17 problems

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…

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…

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…

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 $n$ 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…

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:

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…

Let $F\colon \mathbf{TriAss} \to \mathbf{TriLie}$ be the functor obtained from the operations in Lemma, sending a triassociative algebra to its associated Lie trialgebra. For a Lie…

Let $L$ be a mock-Lie algebra, and let $U(L)$ denote its universal enveloping algebra. The Poincaré–Birkhoff–Witt theorem holds for $L$ when the ordered monomials in a basis of $L$…

Let ${\Bbbk}$ be a field of characteristic $0$, let $L$ be a Lie algebra over ${\Bbbk}$, and let $U(L)$ denote its universal enveloping algebra. Finite-dimensionality conjecture. A…

Let $\mathfrak{g}$ be a Lie algebra with a fixed triangular decomposition … and set $\mathfrak{n}=\mathfrak{n}_+$. Let $\lambda\in(\mathfrak{n}/[\mathfrak{n},\mathfrak{n}])^*$, let…