On the Cohomology of Cyclic Associative Algebras

Abstract: We introduce a cohomology theory for cyclic associative algebras, a subclass of shift associative algebras defined by the identity $(xy)z = x(yz) = y(zx)$. This cohomology, denoted $H^{\bullet_{\mathrm{cyc}}(A,} M)$, is a subtheory of Hochschild cohomology obtained by restricting to cochains that satisfy a cyclic compatibility condition derived from the defining identity. We prove that $H^{2_{\mathrm{cyc}}(A,} M)$ classifies cyclic associative extensions of $A$ by a cyclic bimodule $M$. The universal derivation and the module of differential forms $Ω<sup>\bullet_{\mathbb{F}}(A)$ are constructed, and $(Ω^{\bullet_{\mathbb{F}}(A),} d)$ is shown to be the universal cyclic differential graded algebra over $A$. For trivial coefficients, we establish natural inclusions $$HC^n(A) \hookrightarrow H^{n_{\mathrm{cyc}}(A,} \mathbb{F}) \hookrightarrow HH^n(A, \mathbb{F})$$, placing our theory intermediate between Connes' cyclic cohomology and Hochschild cohomology.

• The paper constructs an enveloping algebra and universal differential forms, then proves that its cyclic cochain conditions are preserved by the Hochschild differential.
• The paper establishes a canonical bijection between second cyclic cohomology and equivalence classes of split abelian extensions, making the theory useful for extension and deformation analysis.
• The paper places cyclic cohomology between Connes’ cyclic and Hochschild cohomology and derives smoothness criteria, although the related Ext interpretation and strictness claims remain partly conjectural or unverified.

Context and motivation

Cyclic associative algebras form a subclass of shift associative algebras, the variety introduced by Abdelwahab, Kaygorodov, and Sartayev and defined by the identity $(xy)z = y(zx)$. A cyclic associative algebra satisfies the stronger condition

$(xy)z = x(yz) = y(zx),$

imposing full cyclic symmetry on triple products. The class contains all commutative associative algebras, but also genuinely non-commutative nilpotent examples appearing already in dimension three (e.g., the algebra with basis $e_1, e_2, e_3$ and $e_1e_2 = -e_2e_1 = e_3$, all other products zero). The paper develops a cohomology theory for this variety, positioned between Hochschild cohomology and Connes' cyclic cohomology.

Cyclic bimodules and the enveloping algebra

The coefficient objects are cyclic bimodules: vector spaces $M$ with left and right $A$-actions satisfying three families of identities (C1)–(C3) that extend the defining identity of $A$ to expressions involving a module element in any position. The author constructs an enveloping algebra

$E(A) = (A \otimes A) \oplus A_l \oplus A_r$

with an explicit associative product, and proves that representations of $A$ are equivalent to left $E(A)$-modules. This reduces representation theory to ordinary module theory over a unital-type associative algebra, which is the technical backbone for everything that follows.

Universal derivation and differential forms

Via the augmentation $(xy)z = x(yz) = y(zx),$0, the module of Kähler-type differentials is defined as $(xy)z = x(yz) = y(zx),$1 where $(xy)z = x(yz) = y(zx),$2, with universal derivation $(xy)z = x(yz) = y(zx),$3. The standard universality property holds:

$(xy)z = x(yz) = y(zx),$4

Higher forms are exterior powers over $(xy)z = x(yz) = y(zx),$5, and the pair $(xy)z = x(yz) = y(zx),$6 is shown to be the universal cyclic differential graded algebra over $(xy)z = x(yz) = y(zx),$7: any morphism from $(xy)z = x(yz) = y(zx),$8 into the degree-zero part of a CDG algebra extends uniquely to a CDG morphism. Consequently $(xy)z = x(yz) = y(zx),$9 identifies with skew-symmetric $e_1, e_2, e_3$0-derivations.

