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K-invariants in the algebra U(g) $\otimes$ C(p) for the group SU(2,1)
Representation Theory · Submitted 27 Dec 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We are going to prove that the algebra of K-invariants in $U(\mathfrak{g}) \otimes C(\mathfrak{p})$ is generated by five explicitly given elements.
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The abstract, sentence by sentence
Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be the Cartan decomposition of the complexified Lie algebra $\mathfrak{g}=\mathfrak{sl}(3,\mathbb{C})$ of the group $G=SU(2,1)$. Let $K=S(U(2) \times U(1))$; so $K$ is a maximal compact subgroup of $G$. Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$, and let $C(\mathfrak{p})$ be the Clifford algebra with respect to the trace form $B(X,Y)=\text{tr}(XY)$ on $\mathfrak{p}$. We are going to prove that the algebra of K-invariants in $U(\mathfrak{g}) \otimes C(\mathfrak{p})$ is generated by five explicitly given elements. This is useful for studying algebraic Dirac induction for $(\mathfrak{g},K)$-modules. Along the way we will also recover the (well known) structure of the algebra $U(\mathfrak{g})^K$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be the Cartan decomposition of the complexified Lie algebra $\mathfrak{g}=\mathfrak{sl}(3,\mathbb{C})$ of the group $G=SU(2,1)$. | background 1.00 | 0.14 | 0.02 | asserted 0.99 |
| s1 | Let $K=S(U(2) \times U(1))$; so $K$ is a maximal compact subgroup of $G$. | background 1.00 | 0.12 | 0.01 | asserted 1.00 |
| s2 | Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$, and let $C(\mathfrak{p})$ be the Clifford algebra with respect to the trace form $B(X,Y)=\text{tr}(XY)$ on $\mathfrak{p}$. | background 0.99 | 0.11 | 0.02 | asserted 0.89 |
| s3 | We are going to prove that the algebra of K-invariants in $U(\mathfrak{g}) \otimes C(\mathfrak{p})$ is generated by five explicitly given elements. | result 0.92 | 0.92 | 0.96 | asserted 1.00 |
| s4 | This is useful for studying algebraic Dirac induction for $(\mathfrak{g},K)$-modules. | interpretation 0.97 | 0.22 | 0.03 | asserted 0.88 |
| s5 | Along the way we will also recover the (well known) structure of the algebra $U(\mathfrak{g})^K$. | result 0.84 | 0.78 | 0.04 | asserted 0.96 |
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Original abstract
Let $\mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ be the Cartan decomposition of the complexified Lie algebra $\mathfrak{g}=\mathfrak{sl}(3,\mathbb{C})$ of the group $G=SU(2,1)$. Let $K=S(U(2) \times U(1))$; so $K$ is a maximal compact subgroup of $G$. Let $U(\mathfrak{g})$ be the universal enveloping algebra of $\mathfrak{g}$, and let $C(\mathfrak{p})$ be the Clifford algebra with respect to the trace form $B(X,Y)=\text{tr}(XY)$ on $\mathfrak{p}$. We are going to prove that the algebra of K-invariants in $U(\mathfrak{g}) \otimes C(\mathfrak{p})$ is generated by five explicitly given elements. This is useful for studying algebraic Dirac induction for $(\mathfrak{g},K)$-modules. Along the way we will also recover the (well known) structure of the algebra $U(\mathfrak{g})^K$.
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