Dominant cyclic highest-weight presentation

Assume the Axiom of Choice. Let $\mathfrak{g}$ be a finite-dimensional complex semisimple Lie algebra with Cartan subalgebra $\mathfrak{h}$ and a chosen positive system with simple roots ${\alpha }_1,\dots ,{\alpha }_r$, let ${\mathfrak{n}}^{+}$ be the span of the positive root spaces (Positive and negative nilpotent subalgebras and the Borel), let $\lambda$ be dominant integral with $m_i=⟨\lambda ,{\alpha }_i^{\vee }⟩\in {\mathbb{Z}}_{\ge 0}$ (Integral, dominant, and strictly dominant weights), and for each $i$ choose $e_i\in {\mathfrak{g}}_{{\alpha }_i}$, $f_i\in {\mathfrak{g}}_{-{\alpha }_i}$ with $[e_i,f_i]=h_{{\alpha }_i}$ (The root sl_2 triple).

In the universal enveloping algebra $U(\mathfrak{g})$ (Universal enveloping algebra) write $\iota ={\iota }_{\mathfrak{g}}:\mathfrak{g}\to U(\mathfrak{g})$ for the canonical linear map and $1_U$ for the unit. No injectivity of $\iota$ is required here. Let $I_{\lambda }$ be the left ideal generated by the union of the three sets $\iota ({\mathfrak{n}}^{+}),\{\iota (H)-\lambda (H)1_U:H\in \mathfrak{h}\},\{\iota (f_i{)}^{m_i+1}:i=1,\dots ,r\}\mathrm{.}$ Define $M_{\mathrm{i}\mathrm{n}\mathrm{t}}(\lambda ):=U(\mathfrak{g})\mathrm{/}{I}_{\lambda },$ the quotient being taken as left $U(\mathfrak{g})$-modules, and write $v_{\lambda }:=1+I_{\lambda }\in M_{\mathrm{i}\mathrm{n}\mathrm{t}}(\lambda )$ for the class of the unit. Then $M_{\mathrm{i}\mathrm{n}\mathrm{t}}(\lambda )$ is called the dominant cyclic highest-weight module with simple-root integrability relations of highest weight $\lambda$, and $v_{\lambda }$ its canonical generator. The subscript ``int'' records these defining relations; it does not assert, at this stage, local finiteness of the whole module.

Well-definedness and the defining relations. For a subset $S$ of an associative algebra, its generated left ideal is $\{{\sum }_{j=1}^k{u}_j{s}_j:k\ge 0,u_j\in U(\mathfrak{g}),s_j\in S\}$; this is the smallest left ideal containing $S$. Thus $I_{\lambda }$ is well defined as a linear subspace of $U(\mathfrak{g})$, and the quotient carries a left $U(\mathfrak{g})$-module structure because $I_{\lambda }$ is a left ideal. The notation $x\cdot v$ for $x\in \mathfrak{g}$ means $\iota (x)v$; the tensor relations defining $U(\mathfrak{g})$ ensure $\iota ([x,y])=\iota (x)\iota (y)-\iota (y)\iota (x)$, so this is a Lie-algebra action. The class $v_{\lambda }$ therefore satisfies $x\cdot v_{\lambda }=0(x\in {\mathfrak{n}}^{+}),H\cdot v_{\lambda }=\lambda (H)v_{\lambda }(H\in \mathfrak{h}),f_i^{m_i+1}\cdot v_{\lambda }=0,$ because the corresponding elements of $U(\mathfrak{g})$ lie in $I_{\lambda }$. The module is generated by $v_{\lambda }$, since the class of $1$ generates $U(\mathfrak{g})\mathrm{/}{I}_{\lambda }$ under left multiplication.

The negative root spaces are one-dimensional by Root spaces of a complex semisimple Lie algebra are one-dimensional, so the nonzero choices $f_i$ differ only by scalars: if $f_i$ is replaced by $c{f}_i$ with $c\in {\mathbb{C}}^{\times }$ and $e_i$ correspondingly by $c^{-1}{e}_i$, then the generator $\iota (f_i{)}^{m_i+1}$ is replaced by the nonzero scalar multiple $c^{m_i+1}\iota (f_i{)}^{m_i+1}$, which generates the same left ideal. Hence $I_{\lambda }$, and with it $M_{\mathrm{i}\mathrm{n}\mathrm{t}}(\lambda )$, does not depend on these choices.

This names the cyclic presentation; nonvanishing, finite dimensionality and integrability of the resulting module require the subsequent highest-weight results. If a coefficient $m_i$ is zero, its defining power is the first power. If the index set of simple roots is empty, the third generating set is empty.