Lie Algebra Homology Research Articles

On measurings of algebras over operads and homology theories

The notion of a coalgebra measuring, introduced by Sweedler, is a kind of generalized ring map between algebras. We begin by studying maps on Hochschild homology induced by coalgebra measurings. We then introduce a notion of coalgebra measuring between Lie algebras and use it to obtain maps on Lie algebra homology. Further, these measurings between Lie algebras satisfy nice adjoint like properties with respect to universal enveloping algebras. More generally, we introduce and undertake a detailed study of the notion of coalgebra measuring between algebras over any operad $\mathcal O$. In case $\mathcal O$ is a binary and quadratic operad, we show that a measuring of $\mathcal O$-algebras leads to maps on operadic homology. In general, for any operad $\mathcal O$, we construct universal measuring coalgebras to show that the category of $\mathcal O$-algebras is enriched over coalgebras. We develop measuring comodules and universal measuring comodules for this theory. We also relate these to measurings of the universal enveloping algebra $U_{\mathcal O}(\mathscr A)$ of an $\mathcal O$-algebra $\mathscr A$ and the modules over it. Finally, we construct the Sweedler product $C\rhd \mathscr A$ of a coalgebra $C$ and an $\mathcal O$-algebra $\mathscr A$. The object $C\rhd \mathscr A$ is universal among $\mathcal O$-algebras that arise as targets of $C$-measurings starting from $\mathscr A$.

Extremal stability for configuration spaces

We study stability patterns in the high dimensional rational homology of unordered configuration spaces of manifolds. Our results follow from a general approach to stability phenomena in the homology of Lie algebras, which may be of independent interest.

The Homology of the Lamplighter Lie Algebra

We show that the associated Lie algebra of the Malcev $\mathbb{Q}$-completion of the lamplighter group is the pronilpotent completion of the lamplighter Lie algebra. We also prove that the homology of this completed Lie algebra is of uncountable dimension on each degree.

On homology of Lie algebras over commutative rings

We study five different types of the homology of a Lie algebra over a commutative ring which are naturally isomorphic over fields. We show that they are not isomorphic over commutative rings, even over Z, and study connections between them. In particular, we show that they are naturally isomorphic in the case of a Lie algebra which is flat as a module. As an auxiliary result we prove that the Koszul complex of a module M over a principal ideal domain that connects the exterior and the symmetric powers 0→ΛnM→M⊗Λn−1M→…→Sn−1M⊗M→SnM→0 is purely acyclic.

On the free Jordan algebras

Let K be a field of characteristic zero. For integers n,D≥1, let Jn(D) be the degree n component of the free Jordan algebra J(D) over D generators. A conjecture for the character (in particular for the dimension) of the GL(D)-module Jn(D) is proposed.Let sl2J(D) be the Tits-Allison-Gao construction of J(D). Two natural conjectures for the homology of Lie algebra sl2J(D) are stated, and each of them implies the previous conjecture.The cyclicity of the Jordan structures, namely that the symmetric group SD+1 acts on the multilinear part of J(D), plays an essential role to connect the Lie algebra homology of sl2J(D) and the character of Jn(D).

Homology of Lie algebras of orthogonal and symplectic generalized Jacobi matrices

In this note, we compute the homology with trivial coefficients of Lie algebras of generalized Jacobi matrices of type B , C and D over an associative unital k -algebra with k being a field of characteristic 0.

Hochschild coniveau spectral sequence and the Beilinson residue

We develop the Hochschild analogue of the coniveau spectral sequence and the Gersten complex. Since Hochschild homology does not have devissage or A^1-invariance, this is a little different from the K-theory story. In fact, the rows of our spectral sequence look a lot like the Cousin complexes in Hartshorne's 1966 "Residues & Duality". Note that these are for coherent cohomology. We prove that they agree by an 'HKR isomorphism with supports'. Using the close ties of Hochschild homology to Lie algebra homology, this gives residue maps in Lie homology, which we show to agree with those \`a la Tate-Beilinson.

Higher enveloping algebras

We provide spectral Lie algebras with enveloping algebras over the operad of little $G$-framed $n$-dimensional disks for any choice of dimension $n$ and structure group $G$, and we describe these objects in two complementary ways. The first description is an abstract characterization by a universal mapping property, which witnesses the higher enveloping algebra as the value of a left adjoint in an adjunction. The second, a generalization of the Poincar\'{e}-Birkhoff-Witt theorem, provides a concrete formula in terms of Lie algebra homology. Our construction pairs the theories of Koszul duality and Day convolution in order to lift to the world of higher algebra the fundamental combinatorics of Beilinson-Drinfeld's theory of chiral algebras. Like that theory, ours is intimately linked to the geometry of configuration spaces and has the study of these spaces among its applications. We use it here to show that the stable homotopy types of configuration spaces are proper homotopy invariants.

