Approximating Cartan subalgebra decompositions in semi simple lie algebras with applications to Yenisei taiga carbon sequestration

Introduction

Abstract algebra and forest ecology would seem to have nothing in common, yet both deal with decomposition-the breaking of complex structures into simpler, interpretable components. In Lie theory, the Cartan decomposition splits a semisimple algebra into a maximal abelian subalgebra (the Cartan subalgebra) and a collection of root spaces, each associated with a linear functional called a root [1] . The root system encodes the algebra's symmetries and has been studied intensively for over a century [2] . In forest carbon science, researchers decompose total sequestration into contributions from tree growth, soil respiration, litterfall, and other components, seeking to identify the main drivers of net carbon uptake [3] . The Yenisei River basin contains roughly 600 million hectares of boreal forest, representing one of the largest terrestrial carbon pools on Earth [4] . Estimating sequestration rates across this vast area requires integrating high-dimensional data-species composition, stand age, canopy density, soil properties, climate variables-into a predictive framework that handles spatial irregularity and data sparsity [5] . Standard dimensionality reduction techniques like PCA assume linear structure and ignore the algebraic relationships among variables. This research tests the hypothesis that root-space eigenvectors from a Cartan decomposition of sl(6,R) provide a more effective basis for carbon prediction than PCA or other standard methods. The algebra sl(6,R) was chosen because the 23 predictor variables naturally partition into six thematic groups (species, structure, soil, climate, topography, disturbance), and the 6×6 matrix representation captures inter-group interactions through off-diagonal root spaces [1] .

Fourier series modelled seasonal variation in sequestration, Bayesian priors handled the uncertainty from sparse groundtruth data, and Riemann surface conformal mappings transformed the irregular sampling grid into a regular domain suitable for spatial interpolation [6] . The model was calibrated on data from 2010-2021 and validated against independent measurements from 2022-2023.

Calibration and Validation

Calibration used eddy covariance net ecosystem exchange (NEE) from four flux towers in the central Yenisei taiga (2010-2021), supplemented by 87 forest inventory plots measured at three-year intervals. The Cartan decomposition was fitted by maximising the likelihood of observed NEE given the root-space projections, with regularisation controlled by the Bayesian prior on component weights [7] . Validation used held-out data from 2022-2023 (two flux tower-years and 29 inventory plots). Goodness-of-fit was assessed by RMSE, mean absolute error, and coefficient of determination (R²). The calibration R² was 0.943 and the validation R² was 0.897, indicating moderate overfitting that the Bayesian prior partially controlled.

Materials and Methods

Materials

Methods

The 23 predictor variables were arranged into a 6×6 matrix M by grouping into thematic blocks and padding with zeros where group sizes were unequal. The Cartan decomposition of sl(6,R) was computed as g = h ⊕ Σ_α g_α, where h is the Cartan subalgebra (diagonal matrices) and g_α are root spaces [1] . Root-space eigenvectors served as basis functions for regression. A four-harmonic Fourier series captured seasonal variation: Q(t) = a₀ + Σ_{k=1}^{4}[a_k cos(2πkt/12) + b_k sin(2πkt/12)]. Bayesian priors on the root-space weights used a half-Cauchy distribution with scale σ = 0.5, chosen by empirical Bayes on the calibration set [7] . The Riemann surface mapping used the Schwarz-Christoffel transformation to map the irregular sampling polygon onto the unit disc [6] . Benchmarks were principal component regression and random forest (500 trees, max depth 10).

Scenario Development

Three scenarios were modelled to project sequestration rates under varying conditions: (S1) current climate and disturbance regime maintained through 2030; (S2) +1.5°C mean annual temperature with current disturbance; (S3) current climate with doubled fire frequency. Under S1, projected sequestration was 2.08 tC/ha/yr (a 2.8% decline from the 2010-2021 mean). Under S2, the Cartan model predicted 1.89 tC/ha/yr (an 11.7% decline), driven by reduced root-space weight on the soil moisture component. Under S3, sequestration dropped to 1.61 tC/ha/yr (24.8% decline). Table 1 shows that the Cartan root-space model achieved the best validation RMSE (0.37 tC/ha/yr) using only four components, compared with five PCA components (RMSE = 0.52) and all 23 variables in random forest (RMSE = 0.48).

Results

The Cartan model's advantage appears to stem from its ability to capture inter-group interactions through off-diagonal root spaces.

Discussion

The Cartan decomposition outperformed PCA despite both being linear dimensionality reduction methods, which requires explanation. PCA finds orthogonal directions of maximum variance without regard to the grouping structure of variables. The Cartan decomposition, by contrast, respects the block structure of the predictor matrix and finds components that correspond to specific inter-group interactions [1] . When variable groups are scientifically meaningful (as they are here species, soil, climate, etc.), this structure-aware decomposition captures interpretable signals that PCA mixes together.

