Hopf algebras and invariants of the Johnson cokernel

Abstract

We show that if H is a cocommutative Hopf algebra, then there is a natural action of Aut(F n ) on H ⊗n which induces an Out(F n ) action on a quotient H ⊗n . In the case when H = T(V) is the tensor algebra, we show that the invariant Tr C of the cokernel of the Johnson homomorphism studied in [5] projects to take values in H vcd (Out(F n ); H ⊗n ). We analyze the n = 2 case, getting large families of obstructions generalizing the abelianization obstructions of [8].arXiv:1509.03236v1 [math.AT] 10 Sep 2015where GL 2 (Z) acts on Sym(L (2) ) ⊗2 ∼ = Sym(L (2) ⊗ k 2 ) via the standard action on k 2 , which induces the action of GL 2 (Z) on Sym(L (2) ) ⊗2 . This latter group can be computed via the methods of [7], leading to many families of obstructions [λ] Sp ⊗ M k and [λ] Sp ⊗ S k , where M k , S k denote spaces of modular (respectively cusp) forms of weight k. The simplest new families of obstructions one gets are [2k − 1, 1 2 ] Sp ⊗ S 2k+2 ⊂ C 2k+5 , and ([2k + 1, 1 2 ] Sp ⊕ [2k, 2, 1] Sp ⊕ [2k, 1 3 ] Sp ) ⊗ M 2k+2 ⊂ C 2k+7 .

The $\mathsf {Lie}$ Lie algebra

“…is even and b > 0, and α(2m, 0) = ⌈ 2m 3 ⌉ − 1. In [8], roughly speaking, we had proven the same fact with α(a, b) replaced by the smaller number s a−b+2 .…”

“…In [8], we proved that H r (H) ∼ = H 2r−3 (Out(F r ); H ⊗r ), where Aut(F r ) acts on H ⊗r via the Hopf algebra structure (see section 2.3), and H ⊗r is a natural quotient on which inner automorphisms act trivially. When H = Sym(V ), H ⊗r = H ⊗r and the action of Out(F r ) factors through the standard GL r (Z) action.…”

An Abelian Quotient of the Symplectic Derivation Lie Algebra of the Free Lie Algebra

“…(3) Find a linear map C : h g,1 (12) Sp ։ Q which annihilates W . (4) Check that C is trivial on the image of the bracket map. 4.1.…”

“…a 3 , a 4 , a 5 , b 4 , a 6 , b 2 , a 7 , b 3 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 4 , a 6 , b 2 , a 7 , b 3 , b 6 , b 5 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b4 , b 2 , a 7 , b 3 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 4 , b 2 , a 7 , b 3 , b 6 , b 5 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 4 , b 3 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 4 , b 3 , b 6 , b 5 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 5 , b 2 , a 7 , b 3 , b 4 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 5 , b 2 , a 7 , b 3 , b 6 , b 4 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 5 , b 3 , b 4 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 5 , b 3 , b 6 , b 4 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 6 , b 3 , b 4 , b 5 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 6 , b 3 , b 5 , b 4 , b 7 },{a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 7 , b 3 , b 4 , b 5 , b 6 }, {a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 7 , b 3 , b 5 , b 4 , b 6 }, {a 2 , a 3 , a 4 , a 5 , b 4 , a 6 , b 2 , b 5 , a 7 , b 3 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , b 4 , a 6 , b 2 , a 7 , b 5 , b 3 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 4 , b 2 , b 5 , a 7 , b 3 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , b 4 , b 2 , a 7 , b 5 , b 3 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 4 , b 5 , b 3 , b 6 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 4 , b 5 , b 6 , b 3 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 4 , b 6 , b 3 , b 5 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 4 , b 6 , b 5 , b 3 , b 7 }, {a 2 , a 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 5 , b 4 , b 6 , b 3 , b 7 }, {a 2 , a 3 , b 3 , a 4 , a 5 , b 4 , a 6 , b 2 , a 7 , b 5 , b 6 , b 7 }, {a 2 , a 3 , b 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 4 , b 5 , b 6 , b 7 }, {a 2 , a 3 , b 3 , a 4 , a 5 , a 6 , a 7 , b 2 , b 4 , b 6 , b 5 , b 7 }, {a 2 , a 3 , a 4 , b 3 , b 4 , a 5 , a 6 , b 2 , a 7 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , b 3 , a 5 , b 4 , a 6 , b 2 , a 7 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , b 3 , a 5 , a 6 , b 4 , b 2 , a 7 , b 5 , b 6 , b 7 }, {a 2 , a 3 , a 4 , b 3 , a 5 , a 6 , a 7 , b 2 , b 4 , b 5 , b 6 , b 7 },…”

Character varieties as a tensor product

“…We say that a Hopf algebra H acts on a Hopf algebra A if A has an algebra action µ : Starting with the action of H on itself by conjugation h·h 1 = h ′ h 1 S(h ′′ ), we get the conjugation action of H on H ⊗n defined in [13]. We can dualize this to get a coaction of a commutative Hopf algebra…”