Representation Homology

• Representation homology is a derived invariant that encodes the structure of representation schemes and moduli spaces using DG algebra resolutions.
• Trace maps link cyclic homology to representation homology, extending classical character maps into derived Procesi theorems.
• The theory applies to noncommutative algebra, topology, and neural networks, offering practical computational methods for detecting deformation obstructions.

Representation homology is a homological invariant governing the structure of representation schemes and moduli spaces of local systems. It appears in both noncommutative algebra and topology, systematically resolving and extending classical representation varieties to derived, obstruction-sensitive functors. Developed through the machinery of model categories and derived algebraic geometry, representation homology provides a powerful toolset unifying invariants from algebra, topology, noncommutative geometry, and even neural network representation theory.

1. Derived Representation Schemes and the Definition of Representation Homology

For an associative $k$-algebra $A$ (over $\operatorname{char} k = 0$), the $n$-dimensional representation scheme of $A$ is the functor $$\operatorname{Rep}_n(A)\colon \operatorname{CommAlg}_k \to \mathrm{Sets}$$ with $$\operatorname{Rep}_n(A)(B) = \operatorname{Hom}_{\mathrm{Alg}_k}(A, M_n(B))$$. This functor is corepresented by a commutative $k$-algebra $A_n = k[\operatorname{Rep}_n(A)]$.

The category of (augmented) DG $k$-algebras, DGA$A$0, admits a standard Quillen model structure. Extending $A$1 to DG algebras, one achieves the left-derived functor $A$2 by replacing $A$3 with a cofibrant resolution $A$4. The derived representation scheme is then

$A$5

The representation homology is defined as

$A$6

and for any DG algebra $A$7 with augmentation, the natural restriction maps $A$8 allow for the stable object $A$9 to be constructed (Berest et al., 2013, Berest et al., 2013, Berest et al., 2011).

For topological spaces, the theory generalizes: for a pointed, connected CW complex $\operatorname{char} k = 0$0 and an affine group scheme $\operatorname{char} k = 0$1, one considers the derived functor of the representation scheme construction on the Kan loop group model $\operatorname{char} k = 0$2: $\operatorname{char} k = 0$3 and defines representation homology as

$\operatorname{char} k = 0$4

In degree zero, this recovers the coordinate ring of the classical representation variety; for $\operatorname{char} k = 0$5, $\operatorname{char} k = 0$6 detects higher-order, derived deformation-theoretic or topological invariants (Berest et al., 2020, Berest et al., 2017, Li, 2024).

2. Trace Maps, Cyclic Homology, and Derived Procesi Theorems

There are canonical “trace maps” from cyclic homology to representation homology. For a DG algebra $\operatorname{char} k = 0$7, let $\operatorname{char} k = 0$8 denote reduced cyclic homology. The trace map is a morphism

$\operatorname{char} k = 0$9

that extends the classical character map in degree zero. These assemble into

$n$0

which in degree zero encodes the classical Procesi theorem (characters generate $n$1-invariants).

In the stable limit $n$2, Berest and Ramadoss proved (Berest et al., 2013): $n$3 (Theorem 4.2), establishing a derived, stable version of Procesi’s result for trace algebras. For finite $n$4, failure of surjectivity is measured by the homology of a kernel complex $n$5, leading to a long exact sequence linking representation homology and cyclic homology.

Koszul duality underpins this structure: the Chevalley–Eilenberg complex $n$6 is Koszul dual to the stable trace subalgebra $n$7 (Berest et al., 2013, Berest et al., 2014, Berest et al., 2011).

3. Geometric, Topological, and Noncommutative Applications

The Kontsevich–Rosenberg principle advocates viewing the family $n$8 as a noncommutative “spectrum” for $n$9. However, $A$0 is generally not a flat functor. The derived functor $A$1 corrects for this inexactness. The higher representation homology groups $A$2 detect obstructions to geometric properties such as smoothness, being a (noncommutative) complete intersection, or Cohen–Macaulayness, for classical representation schemes (Berest et al., 2013, Berest et al., 2013).

Representation homology over topological spaces $A$3 with group coefficients $A$4, $A$5, generalizes representation varieties of $A$6. In the simply connected case, $A$7 is trivial, but higher homology $A$8 captures refined rational homotopy invariants of $A$9, computed via Sullivan and Quillen models (Berest et al., 2020, Berest et al., 2017). For suspensions $$\operatorname{Rep}_n(A)\colon \operatorname{CommAlg}_k \to \mathrm{Sets}$$0, representation homology coincides with higher Hochschild homology.

For Riemann surfaces $$\operatorname{Rep}_n(A)\colon \operatorname{CommAlg}_k \to \mathrm{Sets}$$1 and unipotent groups $$\operatorname{Rep}_n(A)\colon \operatorname{CommAlg}_k \to \mathrm{Sets}$$2, $$\operatorname{Rep}_n(A)\colon \operatorname{CommAlg}_k \to \mathrm{Sets}$$3 is computed via a Koszul complex, and the vanishing of higher degrees is a sharp criterion for when commuting schemes are global complete intersections (Li, 2024).

