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pith:EZFD3J6I

pith:2026:EZFD3J6IV7W4O2SMXKRT5A4J5V

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Rota Baxter Operators on Truncated Polynomial Algebras

Azhar Farooq

Rota-Baxter operators of weights zero and one on the truncated polynomial algebra R = K[x1,…,xn]/m² are classified by nilpotency conditions or by idempotent endomorphisms on the maximal ideal quotient.

arxiv:2605.15670 v1 · 2026-05-15 · math.AC

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Claims

C1strongest claim

We classify all Rota--Baxter operators of weights zero and one on the truncated polynomial algebra R=K[x_1,…,x_n]/m^2. For weight zero, we prove that the Rota--Baxter operators are precisely the linear maps P satisfying P^2=0 and Image(P)⊂m/m^2. For nonzero weight, a standard rescaling reduces the classification to weight one. In this case, the operators split into two disjoint families according to the value of P(1)∈{0,−1}. On the maximal ideal m/m^2, such operators induce an endomorphism L satisfying L^2+L=0, equivalently, −L is idempotent. We further show that each family is isomorphic to the variety of idempotent matrices.

C2weakest assumption

The claim that a standard rescaling reduces the nonzero-weight case to weight one without loss of generality, and that the resulting operators on m/m^2 always induce an endomorphism L with L^2 + L = 0, as stated in the abstract.

C3one line summary

Classifies Rota-Baxter operators on truncated polynomial algebras as maps satisfying P squared equals zero or inducing idempotent endomorphisms on the maximal ideal, with each family corresponding to the variety of idempotent matrices.

References

16 extracted · 16 resolved · 0 Pith anchors

[1] Baxter,An analytic problem whose solution follows from a simple algebraic identity, Pacific J 1960

[2] Rota,Baxter algebras and combinatorial identities I, II, Bull 1969

[3] A. Connes, and D. Kreimer,Renormalization in quantum field theory and the Riemann- Hilbert problem I: the Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys.,210(2000), 249–273 2000

[4] Cartier, On the structure of free Baxter algebras,Adv 1972

[5] A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys.199(1998), 203–242 1998

Formal links

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Receipt and verification

First computed	2026-05-20T00:01:11.355884Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

264a3da7c8afedc76a4cbaa33e8389ed7c8cb9f220efc0506e7b4dabe23689a7

Aliases

arxiv: 2605.15670 · arxiv_version: 2605.15670v1 · doi: 10.48550/arxiv.2605.15670 · pith_short_12: EZFD3J6IV7W4 · pith_short_16: EZFD3J6IV7W4O2SM · pith_short_8: EZFD3J6I

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/EZFD3J6IV7W4O2SMXKRT5A4J5V \
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Canonical record JSON

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    "license": "http://creativecommons.org/licenses/by/4.0/",
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    "submitted_at": "2026-05-15T06:46:50Z",
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