Relative BGG sequences; II. BGG machinery and invariant operators

For a real or complex semisimple Lie group G and two nested parabolic subgroups Q⊂P⊂G, we study parabolic geometries of type (G,Q). Associated to the group P, we introduce the classes of relative natural bundles and of relative tractor bundles and construct some basic invariant differential operators on such bundles. We define a (rather weak) notion of “compressability” for operators acting on relative differential forms with values in a relative tractor bundle. Then we develop a general machinery which converts a compressable operator to an operator on bundles associated to completely reducible representations on relative Lie algebra homology groups.Applying this machinery to a specific compressable invariant differential operator of order one, we obtain a relative version of BGG (Bernstein–Gelfand–Gelfand) sequences. All our constructions apply in the case P=G, producing new and simpler proofs in the case of standard BGG sequences. We characterize cases in which the relative BGG sequences are complexes or even fine resolutions of certain sheaves and describe these sheaves. We show that this gives constructions of new invariant differential operators as well as of new subcomplexes in certain curved BGG sequences. The results are made explicit in the case of generalized path geometries.

Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism

We study the derived representation scheme DRep _n(A) parametrizing the n -dimensional representations of an associative algebra A over a field of characteristic zero. We show that the homology of DRep _n(A) is isomorphic to the Chevalley–Eilenberg homology of the current Lie coalgebra \mathfrak {gl}_n^*(\bar{C}) defined over a Koszul dual coalgebra of A . This gives a conceptual explanation to some of the main results of [BKR] and [BR], relating them (via Koszul duality) to classical theorems on (co)homology of current Lie algebras \mathfrak {gl}_n(A) . We extend the above isomorphism to representation schemes of Lie algebras: for a finite-dimensional reductive Lie algebra \mathfrak g , we define the derived affine scheme DRep _{\mathfrak g}(\mathfrak a) parametrizing the representations (in \mathfrak g ) of a Lie algebra \mathfrak{a} ; we show that the homology of DRep_ {\mathfrak g}(\mathfrak a) is isomorphic to the Chevalley–Eilenberg homology of the Lie coalgebra \mathfrak g^*(\bar{C}) , where C is a cocommutative DG coalgebra Koszul dual to the Lie algebra \mathfrak a . We construct a canonical DG algebra map \Phi_{\mathfrak g}(\mathfrak a): \mathrm {DRep}_{\mathfrak g}(\mathfrak a)^G \to \mathrm {DRep}_{\mathfrak h}(\mathfrak a)^W , relating the G -invariant part of representation homology of a Lie algebra \mathfrak a in \mathfrak g to the W -invariant part of representation homology of \mathfrak a in a Cartan subalgebra of \mathfrak g . We call this map the derived Harish-Chandra homomorphism as it is a natural homological extension of the classical Harish-Chandra restriction map. We conjecture that, for a two-dimensional abelian Lie algebra \mathfrak{a} , the derived Harish-Chandra homomorphism is a quasi-isomorphism. We provide some evidence for this conjecture, including proofs for \mathfrak {gl}_2 and \mathfrak {sl}_2 as well as for \mathfrak {gl}_n, \mathfrak {sl}_n, \mathfrak{so}_n and \mathfrak{sp}_{2n} in the inductive limit as n \to \infty . For any complex reductive Lie algebra \mathfrak g , we compute the Euler characteristic of DRep _{\mathfrak g}(\mathfrak a)^G in terms of matrix integrals over G and compare it to the Euler characteristic of DRep _{\mathfrak h}(\mathfrak a)^W . This yields an interesting combinatorial identity, which we prove for \mathfrak {gl}_n and \mathfrak{sl}_n (for all n ). Our identity is analogous to the classical Macdonald identity, and our quasi-isomorphism conjecture is analogous to the strong Macdonald conjecture proposed in [Ha1, F] and proved in [FGT]. We explain this analogy by giving a new homological interpretation of Macdonald's conjectures in terms of derived representation schemes, parallel to our Harish-Chandra quasi-isomorphism conjecture.

