31 problems

Noetherianity conjecture. The universal enveloping algebra

Duflo conjecture. The algebras

Let $K$ be a field of characteristic zero, let $\mathfrak{g}$ be a Lie algebra over $K$, and let $U(\mathfrak{g})$ denote its universal enveloping algebra. Sierra–Walton conjecture…

Operadic realization conjecture. The expression $\left(1-\sum_{k\geq 1}p_k\right)^{-1}$ is Schur positive and can be expressed in terms of the free algebra over an operad. This con…

Vinberg's lifting conjecture. The Mishchenko–Fomenko algebra can be lifted to a commutative subalgebra of the universal enveloping algebra $U(g)$.

Let $(\mathfrak{g},\sigma)$ be a symmetric pair, with Cartan decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$. Let $U(\mathfrak{g})$ be the universal enveloping algebra,…

Let $T$ be a simple Lie triple system and let $U(T)$ be its universal enveloping algebra. The augmentation ideal conjecture. The only proper ideal of $U(T)$ is its augmentation ide…

Let $\lambda$ be a dominant weight, let $V_\lambda$ be the corresponding representation space, and let $\Omega_\lambda(u)=\Omega_\lambda+u\otimes \operatorname{id}$. Using a basis…

The paper considers the Lie algebra $\mathfrak{gl}_n(\mathbb{C})$, its universal enveloping algebra $U(\mathfrak{gl}_n(\mathbb{C}))$, and the polynomial families parameterized by d…

Let $V$ be a simple Lie triple system and let $U(V)$ be its universal enveloping algebra. The augmentation-ideal conjecture. The only proper ideal of $U(V)$ is its augmentation ide…

Conjecture. Under either of these hypotheses, $I_{\mathfrak{a}}(\mathbb{F})$ has the stated polynomial-algebra structure with the given derivation action.

Let $U_{q,r}(so(N+2))$ be the quantum orthogonal universal enveloping algebra generated by the ordered elements described above, and let a monomial of type … be formed, where the p…

Let $e,f,h$ be the basis elements of $\mathfrak{sl}_2$, and define the $2\times 2$ matrices … For nonnegative integers $x,y,z$, let $f(x,y,z)$ be the sum over all orders of taking…

Let $U(\mathfrak{so}(n))$ be the universal enveloping algebra, and let $\mathfrak{O}_{\overline{0}}$ denote the even subalgebra of the TAMA. For a sequence $A=(a_1,a_2,a_3,a_4)$ of…

Let $\mathfrak{g}=\mathfrak{gl}(m\cdot\1+nP)$, or let $\mathfrak{g}$ be any restricted subalgebra of it, and let $X\in\mathfrak{g}$ with $X\notin\ker d$. Write $X^{[2]}$ for the re…

Let $n$ be a positive integer, let $L$ be an $n$-strong Lie algebra over a field with at least $n$ elements, and let $A(L)$ be its universal enveloping algebra. For a monomial…

Let $(\mathfrak{g},\top_{\mathfrak{g}},[\,\ ])$ be a locality Lie algebra, let $\mathcal{T}_\top(\mathfrak{g})$ be its locality tensor algebra, and let $J_\top(\mathfrak{g})$ be th…

Let $W_{geq -1}$ be the Lie algebra of polynomial vector fields on the affine line with degrees at least $-1$, and let $U(-)$ denote the universal enveloping algebra. Non-noetheria…

Let $\mathfrak g$ be a finite-dimensional nilpotent Lie algebra. Write $Z(\mathfrak g)$ for the center of its universal enveloping algebra, and let the algebra of polynomial invari…

Generation conjecture. Let $T(B)$ be the algebra generated by $T^k(a_1,\ldots,a_k)$ where $a_1,a_2,\ldots,a_k\in B$ and $k\in\mathbb{Z}_{+}$. Then

Let $L$ be an $L_\infty$ algebra. Write $\mathcal C(L)$ for its Chevalley–Eilenberg coalgebra, $\mathcal U L$ for Lada and Markl's universal enveloping $A_\infty$ algebra, and…

Groupwise positive-characteristic isomorphism conjectures. The following three statements should hold:

Positive-characteristic isomorphism conjecture. If $U(L)\cong U(H)$, then $L\cong H$ also holds over fields of positive characteristic.

Let $\sigma$ be an involution of ${\mathfrak g}$, let $H_1,\ldots,H_l$ be the chosen basic invariants with degrees $d_1,\ldots,d_l$, and let $\varpi$ be the symmetrisation map. Let…

Let $\sigma$ be an involution of ${\mathfrak g}$, let $H_1,\ldots,H_l$ be the chosen basic invariants with degrees $d_1,\ldots,d_l$, and write $(H_j)_{(i,d_j-i)}$ for their bihomog…