Irreducible highest weight representations as a graded algebra
Let $L$ be a semisimple Lie algebra and let $V(lambda)$ be a finite dimensional irreducible $L$-module with the highest weight $lambda$. How can we view the sum
begin{align*}
oplus_{ninmathbb{N}}V(nlambda)
end{align*}
as a graded algebra with $n$th homogeneous component $V(nlambda)$? More precisely, if $lambda$ and $mu$ are dominant weights, how do we define multiplication on $V(lambda)V(mu)$ and why $V(lambda)V(mu) subset V(lambda + mu)$?
representation-theory lie-algebras graded-rings graded-algebras
asked Dec 26 at 5:26
NongAm
11818
add a comment |
Let $L$ be a semisimple Lie algebra and let $V(lambda)$ be a finite dimensional irreducible $L$-module with the highest weight $lambda$. How can we view the sum
begin{align*}
oplus_{ninmathbb{N}}V(nlambda)
end{align*}
as a graded algebra with $n$th homogeneous component $V(nlambda)$? More precisely, if $lambda$ and $mu$ are dominant weights, how do we define multiplication on $V(lambda)V(mu)$ and why $V(lambda)V(mu) subset V(lambda + mu)$?
representation-theory lie-algebras graded-rings graded-algebras
asked Dec 26 at 5:26
NongAm
11818
add a comment |
Let $L$ be a semisimple Lie algebra and let $V(lambda)$ be a finite dimensional irreducible $L$-module with the highest weight $lambda$. How can we view the sum
begin{align*}
oplus_{ninmathbb{N}}V(nlambda)
end{align*}
as a graded algebra with $n$th homogeneous component $V(nlambda)$? More precisely, if $lambda$ and $mu$ are dominant weights, how do we define multiplication on $V(lambda)V(mu)$ and why $V(lambda)V(mu) subset V(lambda + mu)$?
representation-theory lie-algebras graded-rings graded-algebras
asked Dec 26 at 5:26
NongAm
11818
Let $L$ be a semisimple Lie algebra and let $V(lambda)$ be a finite dimensional irreducible $L$-module with the highest weight $lambda$. How can we view the sum
begin{align*}
oplus_{ninmathbb{N}}V(nlambda)
end{align*}
as a graded algebra with $n$th homogeneous component $V(nlambda)$? More precisely, if $lambda$ and $mu$ are dominant weights, how do we define multiplication on $V(lambda)V(mu)$ and why $V(lambda)V(mu) subset V(lambda + mu)$?
representation-theory lie-algebras graded-rings graded-algebras
representation-theory lie-algebras graded-rings graded-algebras
asked Dec 26 at 5:26
NongAm
11818
asked Dec 26 at 5:26
NongAm
11818
asked Dec 26 at 5:26
NongAm
11818
asked Dec 26 at 5:26
NongAm
11818
asked Dec 26 at 5:26
NongAm
11818
11818
add a comment |
add a comment |
1 Answer
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If $v_{lambda}in V(lambda)$ and $v_{mu}in V(mu)$ are highest weight vectors, then $v_{lambda}otimes v_{mu}$ is s highest weight vector in $V(lambda)otimes V(mu)$ of weight $lambda+mu$, which spans the weight space for that weight. Hence this spans an irreducible representation isomorphic to $V(lambda+mu)$ and projecting along an invariant complement, one a unique (up to scale) homomorphism $V(lambda)otimes V(mu)to V(lambda+mu)$. If you realize each $V(nlambda)$ as a subrepresentation of $S^nV(lambda)$ in this way, the choice of a $v_lambdain V(lambda)$ gives you a highest weight vector in each $V(nlambda)$ and thus a scale for all the homomorphisms in question. (Viewed in this way, one is actually constructing a subalgebra of the symmetric algebra $S^*V(lambda)$ of $V(lambda)$.
