Abstract
We will give a new relationship between several simple automata and formal power series as word invariants. Such an invariant is derived from certain combinatorics and algebraic structures. We review them, and especially we deal with a connection to Lie theory via through tilings.
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References
Abe, E.: Hopf Algebras. Cambridge Tracts in Math., vol. 74. Cambridge Univ. Press, New York (1980)
Allison, B.N., Azam, S., Berman, S., Gao, Y., Pianzola, A.: Extended Affine Lie Algebras and Their Root Systems. Mem. Amer. Math. Soc., vol. 126 (1997)
Berman, S., Morita, J., Yoshii, Y.: Some factorizations in universal enveloping algebras of three dimensional Lie algebras and generalizations. Can. Math. Bull. 45, 525–536 (2002)
Chiba, H., Guo, J., Morita, J.: A new basis of U(sl 2) and Heisenberg analogue. Hadron. J. 30, 503–512 (2007)
Dobashi, D., Morita, J.: Groups, Lie algebras and Gauss decompositions for one dimensional tilings. Nihonkai Math. J. 17, 77–88 (2006)
Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. GTM, vol. 9. Springer, New York (1972)
Humphreys, J.E.: Linear Algebraic Groups. GTM, vol. 21. Springer, New York (1975)
Kac, V.: Infinite Dimensional Lie Algebras, 3rd edn. Cambridge Univ. Press, New York (1990)
Kellendonk, J.: Noncommutative geometry of tilings and gap labelling. Rev. Math. Phys. 7(7), 1133–1180 (1995)
Kellendonk, J.: The local structure of tilings and their integer group of coinvariants. Commun. Math. Phys. 187, 115–157 (1997)
Kellendonk, J., Putnam, I.F.: Tilings, C ∗-algebras and K-theory. In: Directions in Mathematical Quasicrystals. CRM Monogr. Ser., vol. 13, pp. 177–206. Amer. Math. Soc., Providence (2000)
Lawson, M.: Inverse Semigroups (The Theory of Partial Symmetries). World Scientific, Singapore (1998)
Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge Univ. Press, New York (1995)
Lothaire, M.: Algebraic Combinatorics on Words. Encyclopedia of Mathematics and Its Application, vol. 90. Cambridge Univ. Press, Cambridge (2002)
Masuda, T., Morita, J.: Local properties, bialgebras and representations for one dimensional tilings. J. Phys. A, Math. Gen. 37, 2661–2669 (2004)
Moody, R.V., Pianzola, A.: Lie Algebras with Triangular Decompositions. Canad. Math. Soc. Ser. Monogr. Adv. Text. Wiley, New York (1995)
Morita, J.: Tiling, Lie theory and combinatorics. Contemp. Math. 506, 173–185 (2010)
Morita, J., Plotkin, E.: Gauss decompositions of Kac-Moody groups. Commun. Algebra 27, 465–475 (1999)
Morita, J., Plotkin, E.: Prescribed Gauss decompositions for Kac-Moody groups over fields. Rend. Semin. Mat. Univ. Padova 106, 153–163 (2001)
Morita, J., Terui, A.: Words, tilings and combinatorial spectra. Hiroshima Math. J. 39, 37–60 (2009)
Morita, J., Yoshii, Y.: Locally extended affine Lie algebras. J. Algebra 301, 59–81 (2006)
Acknowledgements
The author wishes to express his hearty thanks to Professor Akira Terui for his valuable advice.
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Morita, J. (2013). Words, Automata and Lie Theory for Tilings. In: Iohara, K., Morier-Genoud, S., Rémy, B. (eds) Symmetries, Integrable Systems and Representations. Springer Proceedings in Mathematics & Statistics, vol 40. Springer, London. https://doi.org/10.1007/978-1-4471-4863-0_14
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DOI: https://doi.org/10.1007/978-1-4471-4863-0_14
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