71 problems

Let $\Lambda$ be a finite-dimensional $\mathbbm{k}$-algebra, and let $\mathcal{N}$ be the ideal of $HH^{\ast}(\Lambda)$ generated by all homogeneous nilpotent elements. Snashall–So…

Căldăraru's HKR module-compatibility conjecture. The isomorphism $\mathrm{IK}$ is compatible with the module structures on differential forms over polyvector fields and on Hochschi…

Let $A$ be the algebra of functions on an affine space, let $V_A$ be the graded Lie algebra of polyvector fields on that affine space, and let $C^{\bullet}(A)$ be the differential…

Let $A$ be a finite-dimensional algebra over a field $K$, and let $\mathcal{N}$ be the ideal of $\operatorname{HH}^*(A)$ generated by all homogeneous nilpotent elements. Hochschild…

Let $\Lambda$ be a monomial triangular algebra, meaning a triangular algebra whose defining relations are paths. Its Hochschild cohomology is denoted by…

Let $\mathcal{M}$ be a von Neumann algebra, and let $H^n(\mathcal{M},\mathcal{M})$ denote its continuous Hochschild cohomology groups with coefficients in itself. Kadison–Ringrose'…

An $A_{\infty}$-algebra is a homotopy associative algebra; its cohomological Hochschild complex is a differential graded Lie algebra. A $2$-algebra is an algebra over the operad…

Kontsevich's tangent-map conjecture. The induced map

Cohen–Jones BV algebra conjecture. The identification $F$ is an isomorphism of BV algebras.

Let $A$ be a finite-dimensional algebra over a field $\k$, let $H^*(A,A)$ denote its Hochschild cohomology ring, and let $\mathcal{N}$ be the ideal generated by the homogeneous nil…

BV-infinity lifting conjecture. The map $\mathrm{HKR}$ should lift to a $BV_\infty$ map.

Happel's conjecture. One has

Ginzburg–Kaledin's conjecture. The isomorphism is multiplicative and extends from symmetric powers to any symplectic orbifold.

Let $H(\underline{t};q)$ be the rank-one double affine Hecke algebra with parameters $q$ and $\underline{t}$. Here $HH^i$ denotes its $i$-th Hochschild cohomology group, and…

Let $M$ be an exact symplectic manifold satisfying Assumption, and let … be the natural open–closed string map from symplectic cohomology to the Hochschild cohomology of the Fukaya…

Let $A$ be a tilted algebra. Its first Hochschild cohomology space is denoted by $H^1(A)$. Simply connectedness conjecture. The algebra $A$ is simply connected if and only if … Thi…

Let ${\cal C}$ be an $A_{\infty}$-category such that $HH^1({\cal C})$ is abelian and the connected automorphism group $G_{\cal C}=Aut^0({\cal C})$ is a finite-dimensional commutati…

Consider a topological string theory obtained by twisting a unitary $N=2$ $d=2$ field theory and coupling it to topological gravity. Let the category of topological D-branes encode…

Let $M$ be a manifold and let $\mathcal{A}$ be the open-string algebra, with Hochschild cohomology $H^*(\operatorname{Hoch}(\mathcal{A}))$, while the closed-string BRST cohomology…

Let $G$ be a finite group, let $k$ be a field whose characteristic divides the order of $G$, and let $B$ be a block of the group algebra $kG$ with defect group $P$. Linckelmann's c…

Let $\mathcal{A}$ and $\mathcal{B}$ be small dg categories, and consider a larger source bicategory whose Hom category for $(\mathcal{A},\mathcal{B})$ is the whole derived category…

Belmans–Smirnov's conjecture. If $X$ is neither (co)minuscule nor (co)adjoint, then $X$ is not Hochschild global; equivalently, for some $i>0$ and some $j$,

Let $V$ be a symmetric monoidal category and let $\mathcal{P}$ be a $V$-enriched $2$-operad. Let $\mathcal{P}_1$ be the $1$-operad obtained by restricting $\mathcal{P}$ to the $1$-…

Open-closed Deligne conjecture. For any open-closed homotopy algebra $(\mathfrak l,\mathfrak q)$, there is a $G_\infty$ algebra structure on the space…

Let $X$ be a log scheme for which the constructions in the paper are defined, let $i:X\to B$ be the local complete intersection morphism used to define logarithmic Hochschild cohom…