250 problems

Let $\overline{\mathfrak{g}}$ be a finite-dimensional simple Lie algebra, and let $\lambda\in \overline{P}_+\backslash\{0\}$ be a non-zero dominant integral weight. The relative as…

Let $p>2$ and let a finite-dimensional Lie algebra or Lie superalgebra have an indecomposable Cartan matrix normalized as in the source's normalization convention. A Chevalley basi…

Let $\mathfrak{h}$ be the Cartan subalgebra and let $\Lambda$ denote the set of parameters for which the corresponding quantization exists, viewed as a semigroup under the semigrou…

Let $\mathrm{Dif}_1$ denote the Lie algebra of differential operators of order at most one on the circle, and let $\Psi_3,\widetilde\Psi_5,\widetilde\Psi_7,\widetilde\Psi_9,\dots$…

Let ${\frak g}$ be a finite-dimensional Lie algebra with a metric and orthonormal basis $\{{\frak g}_a\}_{a=1}^{\dim{\frak g}}$, and let $R$ be a finite-dimensional representation.…

Let ${\cal SW}(V)$ be the Schur–Weyl category, let $V_{\lambda}$ denote its distinguished objects, and let ${\bf g}=sl(V_S)$ act on these objects. A linear map between two objects…

Let $(\Gamma_1,\Gamma_2,\tau)$ be an admissible triple and let $r^0\in\mathfrak h\otimes\mathfrak h$ satisfy the equations referenced in the source. Define $\tilde r^0=r^0-P^0/2$,…

Let $\lambda\in\mathbb C$, let ${\mathfrak{gl}}(\lambda)$ be embedded in $\mathrm{Dif}_1$, and let … be the first page of the associated Feigin Serre–Hochschild spectral sequence.…

Let $\mathfrak{g}$ be a semisimple Lie algebra over an algebraically closed field of characteristic $0$, and let $x\in\mathfrak{g}$. By $C_{\mathfrak{g}}(x)$ denote the centralizer…

Let $\mathfrak{a}$ be a simple Lie algebra, let ${\mathbf d}\geq 1$, and let $\psi_{\mathbf d}$ be the real-valued polynomial obtained by restricting the potential $\Psi$ to the sp…

Let $A_n$ be the free associative algebra on generators $x_1,\dots,x_n$, regarded as a Lie algebra with bracket $[a,b]=a\cdot b-b\cdot a$. For a monomial…

For each $n$, let ${\mathfrak{qtr}}_n$ be the Lie algebra generated by the elements $r_{ij}$, subject to the classical Yang–Baxter relations, and let ${\mathbf{QTr}}_n$ be the grou…

Let ${\cal P}$ be a quadratic Koszul operad. Let ${\cal S}^*_{{\cal P}_\infty}$ be the suboperad of natural operations generated by the restricted class of operations generalizing…

Let ${\cal B}_{\it Lie}^*$ be the differential graded operad of natural operations on the Chevalley–Eilenberg complex of a Lie algebra with coefficients in itself, and let…

Let $L_n$ be the quadratic Lie algebra associated with the pure cactus group, let $\mathbf{L}_n$ be the corresponding braid Lie algebra, and let $\xi_n:L_n\to\mathbf{L}_n$ be defin…

Let $\widehat{\Gamma_n}$ be the prounipotent completion of the pure cactus group $\Gamma_n$, and let…

Let $\Gamma_n$ be the pure cactus group, let $\Gamma_n^k$ denote its lower central series, and let $\mathcal{L}_n$ be the associated graded Lie algebra after quotienting by $2$-tor…

Let $U_n=U(L_n)$, where $L_n$ is the quadratic Lie algebra generated by the elements $\mu_{ijk}$. Consider the natural morphisms $U_{n+1}\to U_n$ and $L_{n+1}\to L_n$ sending…

Let $L_n$ be the quadratic Lie algebra generated by the elements $\mu_{ijk}$, and let $U_n=U(L_n)$ be its universal enveloping algebra. Freeness conjecture. The $\mathbb{Z}$-module…

Intrinsic-bracket conjecture. The homomorphism $I$ induces an isomorphism of operads

Let ${\mathcal L{\it ie}}$ be the Lie operad, and let ${\mathcal B}_{{\mathcal L{\it ie}}}$ be the operad of natural operations on the cohomology of Lie algebras. Lie-operad conjec…

Let $\mathfrak{sp}_{2n}$ be the symplectic Lie algebra, and consider a parabolic subalgebra of type (b) with block parameters $d=(d_1,\ldots,d_r,d_{r+1},d_r,\ldots,d_1)$. Let…

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra, let $E=(B\otimes B)^{\mathfrak{g}}$, and let $p_1,\ldots,p_r$ and $\widehat p_1,\ldots,\widehat p_r$ be as above. Let…

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra with dual Coxeter number $g$. Let $p_1,\ldots,p_r$ be homogeneous generators of $(S\mathfrak{g})^{\mathfrak{g}}$, and…

Let $\mathfrak{g}$ be a simple finite-dimensional Lie algebra over $\mathbb{C}$, let $B$ be the algebra used in the paper, and set $E=(B\otimes B)^{\mathfrak{g}}$. Let…