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2022 | OriginalPaper | Chapter

Loop Grassmannians of Quivers and Affine Quantum Groups

Authors : Ivan Mirković, Yaping Yang, Gufang Zhao

Published in: Representation Theory and Algebraic Geometry

Publisher: Springer International Publishing

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Abstract

We construct for each choice of a quiver Q, a cohomology theory A, and a poset P a “loop Grassmannian” \({\mathcal G}^P(Q,A)\). This generalizes loop Grassmannians of semisimple groups and the loop Grassmannians of based quadratic forms. The addition of a “dilation” torus \({\mathcal D}{ \subseteq } \mathbb {G}_m^2\) gives a quantization \({\mathcal G}^P_{\mathcal D}(Q,A)\). This construction is motivated by the program of introducing an inner cohomology theory in algebraic geometry adequate for the Geometric Langlands program (Mirković, Some extensions of the notion of loop Grassmannians. Rad Hrvat. Akad. Znan. Umjet. Mat. Znan., the Mardešić issue. No. 532, 53–74, 2017) and on the construction of affine quantum groups from generalized cohomology theories (Yang and Zhao, Quiver varieties and elliptic quantum groups, preprint. arxiv1708.01418).

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Appendix
Available only for authorised users
Footnotes
1
When A is de Rham cohomology, then \(\frak {A}_X\) can be viewed as an affinization of the de Rham space XdR of X.
 
2
Here, we do not pay attention to a choice of P, but when P = (1 < ⋅⋅⋅ < m), the fibers at colored points are \({\mathbb P}^m\) and \({\mathcal G}^{1<{\cdot \cdot \cdot }<m}_{{\mathcal D}}(Q, A)\) should “correspond to level m” in the sense that the sections of the standard line bundle \({\mathcal O}(1)\) on this object should be the same as the sections of \({\mathcal O}(m)\) in the case when \(P= \operatorname {\mathrm {{pt}}}\).
 
3
These embedding equations should be integrable hierarchies of differential equations indexed by Q, P and A since this is true in the classical case of \({\mathcal G}(G)\).
 
4
While the grading of a cohomology theory is fundamental, we will disregard it in this paper.
 
5
The terminology of “algebraic cohomology” is also used by Panin-Smirnov for a refinement of the formalism in which the theory is bigraded (to adequately encode the example of motivic cohomology). We will not be concerned with this version.
 
6
What is called Borel-Moore homology here is not quite what this means in classical topology; however, this is just a choice of terminology since the A-setting does contain the precise analogue of Borel-Moore homology. For instance, for smooth X, the more appropriate version would be BMA(X) = ΘA(TX)−1 in terms of the Thom bundle which is defined next.
 
7
By compactly supported cohomology of X, we mean the cohomology of a compactification \( \overline X\) trivialized on the formal neighborhood of the boundary of X in \( \overline X\).
 
8
In general, the derived version of homology \({\mathbb H}_*(X)\) should be the free abelian commutative group object in derived stacks freely generated by X.
 
9
So the connected components of \( \overline {S_0}^T\) are \(( \overline {S_0}\cap \overline {S^-_{-{\alpha }}})\cap { \underline {{\mathcal G}} }(T)_{-\beta }\), for 0 ≤ β ≤ α, identified with the moduli \(\mathcal {H}^{\beta }_{{\alpha }[0]}\) of length β subschemes of α[0].
 
10
One could try replacing a curve by a more general scheme and \({\mathcal H}\) by other notions of powers of a scheme like the Cartesian powers \({\mathcal C}^n_C=C^n\).
 
11
Flatness fails for the poset P = (0 < a, b).
 
12
We denote by \({\mathcal L}_{F/S}\) the direct image of \({\mathcal L}|{ }_F\) to S, etc.
 
13
The modification appears because we use the Hilbert scheme \({ {\mathcal H}_C }\) rather than over powers of curves \({ {\mathcal C}_C }\) (see the Remark 3.3.2(1)).
 
14
While the quoted lemma is stated at a single point of C, we actually need a version of the lemma for the family \(Z^{\alpha }(G)\to {\mathcal H}^{\alpha }_{{C\times I}}\). This is easy using the moduli description of semi-infinite stratifications from Sect. 3.4.5.
 
15
Here, \(\mathbb {G}\) is defined over the ring of constants \(A( \operatorname {\mathrm {{pt}}})\) of the cohomology theory.
 
16
While [YZ17] deals with the case of elliptic cohomology, some of its ideas appear in an earlier paper [YZ14] which was only concerned with affine groups \(\mathbb {G}\). This allowed for a trivialization of Thom line bundles which accounts for a different presentation of functoriality of cohomology in that paper.
 
17
The fibered product has to be derived for the relevant base change to hold unless dp, dq are transversal.
 
18
Placing the numerator of the factor \( \operatorname {\mathrm {fac}}_1\) in Equation [YZ14, (2)] on the denominator to get the corresponding factor in the shuffle formula for \( \underline {U}^+_{{\mathcal D}}(Q,A)\). The homomorphism from (1) is on the level of shuffle algebras the multiplication by the Euler class of \((\frak {u}\oplus \frak {g}/\frak {p})\otimes \omega \).
 
19
Notice that this is stronger than the standard definition of locality which only requires such isomorphism over the regular part of the configuration space where \({\mathcal L}\) happens to trivialize by Sect. 5.2.
 
20
One formal way to say it is that \(U^+_{\mathcal D}(Q,A)_{\alpha }\) is the smallest subsheaf on \(\mathbb {G}^{ ({\alpha })}\) such that its pullback to each refinement \({\mathcal C}_{\gamma }\) contains Lγ.
 
21
For \(C=\mathbb {A}^1\), we have a canonical trivialization of \({\mathcal G}(G)\to { \mathcal {H}_C }\) over \(C=\mathcal {H}^1_C\), as \({ \underline {{\mathcal G}} }(G)\). Now, consider the pullback \({\mathcal G}_{C^2}(G){ { \overset { \mbox{def}}= } } C^2\times _{C ^{ [2] } }{\mathcal G}_{\mathcal {H}_{C\times I}}(G)\) of the restriction of \({\mathcal G}_{\mathcal {H}_{C\times I}}(G)\) to \(\mathcal {H}^2=C^{2}\). The locality identifies it over C2 − ΔC with the constant bundle \({ \underline {{\mathcal G}} }(G)^2\). By fusion of \(u,v\in { \underline {{\mathcal G}} }(G)\), we mean the limit (when it exists) over the diagonal of the constant section (u, v) which is defined off the diagonal.
 
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Metadata
Title
Loop Grassmannians of Quivers and Affine Quantum Groups
Authors
Ivan Mirković
Yaping Yang
Gufang Zhao
Copyright Year
2022
DOI
https://doi.org/10.1007/978-3-030-82007-7_8

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