27 problems

Buchstaber's conjecture. The algebra $H^*(W^+)$ is generated, using some non-trivial Massey products, by its first cohomology $H^1(W^+)$.

Pirashvili's conjecture. A perfect complex Lie algebra is semisimple if and only if

Let $\mathcal{T}$ be the supertranslation algebra associated to a super-Minkowski spacetime, and consider its third Lie algebra cohomology. A 3-cocycle is Lorentz-invariant if it i…

Vogan's conjecture. The induced topologies on the cohomology groups $H^{\text{p}}(\mathfrak{n},M_{\text{glob}})$ are Hausdorff, and there are natural isomorphisms of $L_{0}$-repres…

Let $\mathfrak A$ be an associative algebra with a trace, and let $D_1,\dots,D_{2n}$ be derivations satisfying conditions (i)--(iii) from Section 2.4. Let $\Sigma_{N,2n-1}$ denote…

Let $\mathfrak A$ be an associative algebra with a trace, and let $D_1,\dots,D_i$ be trace-preserving associative derivations satisfying conditions (i)--(iii): each $D_j$ is trace-…

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra, let $\mathfrak{n}$ be a nilpotent subalgebra, and write $C^j$ and $H^j$ for the corresponding cochain and cohomology…

Let $\mathrm{Vect}(\mathbb R^n)$ be the Lie algebra of vector fields on $\mathbb R^n$, let $\Omega(\mathbb R^n)$ denote the space of differential forms, and let…

Feigin–Novikov conjecture. Every cocycle of the vector field algebra is cohomologous to a linear combination of geometric cocycles obtained by integration along the basis cycles in…

Let $\cV$ be a tensor $\lie{witt}_d$-module, with Lie derivative homomorphism … Let $\tau\colon \HH_2^{Lie}(\operatorname{End}\cV)\to\C$ be the functional determined by the univers…

Toral rank conjecture. One has

Lie-algebra cohomology conjecture. (a) If

Invariant Dolbeault cohomology conjecture. The inclusion $\iota$ is a quasi-isomorphism; equivalently, it induces an isomorphism

Let $\mathcal{S}$ be the Lie algebra in the paper, let $\mathcal{K}$ be its central module, and let a $\mathcal{K}$-valued affine cocycle be a bilinear map on $\mathcal{S}$ satisfy…

Let $L$ be a finite-dimensional nilpotent Lie algebra defined over the rational numbers, and let $C(L)$ denote its center. Nilpotent Lie algebra toral rank conjecture. Then … This…

Let $\mathfrak{g}$ be a Lie algebra, and let $HL_p(\mathfrak{g},K)$ denote its Leibniz homology with trivial coefficients. The preceding conditions are the vanishing of…

Let $L$ be a finite-dimensional nilpotent Lie algebra over $\mathbb{R}$, let $Z$ be its centre, and let $H^*(L)$ denote its cohomology ring with trivial coefficients. The action of…

Let $d54a$ be a finite-dimensional complex simple Lie algebra, and let $d54a[t]=d54a\otimes\mathbb{C}[t]$ be its polynomial current algebra. Write $\operatorname{H}^\bullet$ for Li…

Let $d54a$ be a finite-dimensional complex simple Lie algebra, let $sd45e 1$, and set $A=d53e[t]/\langle t^s\rangle$. The corresponding truncated current algebra is…

Let $E$ be a smooth commutative algebra, and let $\mathfrak{gl}_{\infty}(E)$ and $\mathfrak{gl}_n(E)$ be the infinite and finite general linear Lie algebras over $E$, with correspo…

Let $I$ be the ideal in the Lie algebra used to describe the pure-spinor cohomology, and let $H_3(I)$ denote its third homology. For spinor indices $\alpha$ and $\dot{\beta}$, defi…

Let $I$ be the ideal whose abelianization is $I/[I,I]$, and let $W^{\alpha}$ be elements of $I$. The quotient $I/[I,I]$ records the corresponding field strengths, while its dual is…

Let $\mathfrak q\subseteq\mathfrak g$ be a constructible parabolic subalgebra, with nilradical $\mathfrak u$, Levi factor $\mathfrak l$, and associated admissible categories of…

Hanlon's property M conjecture. Property M should hold not only for semisimple Lie algebras, but also for the nilpotent radical of a parabolic subalgebra of a semisimple Lie algebr…

Let $\mathfrak{g}$ be a finite-dimensional complex nilpotent Lie algebra, let $H^{\ast}(\mathfrak{g},\mathbb{C})$ denote its cohomology, and let $Z(\mathfrak{g})$ be its center. To…