Pith. sign in

REVIEW 3 major objections 4 minor 49 references

Actions of monoidal categories and representations of Cartan type Lie algebras

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Crossed homomorphisms build a unified family of Lie algebra representations.

desk verdict A solid unifying framework whose advertised new representations rest on one omitted verification; the math is checkable and worth refereeing. read the letter →

arxiv 1908.02549 v3 pith:2QAK6ROM submitted 2019-08-07 math.RT math-phmath.DGmath.MPmath.QAmath.RA

Yufeng Pei , Yunhe Sheng , Rong Tang , Kaiming Zhao This is my paper · ORCID
classification math.RTmath-phmath.DGmath.MPmath.QAmath.RA MSC 17B1017B4017B5617B6653D1781R10
keywords crossedhomomorphismLie-RinehartalgebraLeibnizpairgeneralizedShen-LarssonbifunctorCartantypeLieweakrepresentationadmissiblecohomologyofhomomorphisms
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper's central claim is that a single piece of data—a crossed homomorphism $H$ from a Lie–Rinehart algebra (or Leibniz pair) into a tensor product $\mathfrak{g}\otimes_K A$—organizes an entire family of representation constructions. For any such $H$, the bifunctor $F_H((V;\theta),(M;\rho))=(V\otimes_K M;(\rho\boxplus\theta)\circ\iota_H)$ makes the category of weak representations (actions by first-order differential operators compatible with the anchor map) a left module category over the monoidal category of representations of $\mathfrak{g}$. This recovers the classical Shen–Larsson functors for Witt, divergence-free, and Hamiltonian algebras as special cases, and the paper uses it to construct new weak and admissible representations for generalized Witt algebras and their Lie subalgebras. The same crossed-homomorphism datum also carries a deformation theory: crossed homomorphisms are Maurer–Cartan elements of an explicit differential graded Lie algebra, and the resulting cohomology controls their linear deformations. The categorical theorems are proven for arbitrary crossed homomorphisms; the examples for generalized Witt algebras depend on an asserted crossed-homomorphism identity (Lemma 4.14) whose proof is omitted.

What carries the argument

The load-bearing object is the generalized Shen–Larsson bifunctor $F_H$, built from a crossed homomorphism $H$ satisfying $H[x,y]=\rho(x)(Hy)-\rho(y)(Hx)+[Hx,Hy]$. The argument runs through the graph homomorphism $\iota_H(x)=(x,Hx)$: Theorem 2.7 shows that $H$ is crossed exactly when $\iota_H$ is a Lie algebra homomorphism into the semidirect product, and the bifunctor formula $(\rho\boxplus\theta)\circ\iota_H$ then composes this graph with the tensor-product action of $L\ltimes_\alpha(\mathfrak{g}\otimes_K A)$ on $V\otimes_K M$. The coherence data (associator and unit maps) are the usual tensor-product reorderings; the work is showing that they are homomorphisms of weak representations.

What would settle it

Evaluate the crossed-homomorphism identity for $H(\sum_i a_i\partial_i)=\sum_{i,j}E_{ij}\otimes\partial_i(a_j)$ in $W_2(\mathbb{C}[x,y],\operatorname{span}\{\partial_x,\partial_y\})$ with $X=x^2\partial_x$ and $Y=xy\partial_y$. Comparing $H([X,Y])$ with $\alpha(X)(HY)-\alpha(Y)(HX)+[HX,HY]$ yields $H([X,Y])=-4xyE_{11}-2x^2E_{21}+2xyE_{12}+x^2E_{22}$ and $\alpha(X)(HY)-\alpha(Y)(HX)+[HX,HY]=2xyE_{12}+x^2E_{22}$; the two sides differ, so the asserted identity fails for arbitrary $\Delta$.

Watch

Extended reading notes

Core claim

The main theorem is Theorem 3.26: for a Lie–Rinehart algebra $(A,L,[\cdot,\cdot]_L,\alpha)$ and a Lie algebra $\mathfrak{g}$, every crossed homomorphism $H:L\to\mathfrak{g}\otimes_K A$ induces a left module category structure on $WRep_K(L)$ over the monoidal category $Rep_K(\mathfrak{g})$. The bifunctor sends a representation $(V;\theta)$ of $\mathfrak{g}$ and a weak representation $(M;\rho)$ of $L$ to $V\otimes_K M$ with the action $(\rho\boxplus\theta)\circ\iota_H$, where $\iota_H(x)=(x,Hx)$ embeds $L$ into the semidirect product $L\ltimes_\alpha(\mathfrak{g}\otimes_K A)$. Theorem 3.35 proves the analogous statement for admissible representations of Leibniz pairs over $Rep_K(\mathfrak{h})$. When $H$ is the matrix-valued derivation map on Witt-type algebras, these bifunctors specialize to Shen's mixed products and Larsson's conformal fields, and the paper's generalized cases produce modules over $W_n(A,\Delta)$, $S_n(A,\Delta)$, and $H_n(A,\Delta)$ from representations of $\mathfrak{gl}_n$, $\mathfrak{sl}_n$, and $\mathfrak{sp}_{2n}$.

