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Questions tagged [loop-groups]

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Groups of loops in a topological groups, such as the group of based loops.

40 questions
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1 answer
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I am studying elliptic curves and came across an interesting pattern. For the elliptic curve: $$ y^2 = x^3 - 219x + 1654. $$ In this elliptic curve there are only $8$ solution couples $(x,y)$, the ...
MD.meraj Khan's user avatar
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2 votes
0 answers
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Let $R$ be a local reduced ring, $G$ split semisimple simply connected over $R$ and $g\in G(R[t,t^{-1}])$. Assume that there exists a dominant cocharacter $\lambda\in X_{*}(T)^{+}$ such that for ...
prochet's user avatar
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8 votes
1 answer
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Is there a version of the Kan loop group that is based on cubic rather than simplicial objects?
Boris Tsygan's user avatar
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5 votes
0 answers
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Consider a rational map $A : S^1 \to U(n)$, i.e. a matrix of rational functions such that evaluation at any $z \in S^1 \subset \mathbf C$ is unitary. These objects show up in digital signal processing ...
amcerbu's user avatar
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Let $G$ be a topological group. For any pointed topological space $X$, define $[X,G]$ to be the group whose underlying topological space is the space of pointed continuous maps from $X$ to $G$, with ...
Dominic Else's user avatar
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1 vote
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Let $(A,\mathfrak{m},\kappa)$ denote a commutative local Artinian ring. Somewhat by accident, I've stumbled across the following interesting decomposition: $$ A(\!(t)\!)^\times = t^\mathbb{Z} \cdot (1 ...
M.G.'s user avatar
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3 votes
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Let $H = L^2(U(1),\mathbb{C})$. The "basic" irreducible projective level 1 representation $\mathcal{H}$ of the loop group $LU(1)$ has underlying Hilbert space isomorphic to $\smash{\hat{\...
lw h's user avatar
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2 votes
1 answer
296 views

Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over $\mathbb{C}$, we have the notion of affinization of $\mathfrak{g}$, which is the central extension of the corresponding loop algebra. ...
Estwald's user avatar
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3 votes
1 answer
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Let us consider a Lie group $G$ with Lie algebra $\mathfrak{g}$ and let $L\mathfrak{g} = C^\infty(S^1, \mathfrak{g})$ the Lie algebra of the loop group $LG$. My question is about continuous Lie ...
Matthias Ludewig's user avatar
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7 votes
2 answers
697 views

In Pressley and Segal's book Loop Groups, they define a "basic inner product" $\langle-,-\rangle$ on a simple Lie algebra to be (minus) the Killing form scaled so that $\langle h_\alpha,h_\...
David Roberts's user avatar
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8 votes
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I have a question about Section 8.6 of Pressley-Segal's Loop groups book. Let $G$ be a compact, connected Lie group. Proposition 8.6.6 concerns the comparison of homotopy type between its polynomial ...
onefishtwofish's user avatar
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Let $G$ be a connected, compact Lie group, $T$ a maximal torus. Let $LG$ be the group of smooth (or polynomial) loops and $X=LG/T$ the affine flag variety ($T$ acts say by right multiplication). In ...
onefishtwofish's user avatar
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6 votes
1 answer
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Let $G$ be a complex reductive group. Let $LG$ and $L^+ G$ denote the formal loop spaces given by maps from the punctured formal disk and the formal disk, respectively, to $G$. The quotient $LG/L^+ G$ ...
G. Gallego's user avatar
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281 views

Let $G=\operatorname{SL}(2)$ and $V_n$ be the n+1 dimensional irreducible representation of $G$. This gives a representation for $G(O)=\operatorname{Map}(\operatorname{Spec} k[[t]], G)$ and hence an ($...
Xu Kai's user avatar
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8 votes
0 answers
201 views

Let $G$ be a compact simply-connected Lie group. Then one can look at the homology $H_*(\Omega G;\mathbb{Z})$ of the based-loop space $\Omega G$ in at least two different ways: (1) Via Bott-Samelson'...
ChiHong Chow's user avatar
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