Dual Hodge decompositions and derived Poisson brackets

We study general properties of Hodge-type decompositions of cyclic and Hochschild homology of universal enveloping algebras of (DG) Lie algebras. Our construction generalizes the operadic construction of cyclic homology of Lie algebras due to Getzler and Kapranov. We give a topological interpretation of such Lie Hodge decompositions in terms of $$S^1$$ -equivariant homology of the free loop space of a simply connected topological space. We prove that the canonical derived Poisson structure on a universal enveloping algebra arising from a cyclic pairing on the Koszul dual coalgebra preserves the Hodge filtration on cyclic homology. As an application, we show that the Chas–Sullivan Lie algebra of any simply connected closed manifold carries a natural Hodge filtration. We conjecture that the Chas–Sullivan Lie algebra is actually graded, i.e. the string topology bracket preserves the Hodge decomposition.

(Co)homology of poset Lie algebras

We investigate the (co)homological properties of two classes of Lie algebras that are constructed from any finite poset: the solvable class $\frak{gl}^\preceq$ and the nilpotent class $\frak{gl}^\prec$. We confirm the conjecture of Jollenbeck that says: every prime power $p^r\!\leq\!n\!-\!2$ appears as torsion in $H_\ast(\frak{nil}_n;\mathbb{Z})$, and every prime power $p^r\!\leq\!n\!-\!1$ appears as torsion in $H_\ast(\frak{sol}_n;\mathbb{Z})$. If $\preceq$ is a bounded poset, then the (co)homology of $\frak{gl}^\preceq$ is \emph{torsion-convex}, i.e. if it contains $p$-torsion, then it also contains $p'$-torsion for every prime $p'\!<\!p$. \par We obtain new explicit formulas for the (co)homology of some families over arbitrary fields. Among them are the solvable non-nilpotent analogs of the Heisenberg Lie algebras from the Cairns & Jambor article, the 2-step Lie algebras from Armstrong & Cairns & Jessup article, strictly block-triangular Lie algebras, etc. The resulting generating functions and the combinatorics of how they are obtained are interesting in their own right. \par All this is done by using AMT (algebraic Morse theory). This article serves as a source of examples of how to construct useful acyclic matchings, each of which in turn induces compelling combinatorial problems and solutions. It also enables graph theory to be used in homological algebra.

Calculations on Lie Algebra of the Group of Affine Symplectomorphisms

We find the image of the affine symplectic Lie algebra gn from the Leibniz homology HL⁎(gn) to the Lie algebra homology H⁎Lie(gn). The result shows that the image is the exterior algebra ∧⁎(wn) generated by the forms wn=∑i=1n(∂/∂xi∧∂/∂yi). Given the relevance of Hochschild homology to string topology and to get more interesting applications, we show that such a map is of potential interest in string topology and homological algebra by taking into account that the Hochschild homology HH⁎-1(U(gn)) is isomorphic to H⁎-1Lie(gn,U(gn)ad). Explicitly, we use the alternation of multilinear map, in our elements, to do certain calculations.

Relative BGG sequences: I. Algebra

We develop a relative version of Kostant's harmonic theory and use this to prove a relative version of Kostant's theorem on Lie algebra (co)homology. These are associated to two nested parabolic subalgebras in a semisimple Lie algebra. We show how relative homology groups can be used to realize representations with lowest weight in one (regular or singular) affine Weyl orbit. In the regular case, we show how all the weights in the orbit can be realized as relative homology groups (with different coefficients). These results are motivated by applications to differential geometry and the construction of invariant differential operators.

(Co)homology of Lie algebras via algebraic Morse theory

The fundamental theorem of cancellation AMT [4] and [11], which is the algebraic generalization of discrete Morse theory [2] for simplicial complexes and smooth Morse theory [10] for differentiable manifolds, is discussed in the context of general chain complexes of free modules.The Chevalley (co)homology table of a Lie algebra is often a tremendous beast. Using AMT, we compute the homology of the Lie algebra of all triangular matrices soln over Q or Zp for large enough primes p. We determine the column and row in the table of Hk(soln;Z) where the p-torsion first appears. Module Hk(soln;Zp) is expressed by the homology of a chain subcomplex for the Lie algebra of all strictly triangular matrices niln, using the Künneth formula. All conclusions are accompanied by computer experiments.Then we generalize some results to Lie algebras of (strictly) triangular matrices gln≺ and gln⪯ with respect to any partial ordering ⪯ on [n]. We determine the multiplicative structure of H⁎(gln⪯) w.r.t. the cup product over fields of zero or sufficiently large characteristic, the result being the exterior algebra.Matchings used here can be analogously defined for other Lie algebra families and in other (co)homology theories; we collectively call them normalization matchings. They are useful for theoretical as well as computational purposes.