The cochain complex and extension classification

The cochains $e_1, e_2, e_3$1 consist of multilinear maps $e_1, e_2, e_3$2 satisfying five compatibility conditions derived by linearizing the cyclic identity; the differential is the classical Hochschild coboundary $e_1, e_2, e_3$3. The key technical result—proved in full detail in an appendix—is that $e_1, e_2, e_3$4 preserves these conditions, so $e_1, e_2, e_3$5 is a subcomplex of the Hochschild complex. Low degrees behave classically: $e_1, e_2, e_3$6 is the invariant submodule $e_1, e_2, e_3$7, and $e_1, e_2, e_3$8.

The central theorem is the classification result:

Theorem. There is a canonical bijection $e_1, e_2, e_3$9, the set of equivalence classes of split abelian extensions of $e_1e_2 = -e_2e_1 = e_3$0 by $e_1e_2 = -e_2e_1 = e_3$1.

The proof follows the Hochschild pattern: a 2-cocycle $e_1e_2 = -e_2e_1 = e_3$2 defines a product on $e_1e_2 = -e_2e_1 = e_3$3 via $e_1e_2 = -e_2e_1 = e_3$4, which is cyclic associative exactly when $e_1e_2 = -e_2e_1 = e_3$5 satisfies both the Hochschild 2-cocycle condition and the additional cyclic constraint; changing the section alters $e_1e_2 = -e_2e_1 = e_3$6 by a coboundary. This justifies the cohomology as the correct deformation/extension tool for the variety.

Smoothness and cohomological dimension

Following Cuntz–Quillen, the paper defines $e_1e_2 = -e_2e_1 = e_3$7 to be almost-free if every abelian extension admits a lifting homomorphism, and proves the equivalence:

1. $e_1e_2 = -e_2e_1 = e_3$8 is almost-free;
2. $e_1e_2 = -e_2e_1 = e_3$9 is projective as an $M$0-bimodule;
3. $M$1 has cohomological dimension $M$2 for $M$3.

A companion theorem characterizes cohomological dimension zero: every derivation being inner is equivalent to $M$4 for all $M$5 and to the $M$6-module $M$7 being projective. Classical smooth commutative algebras and free cyclic associative algebras are almost-free.

An important caveat: the identification of $M$8 with $M$9, and hence the long exact sequence machinery used informally in the proofs, is stated as conjectural—it would require derived functors in a non-unital context, which the paper does not construct. The smoothness theorems therefore rest on arguments "following the same lines as the classical case" rather than on a fully established derived-functor framework.

Comparison with Connes' cyclic cohomology

For trivial coefficients, the paper establishes natural injections

$A$0

placing the new theory strictly between Connes' cyclic cohomology and Hochschild cohomology. The first inclusion uses that a Connes-cyclic cochain ($A$1) satisfies the conditions defining $A$2, and that the Connes operator $A$3 contributes only coboundaries here. Both inclusions are claimed to be strict in general, evidenced by low-dimensional classification examples, though no explicit computation of a specific group where strictness holds is given. For commutative associative $A$4, all three theories coincide up to shift, recovering $A$5.

Limitations and open questions

Several points remain open or assumed:

• The $A$6 interpretation of the cohomology is conjectural; without it, the long exact sequences invoked in the smoothness proofs lack a rigorous foundation.
• Strictness of the inclusions $A$7 is asserted but not demonstrated by an explicit cocycle computation.
• No concrete cohomology groups of specific non-commutative cyclic associative algebras (e.g., the 3-dimensional example above) are computed, despite the availability of the low-dimensional classifications.
• The field is assumed algebraically closed of characteristic $A$8 throughout; behavior in small characteristic is not addressed.
• Whether $A$9 governs formal deformations (not merely abelian extensions) of cyclic associative algebras is not treated.

Conclusion

The paper supplies the standard cohomological infrastructure—enveloping algebra, universal derivation, differential forms, abelian extension classification—for the variety of cyclic associative algebras, and situates the resulting theory between two classical cohomologies. Its main verifiable contributions are the closure of the Hochschild differential on the cyclic cochain subspace and the $A$0 extension classification; its main unresolved dependency is the conjectural $A$1 interpretation underpinning the smoothness results.