Homology of the Lie Algebra of Locally Generated Derivations of a Discrete and Proper Metric Space

Abstract: We associate to each proper discrete metric space $X$ a Lie algebra that acts by locally generated derivations on an infinite tensor product of matrix algebras indexed by the points of $X$. We compute the homology of this Lie algebra with trivial scalar coefficients when $X$ is the integer lattice in $n$-dimensional Euclidean space.

• The paper establishes a rigorous framework for computing the homology of locally generated derivations and unveils recursive relationships among primitive elements in discrete spaces.
• It employs Chevalley–Eilenberg complexes and spectral sequences to derive suspension isomorphisms and demonstrate vanishing homology for half-spaces.
• The results bridge operator algebra, coarse geometry, and quantum lattice models, providing algebraic invariants relevant to finite-range entanglement.

Homology of the Lie Algebra of Locally Generated Derivations on Discrete Proper Metric Spaces

Introduction and Motivation

The paper establishes a rigorous framework for analyzing the Lie algebra of locally generated derivations, denoted $lgd(X)$, associated to infinite tensor products of matrix algebras indexed by the points of a discrete and proper metric space $X$. Locally generated derivations are constructed to capture infinitesimal automorphisms that only act nontrivially within bounded neighborhoods. This construction is motivated both by mathematical interests in coarse geometry, operator algebras, and homological invariants, as well as by their physical relevance in the study of finite-range entanglement in multiparticle quantum systems, although this work restricts itself to purely algebraic techniques.

For the pivotal example where $X = \mathbb{Z}^n$, the paper computes the homology of $lgd(X)$ with trivial coefficients, unveiling precise recursive relationships between the primitive parts of Lie algebra homology for different values of $n$.

Construction of the Infinite Tensor Product Algebra and Locally Generated Derivations

Given a proper discrete metric space $X$, and a function $\alpha: X \to \mathbb{N}$, the algebra $A_\alpha(X)$ is defined as the infinite tensor product $\bigotimes_{x \in X} M_{\alpha(x)}(\mathbb{C})$ via a direct limit over finite subsets. The associated Lie algebra $lgd_\alpha(X)$ consists of derivations $X$0 such that, for some $X$1, there exist local elements $X$2 satisfying

$X$3

for any $X$4. The sum is finite for each $X$5. This local generation ensures the action of $X$6 is tightly supported, reflecting the locality property essential in coarse geometry and condensed matter theory.

Taking the direct limit over all possible $X$7, the global algebra $X$8 and Lie algebra $X$9 are defined. Ideals and subalgebras (such as those fixing a chosen state, $X = \mathbb{Z}^n$0) are constructed via similar limiting procedures.

Homological Analysis and Structural Results

Lie Algebra Homology and Hopf Algebra Structure

Lie algebra homology is computed using the standard Chevalley–Eilenberg complex, and $X = \mathbb{Z}^n$1 is shown to possess a canonical coalgebra structure via the diagonal morphism. Under suitable product and correcting morphisms, $X = \mathbb{Z}^n$2 attains the structure of a commutative, cocommutative, connected Hopf algebra. As a consequence of the Milnor–Moore theorem, the homology is entirely determined by its primitive elements: $X = \mathbb{Z}^n$3 with graded considerations dictating whether the symmetric algebra becomes exterior when primitives are concentrated in odd degrees.

Excision and Mayer–Vietoris Properties

Locally generated derivations admit excision properties, matching the $X = \mathbb{Z}^n$4-excisive decompositions relevant to coarse geometry. The Lie algebra version of the Mayer–Vietoris principle is verified, ensuring that homological invariants of $X = \mathbb{Z}^n$5 align with those of coarse operator algebras.

