: \bigwedge' L \to U(L) by

e(x 1x n)=1n! σε(σ)i(x σ(1))i(x σ(n)),e(x_1\wedge \ldots x_n) = \frac{1}{n!}\sum_\sigma \varepsilon(\sigma)i(x_{\sigma(1)})\ldots i(x_{\sigma(n)}),

where the sum is over all permutations and ε(σ)\varepsilon(\sigma) is the Koszul sign.

Theorem

(Poincaré-Birkhoff-Witt)(cf. Quillen)

The mapping ee is an isomorphism of pre-dgcas.

Corollary

i:LU(L)i : L \to U(L) defines an isomorphism between LL and the space of primitives of U(L)U(L).

Corollary

The natural map UH(L,)H(U(L,)UH(L,\partial)\to H(U(L,\partial) is an isomorphism of cocommutative pre-ghas.

The Lie algebra of primitives, PP

Let (A,)(A,\partial) be a cocommutative pre-dgha. The vector space P(A)P(A) of primitive elements (for the coalgebra structure, cf. differential graded coalgebra), is not stable under the multiplication, however the commutator [α,β][\alpha,\beta] of two elements of P(A)P(A) is again in P(A)P(A). This defines a pre-gla structure on P(A)P(A) and we can put the induced differential on it to obtain P(A,)P(A,\partial).

The inclusion P(A)AP(A)\to A extends to a morphism of cocommutative pre-dghas σ:UP(A)A.\sigma: UP(A)\to A.

Theorem (Quillen, Quillen)

If AA is conilpotent, σ\sigma is an isomorphism.

The above theorem and earlier corollary show that UU and PP are inverse equivalences between the category, preDGLApre DGLA and that of cocommutative, conilpotent pre-dghas.

Remark

The enveloping algebra of a free Lie algebra 𝕃(V)\mathbb{L}(V) coincides with the tensor algebra, T(V)T(V). It is conilpotent from which one gets PT(V)=𝕃(V)PT(V) = \mathbb{L}(V).

References

The source used for this lexicon was

D. Tanré, Homotopie rationnelle: Modèles de Chen, Quillen, Sullivan, Lecture Notes in Maths No. 1025, Springer, 1983.

Much of the material there was based on Quillen’s paper:

D. Quillen, Rational Homotopy Theory, Ann. of Math., (2) 90 (1969), 205-295.

Revised on December 30, 2015 12:40:00 by Tim Porter (95.145.121.203)