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 Mobius 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.
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