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Register a new userRepresentations of Quantum Minimal Surface Algebrasvia Kac-Moody-theory
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Added by
Ralf K\\\"ohl (n\\'e Gramlich)
Publication date
2021
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and research's language is
English
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We consider epimorphisms from quantum minimal surface algebras onto involutroy subalgebras of split real simply-laced Kac-Moody algebras and provide examples of affine and finite type. We also provide epimorphisms onto such Kac-Moody algebras themselves, where reality of the construction is important. The results extend to the complex situation.
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We consider the subalgebras of split real, non-twisted affine Kac-Moody Lie algebras that are fixed by the Chevalley involution. These infinite-dimensional Lie algebras are not of Kac-Moody type and admit finite-dimensional unfaithful representations. We exhibit a formulation of these algebras in terms of $mathbb{N}$-graded Lie algebras that allows the construction of a large class of representations using the techniques of induced representations. We study how these representations relate to previously established spinor representations as they arise in the theory of supergravity.
We study equivariant localization of intersection cohomology complexes on Schubert varieties in Kashiwaras flag manifold. Using moment graph theory, we establish a link to the representation theory of Kac-Moody algebras and give a new proof of the Kazhdan-Lusztig conjecture for blocks containing an antidominant element.
We give a topological interpretation of the highest weight representations of Kac-Moody groups. Given the unitary form G of a Kac-Moody group (over C), we define a version of equivariant K-theory, K_G on the category of proper G-CW complexes. We then study Kac-Moody groups of compact type in detail (see Section 2 for definitions). In particular, we show that the Grothendieck group of integrable hightest weight representations of a Kac-Moody group G of compact type, maps isomorphically onto K_G^*(EG), where $EG$ is the classifying space of proper G-actions. For the affine case, this agrees very well with recent results of Freed-Hopkins-Teleman. We also explicitly compute K_G^*(EG) for Kac-Moody groups of extended compact type, which includes the Kac-Moody group E_{10}.
We study a correction factor for Kac-Moody root systems which arises in the theory of $p$-adic Kac-Moody groups. In affine type, this factor is known, and its explicit computation is the content of the Macdonald constant term conjecture. The data of the correction factor can be encoded as a collection of polynomials $m_lambda in mathbb{Z}[t]$ indexed by positive imaginary roots $lambda$. At $t=0$ these polynomials evaluate to the root multiplicities, so we consider $m_lambda$ to be a $t$-deformation of $mathrm{mult} (lambda)$. We generalize the Peterson algorithm and the Berman-Moody formula for root multiplicities to compute $m_lambda$. As a consequence we deduce fundamental properties of $m_lambda$.
We construct a (bi)cyclic sieving phenomenon on the union of dominant maximal weights for level $ell$ highest weight modules over an affine Kac-Moody algebra with exactly one highest weight being taken for each equivalence class, in a way not depending on types, ranks and levels. In order to do that, we introduce $textbf{textit{S}}$-evaluation on the set of dominant maximal weights for each highest modules, and generalize Sagans action by considering the datum on each affine Kac-Moody algebra. As consequences, we obtain closed and recursive formulae for cardinality of the number of dominant maximal weights for every highest weight module and observe level-rank duality on the cardinalities.
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