This repository contains Mathematica packages for working with tensor algebras, free Lie algebras, shuffle products, and expected signatures of stochastic processes.
The Tensor package provides fundamental operations for working with tensor algebras and related structures:
CircleTimes: Abstract tensor product with standard algebraic propertiesw: Word marker for representing words in tensor spacesConc: Concatenation operator for tensors of wordsShuf: Shuffle product between wordsQShuf: Quasi-shuffle productDec,DecR: Deconcatenation coproduct (standard and reduced versions)Eulerian: Shuffle Eulerian idempotentUnshuf,UnshufR: Unshuffle coproduct (standard and reduced versions)Proj: Canonical projection onto the free Lie algebraAntipode: Antipode map for the Hopf algebra structureDynkin: Dynkin idempotent in the tensor algebrac: Contraction operator for wordsLogES: Log expected signatureESig: Expected signature of Stratonovich Brownian motionip: Inner product based on Stratonovich expected signatureIESig: Expected signature of Itô Brownian motioniip: Inner product based on Itô expected signatureHExp,HLog: Hoffman's exponential and logarithm for words
- Place the
.wlfiles in a directory where Mathematica can find them - Load the packages using:
<< Tensor`(* Create words *)
w[1, 2, 3]
w[4, 5]
(* Tensor product *)
w[1, 2] ⊗ w[3, 4]
(* Concatenation *)
Conc[w[1, 2] ⊗ w[3, 4]] (* Produces w[1, 2, 3, 4] *)
(* Shuffle product *)
Shuf[w[1, 2] ⊗ w[3, 4]]
(* Quasi-shuffle product *)
QShuf[w[1, 2] ⊗ w[3, 4]]This code implements fundamental operations from:
- Tensor algebra and free Lie algebra theory
- Shuffle algebra and its dual structures
- Expected signature theory for stochastic processes
The packages are useful for researchers in algebraic combinatorics, Lie theory, rough path theory, and stochastic analysis.
- Mathematica (tested on version 12.0 or later)
If using this code in research, please cite appropriately.