Abstract
Suppose that L is a free Lie algebra generated by k elements. This is a homogeneous algebra and dimensions of its homogeneous components are given by Witt’s formula dim \({L_n} = \frac{1}{n}{\Sigma _{a|n}}\mu \left( a \right){k^{n/a}}\) where μ(n) is the Möbius function. There are versions of this formula for multihomogeneous components as well as formulae for components of free Lie superalgebras. Now we suggest to use formal power series for the whole of the free Lie superalgebra. These series yield some new dimension formulas. Main application of our functions concerns invariants of actions of finite groups on free Lie superalgebras. We find generating functions for free generating sets of the invariant subalgebras. Some examples are given.
partially supported by grants RFBR 98-01-01020, 99-01-00233, grant of Ministry of Education of Russia 98–7
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References
Baliturin, Yu. A.: Identical relations in Lie algebras. VNV Science Press, Utrecht, 1987
Bahturin, Yu. A., Mikhalev, A. A., Petrogradsky, V. M., Zaicev, M. V.: Infinite dimensional Lie superalgebras. de Gruyter Exp. Math. 7. de Gruyter, Berlin, 1992
Bourbaki, N.: Groupes et algèbres de Lie. Chapters 1,2,3. Hermann. Paris. 1968
Bourbaki, N.: Groupes et algèbres de Lie. Chapters 4,5,6. Hermann. Paris. 1972
Drensky, V.: Fixed algebras of residually nilpotent Lie algebras. Proc. Amer. Math. Soc. 120 (1994) no. 4, 1021–1028
Formanek, E.: Noncommutative invariant theory, Group actions on rings. Contemp. Math., 43, Amer. Math. Soc., Providence, R.I., (1985) 87–119
Kang, Seok-Jin: Graded Lie superalgebras and the superdimension formula. J. Algebra 204 (1998) no. 2, 597–655
Kang, Seok-Jin, Kim, Myung-Hwan: Free Lie algebras, generalized Witt formula, and the denominator identity. J. Algebra 183 (1996) no. 2, 560–594
Molev, A. I., Tsalenko, L. M.: Representation of the symmetric group in the free Lie (super) algebra and in the space of harmonic polynomials. Funct. Anal. Appl. 20 (1986) 150–152
Molien, T.: Über die Invarianten der linearen Substitutionsgruppen, Sitzungsber. König. Preuss. Akad. Wiss. (1897) 1152–1156
Petrogradsky, V. M.: On invariants of actions of finite groups on free Lie algebras. Sibirsk. Mat. Zh. (to appear)
Petrogradsky, V. M.: Characters and invariants of free Lie superalgebras
Stanley, R. P.: Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.) 1 (1979) no. 3, 475–511
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Petrogradsky, V.M. (2000). On Witt’s Formula and Invariants for Free Lie Superalgebras. In: Krob, D., Mikhalev, A.A., Mikhalev, A.V. (eds) Formal Power Series and Algebraic Combinatorics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-04166-6_52
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DOI: https://doi.org/10.1007/978-3-662-04166-6_52
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