Additional examples include:

• Algebras of dual numbers, polynomial rings, and universal enveloping algebras;
• Link complements in $$\operatorname{Rep}_n(A)\colon \operatorname{CommAlg}_k \to \mathrm{Sets}$$4 and lens spaces, via explicit algebraic models (Berest et al., 2017, Li, 2024);
• Relation to symmetric homology: For $$\operatorname{Rep}_n(A)\colon \operatorname{CommAlg}_k \to \mathrm{Sets}$$5 an associative algebra, the symmetric homology $$\operatorname{Rep}_n(A)\colon \operatorname{CommAlg}_k \to \mathrm{Sets}$$6 coincides with the one-dimensional representation homology $$\operatorname{Rep}_n(A)\colon \operatorname{CommAlg}_k \to \mathrm{Sets}$$7 (Berest et al., 2022).

4. Computational Approaches and Invariants

Practical computations of representation homology are feasible using small DG algebra models built from the presentation of spaces as homotopy colimits. For example, for a closed orientable surface of genus $$\operatorname{Rep}_n(A)\colon \operatorname{CommAlg}_k \to \mathrm{Sets}$$8 and matrix group $$\operatorname{Rep}_n(A)\colon \operatorname{CommAlg}_k \to \mathrm{Sets}$$9, the DG algebra

$$\operatorname{Rep}_n(A)(B) = \operatorname{Hom}_{\mathrm{Alg}_k}(A, M_n(B))$$0

computes $$\operatorname{Rep}_n(A)(B) = \operatorname{Hom}_{\mathrm{Alg}_k}(A, M_n(B))$$1 as the homology of $$\operatorname{Rep}_n(A)(B) = \operatorname{Hom}_{\mathrm{Alg}_k}(A, M_n(B))$$2. The computational package RepHomology for Macaulay2 automates these constructions for surfaces, link complements, and Lie algebra analogues (Li, 2024). Representation homology of a pair $$\operatorname{Rep}_n(A)(B) = \operatorname{Hom}_{\mathrm{Alg}_k}(A, M_n(B))$$3 for neural networks can be computed efficiently via linear programming and union-find algorithms on overlap decompositions, yielding Betti numbers and tracking non-injectivity without reference to an external metric (Beshkov, 3 Feb 2025).

Key invariants derivable from representation homology include:

• Euler characteristics and their stabilization in bigraded settings, from explicit product formulas;
• Combinatorial identities, e.g., Macdonald-type formulas relating characters of symmetric algebras to stable homology (Berest et al., 2013, Berest et al., 2014).

5. Representation Homology in Lie Theory, Neural Representations, and Beyond

In the Lie algebra setting, derived representation schemes $$\operatorname{Rep}_n(A)(B) = \operatorname{Hom}_{\mathrm{Alg}_k}(A, M_n(B))$$4 for a Lie algebra $$\operatorname{Rep}_n(A)(B) = \operatorname{Hom}_{\mathrm{Alg}_k}(A, M_n(B))$$5 and reductive $$\operatorname{Rep}_n(A)(B) = \operatorname{Hom}_{\mathrm{Alg}_k}(A, M_n(B))$$6 lead to representation homology isomorphic to Chevalley–Eilenberg homology of appropriate current Lie coalgebras. Derived Harish-Chandra maps extend classical restriction maps to the derived, homological setting. Macdonald identities emerge as Euler characteristics of representation homology in specific cases, and the strong Macdonald conjecture obtains a homological reformulation in terms of invariants of representation homology (Berest et al., 2014, Berest et al., 2020).

Representation homology also connects to neural network theory. For ReLU networks, the piecewise-linear structure induces a polyhedral stratification of the input, and the topology of image representations is determined by the relative homology $$\operatorname{Rep}_n(A)(B) = \operatorname{Hom}_{\mathrm{Alg}_k}(A, M_n(B))$$7, where $$\operatorname{Rep}_n(A)(B) = \operatorname{Hom}_{\mathrm{Alg}_k}(A, M_n(B))$$8 encodes loci of non-injectivity (overlaps) in the stratification. This approach gives a purely topological rather than geometric characterization of Betti numbers in network representations, decoupled from any choice of metric, and it admits a continuous-exact computational pipeline (Beshkov, 3 Feb 2025).

6. Relations to Other Invariants and Homology Theories

Representation homology relates directly to and generalizes several classical invariants:

• Cyclic homology: via trace maps, representation homology includes cyclic homology as a source of generators for its stable invariants (Berest et al., 2011, Berest et al., 2013).
• Higher Hochschild homology: representation homology recovers higher Hochschild invariants of suspensions, aligning with Loday–Pirashvili functorial constructions (Berest et al., 2017).
• Symmetric homology: as established for characteristic zero, symmetric homology is isomorphic to one-dimensional representation homology; these theories intertwine through bar constructions and their topological models (Berest et al., 2022).
• Lie algebra homology: stable Koszul duality connects representation homology of algebras to Chevalley–Eilenberg (co)homology of current Lie algebras and coalgebras (Berest et al., 2014, Berest et al., 2013).
• Rational homotopy theory: in the simply connected case, representation homology can be computed in terms of Quillen and Sullivan models, illuminating its functorial dependence on rational homotopy type (Berest et al., 2020).

Representation homology, as a unifying derived invariant, thus plays a central role in noncommutative algebraic geometry, topological invariants, and algebraic representations, equipped with explicit computational models, deep connections to Koszul duality, and broad applicability across disciplines.