Lambda Algebra Tools

This repository provides an experimental Haskell implementation of the mod-p Lambda algebra and related constructions in algebraic topology.
It builds on the classic work of:

• A. K. Bousfield, E. B. Curtis, D. M. Kan, D. G. Quillen, D. L. Rector, and J. W. Schlesinger,
  The mod-p lower central series and the Adams spectral sequence, Topology 12 (1973), 97–118.
• E. B. Curtis, Simplicial Homotopy Theory, Lecture Notes in Mathematics 11, Springer, 1967.

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Features

• Geometric λ-words
  A toy model of λ-operations via shuffle products on suspensions (s_0, s_1, …) in the style of the original Haskell prototype.
  Only λ is meaningful here; μ does not admit such a geometric expression.

• Lambda algebra over an odd prime p

  • Generators lambdaN, muN corresponding to \lambda(i-1) and \mu(i-1) in [BCKQRS, Appendix].
  • Differential d^1 defined on generators by the binomial coefficient formulas (2.4′(iv)), extended via the Leibniz rule.
  • Adem relations (2.4′(iii)) implemented as rewriting rules.
  • Reduction to Curtis admissible form: leftmost–innermost strategy, canonical lex order (λ < μ, indices decrease left to right).
  • Polynomials normalized over F_p (coefficients reduced mod p, zero terms removed).

• Basis generation

  • Enumerates all admissible words up to a given degree cap.
  • Groups them by degree and optionally prints ( d^1 ) in admissible form.
  • Implemented with Data.Set, so each admissible word appears exactly once.

• Examples included

  • Geometric λ-words for small suspensions.
  • Computations of ( d^1 ) on λ and μ generators.
  • Adem reduction samples.
  • Basis tables for ( p=3,5 ) up to degree 30.

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Theory

Free Lie algebras and lower central series

For a free group F, the graded quotients of the lower central series

$$\gamma_n F / \gamma_{n+1} F$$

assemble into the free Lie algebra L. Over F_p this structure is enriched with a restricted Lie algebra operation (.)^{[p]}.

The Lambda algebra

The Lambda algebra Lambda is a combinatorial model that encodes this structure together with the action of Steenrod operations. It provides the E^1–page of the Adams spectral sequence for spheres at odd primes.

• Generators:

  • lambda(i-1) in degree 2*i*(p-1) - 1, corresponding to Steenrod operations P^i.
  • mu(i-1) in degree 2*i*(p-1), corresponding to the Bockstein applied to P^i, i.e. βP^i.

• Relations (Adem-type, [BCKQRS 2.4′(iii)]):
  Products of λ–λ, λ–μ, μ–λ, μ–μ with “bad” index patterns are rewritten as sums of admissible words with binomial coefficients mod p.

• Differential ( [BCKQRS 2.4′(iv)] ):

  $$d(\lambda_{n-1}) \;=\; \sum_{i+j=n} \binom{i+j}{i}\, \lambda_{i-1}\lambda_{j-1}, \quad n\ge 2$$ $$d(\mu_{n-1}) \;=\; \sum_{i+j=n} \binom{i+j}{i}\,(\lambda_{i-1}\mu_{j-1} - \mu_{i-1}\lambda_{j-1}), \quad n\ge 1$$

  and extended to all words by the Leibniz rule.

• Admissible form (Curtis):
  Words are reduced by Adem relations to a canonical basis where indices decrease left to right and no forbidden pairs occur.

Thus, λ corresponds to the primary Steenrod operations, μ to their Bocksteins, and the whole algebra governs the structure of the Adams spectral sequence.

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How to Run

The main file is mu.hs. Compile or run with GHC:

runhaskell mu.hs

or

ghc mu.hs && ./mu

This will print:

• Geometric λ examples,
• Differential computations in the Lambda algebra,
• Adem reductions,
• Basis tables up to degree 30 (with differentials).

All output is ASCII only (s_3, lambdaN, muN) for portability.

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Example Output

=== Geometric λ part (ASCII indices) ===
lambda1          = [s_1 i_1, s_0 i_1]
lambda1^2        = [[s_1s_1 i_1, s_1s_0 i_1], [s_0s_1 i_1, s_0s_0 i_1]]
lambda2·lambda1  = [[s_3s_2s_1 i_1, s_3s_2s_0 i_1], [s_1s_0s_1 i_1, s_1s_0s_0 i_1]] + [[s_3s_1s_1 i_1, s_3s_1s_0 i_1], [s_2s_0s_1 i_1, s_2s_0s_0 i_1]] + [[s_2s_1s_1 i_1, s_2s_1s_0 i_1], [s_3s_0s_1 i_1, s_3s_0s_0 i_1]]
lambda1^3        = [[[s_1s_1s_1 i_1, s_1s_1s_0 i_1], [s_1s_0s_1 i_1, s_1s_0s_0 i_1]], [[s_0s_1s_1 i_1, s_0s_1s_0 i_1], [s_0s_0s_1 i_1, s_0s_0s_0 i_1]]]

=== Lambda algebra over F_3 (odd p) ===
d^1(lambda1):
2·lambda1 lambda1
d^1(lambda1·lambda1) (Leibniz):
lambda2 lambda1 lambda1 + 2·lambda1 lambda1 lambda2
d^1(lambda2·lambda1) (Leibniz):
lambda3 lambda1 lambda1
d^1(mu0):
0
d^1(mu1·mu0) (Leibniz):
2·lambda1 mu1 mu1 + mu1 lambda1 mu1

Adem reduction (admissible form):
reduce( lambda2 · lambda_{1+p} ) = lambda3 lambda4
reduce( lambda_{1+p} · mu0 ) = lambda4 mu1
reduce( mu0 · lambda_{1+p} ) = mu1 lambda4

=== Lambda algebra over F_5 (odd p) ===
d^1(lambda1):
2·lambda1 lambda1
d^1(lambda1·lambda1) (Leibniz):
3·lambda2 lambda1 lambda1 + 2·lambda1 lambda1 lambda2
d^1(lambda2·lambda1) (Leibniz):
3·lambda3 lambda1 lambda1 + 3·lambda2 lambda1 lambda2 + 3·lambda1 lambda2 lambda2
d^1(mu0):
0
d^1(mu1·mu0) (Leibniz):
2·lambda1 mu1 mu1 + 3·mu1 lambda1 mu1

Adem reduction (admissible form):
reduce( lambda2 · lambda_{1+p} ) = lambda3 lambda6
reduce( lambda_{1+p} · mu0 ) = lambda6 mu1
reduce( mu0 · lambda_{1+p} ) = mu1 lambda6