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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

   

 

Gaudin Hamiltonians generate the Bethe algebra of a tensor power of the vector representation of $ \mathfrak{gl}_N$


Authors: E. Mukhin, V. Tarasov and A. Varchenko
Original publication: Algebra i Analiz, tom 22 (2010), nomer 3.
Journal: St. Petersburg Math. J. 22 (2011), 463-472
MSC (2010): Primary 82C23, 81Q12
Published electronically: March 18, 2011
MathSciNet review: 2729945
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Abstract: It is shown that the Gaudin Hamiltonians $ H_1$, ..., $ H_n$ generate the Bethe algebra of the $ n$-fold tensor power of the vector representation of $ \mathfrak{gl}_N$. Surprisingly, the formula for the generators of the Bethe algebra in terms of the Gaudin Hamiltonians does not depend on $ N$. Moreover, this formula coincides with Wilson's formula for the stationary Baker-Akhiezer function on the adelic Grassmannian.


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Additional Information

E. Mukhin
Affiliation: Department of Mathematical Sciences, Indiana University — Purdue University Indianapolis, 402 North Blackford St., Indianapolis, Indiana 46202-3216
Email: mukhin@math.iupui.edu

V. Tarasov
Affiliation: Department of Mathematical Sciences, Indiana University — Purdue University Indianapolis, 402 North Blackford St., Indianapolis, Indiana 46202-3216
Address at time of publication: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: vt@math.iupui.edu, vt@pdmi.ras.ru

A. Varchenko
Affiliation: Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, North Carolina 27599-325
Email: anv@email.unc.edu

DOI: http://dx.doi.org/10.1090/S1061-0022-2011-01152-5
Keywords: Gaudin model, Bethe algebra, Calogero–Moser space
Received by editor(s): November 15, 2009
Published electronically: March 18, 2011
Additional Notes: E. Mukhin was supported in part by NSF grant DMS-0900984
V. Tarasov was supported by NSF grant DMS-0901616
A. Varchenko was supported by NSF grant DMS-0555327
Dedicated: To Ludwig Dmitrievich Faddeev on his 75th birthday
Article copyright: © Copyright 2011 American Mathematical Society