Erratum to: Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

Let H be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold X and a real algebraic bundle $${\mathcal {E}}$$ on X. Let $$\mathfrak {h}$$ be the Lie algebra of H. Let $$\mathcal {S}(X,{\mathcal {E}})$$ be the space of Schwartz sections of $${\mathcal {E}}$$ . We prove that $$\mathfrak {h}\mathcal {S}(X,{\mathcal {E}})$$ is a closed subspace of $$\mathcal {S}(X,{\mathcal {E}})$$ of finite codimension. We give an application of this result in the case when H is a real spherical subgroup of a real reductive group G. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let $$\pi $$ be a Casselman–Wallach representation of G and V be the corresponding Harish–Chandra module. Then the natural morphism of coinvariants $$V_{\mathfrak {h}}\rightarrow \pi _{\mathfrak {h}}$$ is an isomorphism if and only if any linear $$\mathfrak {h}$$ -invariant functional on V is continuous in the topology induced from $$\pi $$ . The latter statement is known to hold in two important special cases: if H includes a symmetric subgroup, and if H includes the nilradical of a minimal parabolic subgroup of G.

Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture

Let $H$ be a real algebraic group acting equivariantly with finitely many orbits on a real algebraic manifold $X$ and a real algebraic bundle $\mathcal{E}$ on $X$. Let $\mathfrak{h}$ be the Lie algebra of $H$. Let $\mathcal{S}(X,\mathcal{E})$ be the space of Schwartz sections of $\mathcal{E}$. We prove that $\mathfrak{h}\mathcal{S}(X,\mathcal{E})$ is a closed subspace of $\mathcal{S}(X,\mathcal{E})$ of finite codimension. We give an application of this result in the case when $H$ is a real spherical subgroup of a real reductive group $G$. We deduce an equivalence of two old conjectures due to Casselman: the automatic continuity and the comparison conjecture for zero homology. Namely, let $\pi$ be a Casselman-Wallach representation of $G$ and $V$ be the corresponding Harish-Chandra module. Then the natural morphism of coinvariants $V_{\mathfrak{h}}\to \pi_{\mathfrak{h}}$ is an isomorphism if and only if any linear $\mathfrak{h}$-invariant functional on $V$ is continuous in the topology induced from $\pi$. The latter statement is known to hold in two important special cases: if $H$ includes a symmetric subgroup, and if $H$ includes the nilradical of a minimal parabolic subgroup of $G$.

Homology of depth-graded motivic Lie algebras and koszulity

The Broadhurst-Kreimer (BK) conjecture describes the Hilbert series of a bigraded Lie algebra A related to the multizeta values. Brown proposed a conjectural description of the homology of this Lie algebra (homological conjecture (HC)), and showed it implies the BK conjecture. We show that a part of HC is equivalent to a presentation of A, and that the remaining part of HC is equivalent to a weaker statement. Finally, we prove that granted the first part of HC, the remaining part of HC is equivalent to either of the following equivalent statements: (a) the vanishing of the third homology group of a Lie algebra with quadratic presentation, constructed out of the period polynomials of modular forms; (b) the koszulity of the enveloping algebra of this Lie algebra.

Hopf Cyclic Cohomology in Non-symmetric Monoidal Categories

First, referring to our previous work, 'Hopf cyclic cohomology in braided monoidal categories', we reduce the restriction of the ambient category C being symmetric. We let C to be non-symmetric but assume only the restriction, ψ 2 = id, on the braid map correspond to the Hopf algebra H, which is the main player in the theory. We define a family of examples of such desired braided Hopf algebras, H, living in the category of anyonic vector spaces. Next, on one hand, we will prove that these anyonic Hopf algebras are the enveloping (Hopf) algebras of particular quantum Lie algebras, which we will construct. On the other hand, we will show that, analogous to the non-super and the super case, the well known relationship between the periodic Hopf cyclic cohomology of an enveloping (super) algebra and the (super) Lie algebra homology also holds for these particular quantum Lie algebras.

Leibniz homology of Lie algebras as functor homology

We prove that Leibniz homology of Lie algebras can be described as functor homology in the category of linear functors from a category associated with the Lie operad.