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Dimensional analysis can be viewed as the study of graded objects in algebra. The grading then corresponds to "counting the units" in a precise way. There are of course many examples and I believe that morally, every graded (say $\mathbb{Z}$-graded) algebra/vector space/etc can be viewed as collection of objects having an inrinsic intrinsic dimension. If you have a graded algebra then this just means that the dimensions mutliply correctly. If you have a grading on a vector space and a bilinear map (product, Lie bracket...) which is still homogeneous but not respecting the grading additively then the bilinear map itself carries a fixed dimension:

On example is the polynomial algebra $\mathbb{C}[x,y]$ with its usual $\mathbb{Z}$-grading by the total polynomial degree. Then the canonical Poisson bracket is determined by ${x, \lbrace x, y } \rbrace = 1$ and hence has dimension $-2$ times the units of the generators. Relabelling then into $q$ and $p$, you have the Poisson bracket of classical mechanics with its usual dimension being that of an "inverse action".

More sophisticated examples can be found e.g. in differential geometry where you have zillions of graded algebras/spaces arising naturally. Staying in the realm of Poisson geometry, the canonical Poisson bracket on a cotangent bundle has "momentum degree $-1$": Mathematically, this means that one first has an Euler vector field $E = \sum_i p_i \frac{\partial}{\partial p_i}$ on $T^*M$ as on every vector bundle (where the $p_i$ are fiber coordinates). HEuristicallyHeuristically, this vector field "counts" how many $p$'s you have. Then the Poisson tensor is something like $\pi = \sum_i \frac{\partial}{\partial q^i} \wedge \frac{\partial}{\partial p_i}$ which satisfies $\mathcal{L}_E \pi = - \pi$, making the above statement precise.

In general, associated to a grading one can attach the grading "derivation" (or better just operator) which satisfies $deg = k id$ on the homogeneous components $V^k$ of the graded space $V$. In the above case, $deg$ is just the Euler vector field and the graded spaces are the tensor fields on $T^*M$...

I hope that these examples have convinced you that a "dimensional analysis" happens quite often and naturally in many areas of mathematics. The associated grading opertors operators usually play an important role and help a lot. The least is certainly a "self-correcting" aspect, that handling dimensions correctly avoids stupid errors.

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Dimensional analysis can be viewed as the study of graded objects in algebra. The grading then corresponds to "counting the units" in a precise way. There are of course many examples and I believe that morally, every graded (say $\mathbb{Z}$-graded) algebra/vector space/etc can be viewed as collection of objects having an inrinsic dimension. If you have a graded algebra then this just means that the dimensions mutliply correctly. If you have a grading on a vector space and a bilinear map (product, Lie bracket...) which is still homogeneous but not respecting the grading additively then the bilinear map itself carries a fixed dimension:

On example is the polynomial algebra $\mathbb{C}[x,y]$ with its usual $\mathbb{Z}$-grading by the total polynomial degree. Then the canonical Poisson bracket is determined by ${x, y} = 1$ and hence has dimension $-2$ times the units of the generators. Relabelling then into $q$ and $p$, you have the Poisson bracket of classical mechanics with its usual dimension being that of an "inverse action".

More sophisticated examples can be found e.g. in differential geometry where you have zillions of graded algebras/spaces arising naturally. Staying the realm of Poisson geometry, the canonical Poisson bracket on a cotangent bundle has "momentum degree $-1$": Mathematically, this means that one first has an Euler vector field $E = \sum_i p_i \frac{\partial}{\partial p_i}$ on $T^*M$ as on every vector bundle (where the $p_i$ are fiber coordinates). HEuristically, this vector field "counts" how many $p$'s you have. Then the Poisson tensor is something like $\pi = \sum_i \frac{\partial}{\partial q^i} \wedge \frac{\partial}{\partial p_i}$ which satisfies $\mathcal{L}_E \pi = - \pi$, making the above statement precise.

In general, associated to a grading one can attach the grading "derivation" (or better just operator) which satisfies $deg = k id$ on the homogeneous components $V^k$ of the graded space $V$. In the above case, $deg$ is just the Euler vector field and the graded spaces are the tensor fields on $T^*M$...

I hope that these examples have convinced you that a "dimensional analysis" happens quite often and naturally in many areas of mathematics. The associated grading opertors usually play an important role and help a lot. The least is certainly a "self-correcting" aspect, that handling dimensions correctly avoids stupid errors.