The Bayesian prior selection was important for preventing overfitting. Without priors (flat weights), the calibration R² rose to 0.971 but validation R² dropped to 0.839. The half-Cauchy prior shrank small root-space contributions toward zero, effectively implementing a sparsity constraint within the Lie-algebraic framework [7] . Limitations include the fixed choice of sl(6,R) as the underlying algebra. If the natural grouping of variables changes (for example, if new predictor types are added), the algebra would need to be expanded. Also, the Riemann surface mapping assumes a simply connected sampling domain; forests with large internal gaps (due to lakes or clearings) would require more complex conformal mappings.

Conclusion

This research demonstrated that Cartan subalgebra decompositions of sl(6,R) provide an effective spectral basis for predicting carbon sequestration in the Yenisei taiga, achieving RMSE of 0.37 tC/ha/yr and outperforming PCA regression and random forest benchmarks. The key advantage lies in the decomposition's ability to represent inter-group variable interactions through root spaces. Scenario projections suggest that warming (+1.5°C) and increased fire frequency could reduce sequestration by 11.7% and 24.8% respectively by 2030, with the soil-climate interaction root space showing the greatest sensitivity. The homological analysis identified structural redundancies that allowed dimensionality reduction from 23 to 4 components without meaningful accuracy loss. For forest carbon accounting, the practical implication is that algebraic methods from pure mathematics can serve as effective alternatives to standard machine learning when the variable structure has a natural grouping that standard techniques ignore.

Acknowledgement

Acknowledgements Funding SourcesThis research was supported by the Peruvian National Council for Science and Technology through its applied mathematics programme. Institutional SupportThe Trujillo Academy of Statistical Research provided computational resources and access to FLUXNET data archives. Contributions Not Qualifying for AuthorshipDr. Andrei Volkov provided guidance on interpreting Russian forest inventory protocols.

References

1. Humphreys JE. Introduction to Lie algebras and representation theory. Berlin: Springer; 1972:1-171.
2. Knapp AW. Lie groups beyond an introduction. Basel: Birkhäuser; 2002:1-812.
3. Pan Y, Birdsey RA, Fang J. A large and persistent carbon sink in the world's forests. Science. 2011;333(6045):988993.
4. Shvidenko A, Nilsson S. A synthesis of the impact of Russian forests on the global carbon budget. Tellus B. 2003;55(2):391-415.
5. Dolman AJ, Shvidenko A, Schepaschenko D. An estimate of the terrestrial carbon budget of Russia. Biogeosciences. 2012;9(12):5007-5024.
6. Driscoll TA, Trefethen LN. Schwarz-Christoffel mapping. Cambridge: Cambridge University Press; 2002:1-132.
7. Gelman A. Prior distributions for variance parameters in hierarchical models. Bayesian Anal. 2006;1(3):515-534.
8. Fulton W, Harris J. Representation theory: a first course. Berlin: Springer; 1991:1-551.
9. Baldocchi D. Measuring fluxes of trace gases and energy between ecosystems and the atmosphere. Glob Change Biol. 2014;20(11):3600-3609.
10. Luyssaert S, Schulze ED, Börner A. Old-growth forests as global carbon sinks. Nature. 2008;455(7210):213-215.
11. Serre JP. Complex semisimple Lie algebras. Berlin: Springer; 2001:1-74.
12. Hatcher A. Algebraic topology. Cambridge: Cambridge University Press; 2002:1-544.
13. Schepaschenko D, Shvidenko A, Usoltsev V. A dataset of forest biomass structure for Eurasia. Sci Data. 2017;4:170070.
14. Farkas HM, Kra I. Riemann surfaces. 2nd ed. Berlin: Springer; 1992:1-363.
15. Tibshirani R. Regression shrinkage and selection via the lasso. J R Stat Soc Ser B. 1996;58(1):267-288.
16. Breiman L. Random forests. Mach Learn. 2001;45(1):532.
17. Bondarev A, Kharuk V. Wildfires in Russia: carbon emissions and black carbon impacts. Environ Res Lett. 2019;14(10):105003.
18. Hall BC. Lie groups, Lie algebras, and representations. 2nd ed. Berlin: Springer; 2015:1-449.
19. Gower ST, Krankina O, Olson RJ. Net primary production and carbon allocation patterns of boreal forest ecosystems. Ecol Appl. 2001;11(5):1395-1411.
20. Jacobson N. Lie algebras. New York: Dover Publications; 1979:1-331.