Hochschild–Serre Spectral Sequence and Primitive Element Theorem

A central part of the analysis involves the Hochschild–Serre spectral sequence for an ideal $X = \mathbb{Z}^n$6. Under conditions where the quotient acts trivially in homology and the ideal is acyclic in positive degrees, the primitive elements of the homology of the quotient are shown to correspond, after a degree shift, to those of the ideal: $X = \mathbb{Z}^n$7

Explicit Computations: Homology of $X = \mathbb{Z}^n$8

Primitive Part and Suspension Isomorphism

The main result is the explicit computation of the primitive part of the homology for $X = \mathbb{Z}^n$9. By induction and leveraging spectral sequence arguments, the primitive elements are determined: $lgd(X)$0

This demonstrates a "suspension" isomorphism: the primitive part in $lgd(X)$1-space is isomorphic to that in $lgd(X)$2-space shifted up by one degree. The recursive structure mirrors that of $lgd(X)$3-theory for coarse $lgd(X)$4-algebras and stable homotopy theory ($lgd(X)$5-spectra).

Eilenberg Swindle: Vanishing of Homology in Half-Spaces

Using an abstract Eilenberg swindle, the paper shows that for the half space $lgd(X)$6, the homology $lgd(X)$7 vanishes for all $lgd(X)$8. This is realized by constructing an "infinitary convolution" of shift and correcting morphisms that, via homological algebra, forces primitive elements to zero.

Homology for a Point

For the case $lgd(X)$9, $n$0 is a direct limit of $n$1. Classical invariant theory and results of Dynkin, Hopf, and Chevalley–Eilenberg yield

$n$2

The "fixed state" subalgebra $n$3 recovers primitives in all odd degrees starting from one.

Lie Algebra Morphisms and Actions on Homology

A rich family of endomorphisms and automorphisms (inner by locally nilpotent elements, flip automorphisms exchanging tensor factors, shift maps, and correcting morphisms) are constructed. Via explicit algebraic arguments, all such morphisms are shown to induce the identity in homology, amplifying the robustness of homological invariants under a variety of local transformations.

Implications and Connections

Theoretical Implications

The recursive structure of primitive homology for $n$4 places these Lie algebras in direct analogy to the $n$5-homology of discrete metric spaces, strengthening the bridge between controlled topology, operator algebraic index theory, and homological invariants in infinite tensor contexts. The alignment with the homological invariants encountered in coarse geometry highlights potential avenues for further explorations in noncommutative geometry and the classification of phases of matter. The stability of homological invariants under local automorphisms suggests robustness for physical models using locally generated dynamics.

Practical Implications and Future Directions

Although the paper is algebraic, the connection to quantum spin systems and finite-range entanglement (cf. [Kubota 2025], [Kapustin, Sopenko, Yang 2021]) suggests practical relevance for condensed matter theory, particularly in the classification of invertible quantum phases and studies of topological quantum computation. The homological invariants computed here could, in further analytic settings, serve as algebraic proxies for invariants of quantum phases or operator algebras in infinite volume.

Future work may extend to completed Lie algebras (suited for analytic applications), investigate further connections with cyclic homology (cf. [Loday–Quillen 1984]), and explore the cohomological duals. Speculative directions include the use of these results in stable homotopy or $n$6-spectrum constructions in mathematical physics, and formalizing the links with coarse index theory and topological invariants for quantum lattice systems.

Conclusion

The paper provides a comprehensive analysis of the homology of the Lie algebra of locally generated derivations for infinite tensor products over discrete proper metric spaces. The explicit computation for $n$7, the suspension isomorphism for primitive homology, and the vanishing results in half-spaces collectively solidify the algebraic structure and homological invariants of these Lie algebras, aligning them with the broader paradigm of coarse geometric invariants and stable homotopy theory. The technical machinery developed—excision, spectral sequences, and action of morphisms—positions the results as foundational for further research in operator algebraic topology, mathematical physics, and noncommutative geometry.

References:

• "Homology of the Lie Algebra of Locally Generated Derivations of a Discrete and Proper Metric Space" (2607.08455)
• "Stable homotopy theory of invertible gapped quantum spin systems I: Kitaev's $n$8-spectrum" (Kubota, 16 Mar 2025)
• "A classification of invertible phases of bosonic quantum lattice systems in one dimension" (Kim et al., 2021)
• "Cyclic homology and the Lie algebra homology of matrices" [Loday–Quillen 1984]
• "On the structure of Hopf algebras" [Milnor–Moore 1965]