Abstract
We compute the dimensions of the components for the operad of two compatible brackets and for the bi-Hamiltonian operad. We also obtain character formulas for the representations of symmetric groups and SL 2 in these spaces.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Bershtein, Private communication, 2004.
M. Bershtein, V. Dotsenko, and A. Khoroshkin, “Quadratic algebras related to the bi-Hamiltonian operad,” Int. Math. Res. Notices (to appear).
R. Graham, D. Knuth, and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, Massachusetts, 1994.
A. A. Klyachko, “Lie elements in the tensor algebra,” Sibirsk. Mat. Zh., 15:6 (1974), 1296–1304.
I. Macdonald, Symmetric Functions and Hall Polynomials, Oxford University Press, 1995.
V. A. Smirnov, Simplicial and Operadic Methods in Homotopy Theory [in Russian], Factorial Press, Moscow, 2002.
R. Stanley, Enumerative Combinatorics, vol. 1, Cambridge University Press, 1997.
A. Khoroshkin, “Koszul operads and distributive lattices” (to appear).
A. Björner, M. Wachs, “On lexicographically shellable posets,” Trans. Amer. Math. Soc., 277:1 (1983), 323–341.
A. J. Brandt, “The free Lie ring and Lie representations of the full linear group,” Trans. Amer. Math. Soc., 56:3 (1944), 528–536.
F. Chapoton and B. Vallette, “Pointed and multi-pointed partitions of types A and B,” J. Algebraic Combin., 23:4, 295–316; http://arxiv.org/math.QA/0410051.
B. Fresse, “Koszul duality of operads and homology of partition posets,” Contemp. Math., 346 (2004), 115–215.
V. Ginzburg and M. Kapranov, “Koszul duality for operads,” Duke Math. J., 76:1 (1994), 203–272.
M. Haiman, Vanishing theorems and character formulas for the Hilbert scheme of points in the plane, http://arxiv.org/math.AG/0201148.
M. Markl, “Distributive laws and Koszulness,” Ann. Inst. Fourier (Grenoble), 40:2 (1996), 307–323; http://arxiv.org/hep-th/9409192.
M. Markl, S. Shnider, and J. D. Stasheff, Operads in Algebra, Topology and Physics, Math. Surveys Monographs, vol. 96, Amer. Math. Soc., Providence, RI, 2002.
B. Vallette, “Homology of generalized partition posets,” J. Pure Appl. Algebra, 208:2 (2007), 699–725; http://arxiv.org/math.AT/0405312.
Author information
Authors and Affiliations
Additional information
__________
Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 41, No. 1, pp. 1–22, 2007
Original Russian Text Copyright © by V. V. Dotsenko and A. S. Khoroshkin
The research of the first author is supported by President of the Russian Federation grant no. NSh-2044.2003.2 and by INTAS grant no. 03-3350. The research of the second author is supported by RFBR grant no. 04-01-00637 and INTAS grant no. 03-3350.
Rights and permissions
About this article
Cite this article
Dotsenko, V.V., Khoroshkin, A.S. Character formulas for the operad of two compatible brackets and for the bi-Hamiltonian operad. Funct Anal Its Appl 41, 1–17 (2007). https://doi.org/10.1007/s10688-007-0001-3
Received:
Issue Date:
DOI: https://doi.org/10.1007/s10688-007-0001-3