On the Harishchandra homomorphism for infinite-dimensional Lie algebras

Chari, Vyjayanthi; Ilangovan, Sheila

doi:10.1016/0021-8693(84)90185-6

Cited by 11 publications

(3 citation statements)

References 14 publications

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“…But in affine types the centre is much smaller. Namely, the centre of the (completed, as in §5.1 below) envelope of an affine Kac-Moody algebra is isomorphic to the graded polynomial algebra in only two generators, k and C (of degrees 0 and 2; the definition of C in the affine case is in (5.7) below) [CI84]. Thus, one should not expect to find meromorphic functions S k (z), indexed by the positive exponents k ∈ E, such that they commute with the diagonal action of g for each z ∈ X = C \ {z 1 , .…”

Section: Conjectures On Affine Gaudin Hamiltoniansmentioning

confidence: 99%

Affine Gaudin models and hypergeometric functions on affine opers

Lacroix

Young

2019

Advances in Mathematics

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We conjecture that quantum Gaudin models in affine types admit families of higher Hamiltonians, labelled by the (countably infinite set of) exponents, whose eigenvalues are given by functions on a space of meromorphic opers associated with the Langlands dual Lie algebra. This is in direct analogy with the situation in finite types. However, in stark contrast to finite types, we prove that in affine types such functions take the form of hypergeometric integrals, over cycles of a twisted homology defined by the levels of the modules at the marked points. That result prompts the further conjecture that the Hamiltonians themselves are naturally expressed as such integrals.We go on to describe the space of meromorphic affine opers on an arbitrary Riemann surface. We prove that it fibres over the space of meromorphic connections on the canonical line bundle Ω. Each fibre is isomorphic to the direct product of the space of sections of the square of Ω with the direct product, over the exponents j not equal to 1, of the twisted cohomology of the j th tensor power of Ω. 32 7. Twisted homology and the integral pairing 39 8. Discussion 41 Appendix A. Hyperplane arrangements and quadratic Hamiltonians 47 References 51

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Section: Conjectures On Affine Gaudin Hamiltoniansmentioning

confidence: 99%

Affine Gaudin models and hypergeometric functions on affine opers

Lacroix

Young

2019

Advances in Mathematics

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“…In [CI84, Section 2], the authors defined an algebra U(R, C ) which can be associated to any ring R and a full subcategory C of the category of R-modules. Of specific interest in [CI84] was the case where R = U(g) and C is taken to be the category O for the Kac-Moody algebra g (to this effect, see also [Kum86]). However, when one takes instead R = Y (g) and C to be the category O for Y (g), one arrives at an algebra which is closely related to Y (g) as a left Y (g)-module, but has a different multiplication.…”

Section: Coproduct and Completions Of Yangiansmentioning

confidence: 99%

Coproduct for Yangians of affine Kac–Moody algebras

Guay

Nakajima

Wendlandt

2018

Advances in Mathematics

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“…It remains to note that the elements Z x r−1 with r ≥ 1 generate the center of U(gl n [x]). The latter follows from the fact that the center of U(sl n [x]) is trivial [22]; see also [119,Proposition 2.12].…”

Section: 7mentioning

confidence: 99%