answered Dec 26 at 9:13
Andreas Cap
11k923
add a comment |
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1 Answer
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If $v_{lambda}in V(lambda)$ and $v_{mu}in V(mu)$ are highest weight vectors, then $v_{lambda}otimes v_{mu}$ is s highest weight vector in $V(lambda)otimes V(mu)$ of weight $lambda+mu$, which spans the weight space for that weight. Hence this spans an irreducible representation isomorphic to $V(lambda+mu)$ and projecting along an invariant complement, one a unique (up to scale) homomorphism $V(lambda)otimes V(mu)to V(lambda+mu)$. If you realize each $V(nlambda)$ as a subrepresentation of $S^nV(lambda)$ in this way, the choice of a $v_lambdain V(lambda)$ gives you a highest weight vector in each $V(nlambda)$ and thus a scale for all the homomorphisms in question. (Viewed in this way, one is actually constructing a subalgebra of the symmetric algebra $S^*V(lambda)$ of $V(lambda)$.
answered Dec 26 at 9:13
Andreas Cap
11k923
add a comment |
If $v_{lambda}in V(lambda)$ and $v_{mu}in V(mu)$ are highest weight vectors, then $v_{lambda}otimes v_{mu}$ is s highest weight vector in $V(lambda)otimes V(mu)$ of weight $lambda+mu$, which spans the weight space for that weight. Hence this spans an irreducible representation isomorphic to $V(lambda+mu)$ and projecting along an invariant complement, one a unique (up to scale) homomorphism $V(lambda)otimes V(mu)to V(lambda+mu)$. If you realize each $V(nlambda)$ as a subrepresentation of $S^nV(lambda)$ in this way, the choice of a $v_lambdain V(lambda)$ gives you a highest weight vector in each $V(nlambda)$ and thus a scale for all the homomorphisms in question. (Viewed in this way, one is actually constructing a subalgebra of the symmetric algebra $S^*V(lambda)$ of $V(lambda)$.
answered Dec 26 at 9:13
Andreas Cap
11k923
add a comment |
If $v_{lambda}in V(lambda)$ and $v_{mu}in V(mu)$ are highest weight vectors, then $v_{lambda}otimes v_{mu}$ is s highest weight vector in $V(lambda)otimes V(mu)$ of weight $lambda+mu$, which spans the weight space for that weight. Hence this spans an irreducible representation isomorphic to $V(lambda+mu)$ and projecting along an invariant complement, one a unique (up to scale) homomorphism $V(lambda)otimes V(mu)to V(lambda+mu)$. If you realize each $V(nlambda)$ as a subrepresentation of $S^nV(lambda)$ in this way, the choice of a $v_lambdain V(lambda)$ gives you a highest weight vector in each $V(nlambda)$ and thus a scale for all the homomorphisms in question. (Viewed in this way, one is actually constructing a subalgebra of the symmetric algebra $S^*V(lambda)$ of $V(lambda)$.
answered Dec 26 at 9:13
Andreas Cap
11k923
If $v_{lambda}in V(lambda)$ and $v_{mu}in V(mu)$ are highest weight vectors, then $v_{lambda}otimes v_{mu}$ is s highest weight vector in $V(lambda)otimes V(mu)$ of weight $lambda+mu$, which spans the weight space for that weight. Hence this spans an irreducible representation isomorphic to $V(lambda+mu)$ and projecting along an invariant complement, one a unique (up to scale) homomorphism $V(lambda)otimes V(mu)to V(lambda+mu)$. If you realize each $V(nlambda)$ as a subrepresentation of $S^nV(lambda)$ in this way, the choice of a $v_lambdain V(lambda)$ gives you a highest weight vector in each $V(nlambda)$ and thus a scale for all the homomorphisms in question. (Viewed in this way, one is actually constructing a subalgebra of the symmetric algebra $S^*V(lambda)$ of $V(lambda)$.
answered Dec 26 at 9:13
Andreas Cap
11k923
answered Dec 26 at 9:13
Andreas Cap
11k923
answered Dec 26 at 9:13
Andreas Cap
11k923
answered Dec 26 at 9:13
Andreas Cap
11k923
11k923
add a comment |
add a comment |
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