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Francesco Polizzi
Francesco Polizzi
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Let $g$ be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let $\{u_j\}$ and $\{u^j\}$ be bases of $g$ dual with respect to a nondegenerate invariant bilinear form $(\cdot|\cdot)$ on $g$, and consistent with the triangular decomposition of $g$. Let $L(\Lambda)$ be an integrable representation of $g$ with highest weight $\Lambda$, and let $v_\Lambda$ be its highest weight vector. Denote the Casimir $\Omega=\sum_j u_j\otimes u^j$.

I want to know why $\Omega(v_\Lambda\otimes v_\Lambda)=(\Lambda|\Lambda)v_\Lambda\otimes v_\Lambda$? Could someone give some explanation or some references?

Question. Why $\Omega(v_\Lambda\otimes v_\Lambda)=(\Lambda|\Lambda)v_\Lambda\otimes v_\Lambda$? Could someone give some explanation or some references?

Let $g$ be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let $\{u_j\}$ and $\{u^j\}$ be bases of $g$ dual with respect to a nondegenerate invariant bilinear form $(\cdot|\cdot)$ on $g$, and consistent with the triangular decomposition of $g$. Let $L(\Lambda)$ be an integrable representation of $g$ with highest weight $\Lambda$, and let $v_\Lambda$ be its highest weight vector. Denote the Casimir $\Omega=\sum_j u_j\otimes u^j$.

I want to know why $\Omega(v_\Lambda\otimes v_\Lambda)=(\Lambda|\Lambda)v_\Lambda\otimes v_\Lambda$? Could someone give some explanation or some references?

Let $g$ be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let $\{u_j\}$ and $\{u^j\}$ be bases of $g$ dual with respect to a nondegenerate invariant bilinear form $(\cdot|\cdot)$ on $g$, and consistent with the triangular decomposition of $g$. Let $L(\Lambda)$ be an integrable representation of $g$ with highest weight $\Lambda$, and let $v_\Lambda$ be its highest weight vector. Denote the Casimir $\Omega=\sum_j u_j\otimes u^j$.

Question. Why $\Omega(v_\Lambda\otimes v_\Lambda)=(\Lambda|\Lambda)v_\Lambda\otimes v_\Lambda$? Could someone give some explanation or some references?

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gmvh
gmvh
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Let g$g$ be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let $\{u_j\}$ and $\{u^j\}$ be bases of g$g$ dual with respect to a nondegenerate invariant bilinear form $(\cdot|\cdot)$ on g$g$, and consistent with the triangular decomposition of g$g$. Let $L(\Lambda)$ be an integrable representation of g$g$ with highest weight $\Lambda$, and let $v_\Lambda$ be its highest weight vector. Denote the Casimir $\Omega=\sum_j u_j\otimes u^j$.

I want to know why $\Omega(v_\Lambda\otimes v_\Lambda)=(\Lambda|\Lambda)v_\Lambda\otimes v_\Lambda$? Could someone give some explanation or some references?

Let g be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let $\{u_j\}$ and $\{u^j\}$ be bases of g dual with respect to a nondegenerate invariant bilinear form $(\cdot|\cdot)$ on g, and consistent with the triangular decomposition of g. Let $L(\Lambda)$ be an integrable representation of g with highest weight $\Lambda$, and let $v_\Lambda$ be its highest weight vector. Denote the Casimir $\Omega=\sum_j u_j\otimes u^j$.

I want to know why $\Omega(v_\Lambda\otimes v_\Lambda)=(\Lambda|\Lambda)v_\Lambda\otimes v_\Lambda$? Could someone give some explanation or some references?

Let $g$ be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let $\{u_j\}$ and $\{u^j\}$ be bases of $g$ dual with respect to a nondegenerate invariant bilinear form $(\cdot|\cdot)$ on $g$, and consistent with the triangular decomposition of $g$. Let $L(\Lambda)$ be an integrable representation of $g$ with highest weight $\Lambda$, and let $v_\Lambda$ be its highest weight vector. Denote the Casimir $\Omega=\sum_j u_j\otimes u^j$.

I want to know why $\Omega(v_\Lambda\otimes v_\Lambda)=(\Lambda|\Lambda)v_\Lambda\otimes v_\Lambda$? Could someone give some explanation or some references?

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tudong
tudong
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Action of the Casimir on highest weight modules for Kac-Moody algebra

Let g be a Kac-Moody algebra with a symmetrizable Cartan matrix, and let $\{u_j\}$ and $\{u^j\}$ be bases of g dual with respect to a nondegenerate invariant bilinear form $(\cdot|\cdot)$ on g, and consistent with the triangular decomposition of g. Let $L(\Lambda)$ be an integrable representation of g with highest weight $\Lambda$, and let $v_\Lambda$ be its highest weight vector. Denote the Casimir $\Omega=\sum_j u_j\otimes u^j$.

I want to know why $\Omega(v_\Lambda\otimes v_\Lambda)=(\Lambda|\Lambda)v_\Lambda\otimes v_\Lambda$? Could someone give some explanation or some references?