Coordinate superalgebra
$begingroup$
Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $lambda$ be a dominant integral weight. It follows from the Borel-Weil-Bott theorem that the homogeneous coordinate ring $A_lambda(G/P)$ is a sum of highest weight representations. Namely,
begin{align*}
A_lambda(G/P) = oplus_{n in mathbb{N}} V(nlambda).
end{align*}
Lie superalgebra: Do we have an analogue result of this in the super case? More precisely, let $G$ be a simply connected analytic supergroup with $mathfrak{g} = Lie(G)$ (a basic classical Lie superalgebra) and a Borel subsupergroup $B$. Let $P$ be a subsupergroup containing $B$. I know that there is an analogue of Borel-Weil-Bott theorem for supercase, but I wonder if there is a result for the coordinate ring as above. I guess we might need to assume $lambda$ is a typical weight.
algebraic-geometry representation-theory lie-groups lie-superalgebras
asked Jan 1 at 3:18
NongAmNongAm
13518
$endgroup$
add a comment |
$begingroup$
Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $lambda$ be a dominant integral weight. It follows from the Borel-Weil-Bott theorem that the homogeneous coordinate ring $A_lambda(G/P)$ is a sum of highest weight representations. Namely,
begin{align*}
A_lambda(G/P) = oplus_{n in mathbb{N}} V(nlambda).
end{align*}
Lie superalgebra: Do we have an analogue result of this in the super case? More precisely, let $G$ be a simply connected analytic supergroup with $mathfrak{g} = Lie(G)$ (a basic classical Lie superalgebra) and a Borel subsupergroup $B$. Let $P$ be a subsupergroup containing $B$. I know that there is an analogue of Borel-Weil-Bott theorem for supercase, but I wonder if there is a result for the coordinate ring as above. I guess we might need to assume $lambda$ is a typical weight.
algebraic-geometry representation-theory lie-groups lie-superalgebras
asked Jan 1 at 3:18
NongAmNongAm
13518
$endgroup$
add a comment |
$begingroup$
Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $lambda$ be a dominant integral weight. It follows from the Borel-Weil-Bott theorem that the homogeneous coordinate ring $A_lambda(G/P)$ is a sum of highest weight representations. Namely,
begin{align*}
A_lambda(G/P) = oplus_{n in mathbb{N}} V(nlambda).
end{align*}
Lie superalgebra: Do we have an analogue result of this in the super case? More precisely, let $G$ be a simply connected analytic supergroup with $mathfrak{g} = Lie(G)$ (a basic classical Lie superalgebra) and a Borel subsupergroup $B$. Let $P$ be a subsupergroup containing $B$. I know that there is an analogue of Borel-Weil-Bott theorem for supercase, but I wonder if there is a result for the coordinate ring as above. I guess we might need to assume $lambda$ is a typical weight.
algebraic-geometry representation-theory lie-groups lie-superalgebras
asked Jan 1 at 3:18
NongAmNongAm
13518
$endgroup$
Lie algebra: Let $G$ be a semisimple, simply connected linear algebraic group with a fixed Borel subgroup $B$. Let $P$ be a parabolic subgroup containing $B$. Let $lambda$ be a dominant integral weight. It follows from the Borel-Weil-Bott theorem that the homogeneous coordinate ring $A_lambda(G/P)$ is a sum of highest weight representations. Namely,
begin{align*}
A_lambda(G/P) = oplus_{n in mathbb{N}} V(nlambda).
end{align*}
Lie superalgebra: Do we have an analogue result of this in the super case? More precisely, let $G$ be a simply connected analytic supergroup with $mathfrak{g} = Lie(G)$ (a basic classical Lie superalgebra) and a Borel subsupergroup $B$. Let $P$ be a subsupergroup containing $B$. I know that there is an analogue of Borel-Weil-Bott theorem for supercase, but I wonder if there is a result for the coordinate ring as above. I guess we might need to assume $lambda$ is a typical weight.
algebraic-geometry representation-theory lie-groups lie-superalgebras
algebraic-geometry representation-theory lie-groups lie-superalgebras
asked Jan 1 at 3:18
NongAmNongAm
13518
asked Jan 1 at 3:18
NongAmNongAm
13518
asked Jan 1 at 3:18
NongAmNongAm
13518
asked Jan 1 at 3:18
NongAmNongAm
13518
asked Jan 1 at 3:18
NongAmNongAm
13518
13518
add a comment |
add a comment |
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