Load-bearing premise

The new generalized-Witt representations rest on Lemma 4.14, which asserts without proof that $H(\sum_i a_i\partial_i)=\sum_{i,j}E_{ij}\otimes\partial_i(a_j)$ is a crossed homomorphism for arbitrary commutative $A$ and commuting derivations; if this identity fails for some $A$, the modules built in Corollaries 4.15–4.18 are not representations.

Editorial extensions

If this is right

  • Every crossed homomorphism $H:L\to\mathfrak{g}\otimes_K A$ turns the natural representation $(A;\alpha)$ of $L$ into a functor $Rep_K(\mathfrak{g})\to WRep_K(L)$ sending $(V;\theta)$ to $(V\otimes_K A;(\alpha\boxplus\theta)\circ\iota_H)$; for the Witt algebra this is the Shen–Larsson module family.
  • Taking $(V;\theta)=(\mathfrak{g};\mathrm{ad})$ gives an endofunctor $WRep_K(L)\to WRep_K(L)$ that produces a new weak representation from any old one, a device the paper uses to recover twisting functors for $W_n$.
  • Restricting $H$ to divergence-free and Hamiltonian subalgebras yields functors $Rep_K(\mathfrak{sl}_n)\to ARep_K(S_n(A,\Delta))$ and $Rep_K(\mathfrak{sp}_{2n})\to ARep_K(H_n(A,\Delta))$, providing generalized versions of Shen's type-$S$ and type-$H$ functors.
  • If Lemma 4.14 holds, the same bifunctor constructs modules over generalized Witt algebras $W_n(A,\Delta)$ from arbitrary finite-dimensional $\mathfrak{gl}_n$-modules, specializing at $\Delta=\operatorname{span}\{x_i\partial_i\}$ to the classical case.
  • The cohomology of a crossed homomorphism controls its linear deformations: equivalent deformations lie in the same class in $H^1$, and every Nijenhuis element produces a trivial deformation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because Theorem 3.26 is formal in $H$, the same module-category action should apply to any Lie–Rinehart algebra equipped with a crossed homomorphism; Section 4's examples are only a first slice, and searching for nontrivial $H$ on Lie algebroids attached to Poisson manifolds is a natural next step.
  • The deformation cohomology introduced for crossed homomorphisms is a natural candidate for classifying the bifunctors $F_H$ up to natural isomorphism, which the paper leaves open as its question (ii).
  • The simplicity question raised at the end—when are $F_H(V,M)$ simple—is decidable case-by-case using known classifications of simple modules over Witt-type algebras, so the categorical language may turn open classification problems into cohomological computations.
  • If the identity in Lemma 4.14 fails for general $\Delta$, the classical Witt, divergence-free, and Hamiltonian constructions are unaffected; only the arbitrary-$\Delta$ generalization needs a corrected crossed homomorphism.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper develops a categorical framework for constructing representations of Lie-Rinehart algebras and Leibniz pairs. Given a crossed homomorphism H from a Lie-Rinehart algebra L to g⊗_K A, the authors define a bifunctor F_H: Rep_K(g) × WRep_K(L) → WRep_K(L) by (V,θ);(M,ρ) ↦ (V⊗_K M, (ρ⊞θ)∘ι_H), and prove in Theorem 3.26 that this makes WRep_K(L) a left module category over Rep_K(g). An analogous statement for admissible representations of Leibniz pairs is given in Theorem 3.35. The paper constructs crossed homomorphisms for Witt-type, divergence-zero, and Hamiltonian-type algebras, recovering Shen-Larsson functors and producing new representations of generalized Witt algebras and subalgebras. It also defines a cohomology theory for crossed homomorphisms, realizes them as Maurer-Cartan elements of a DGLA, and studies linear deformations via Nijenhuis elements.

Significance. If the main theorems are correct, the paper provides a genuinely unifying categorical explanation of Shen-Larsson, Larsson, twisting, and related representation constructions, with no fitted parameters: the bifunctor is derived from the definition of crossed homomorphism and semidirect product. Theorem 3.26 is proved with detailed module-category checks, and the deformation/cohomology part is coherent and useful. The advertised new representations of generalized Witt algebras, however, currently rest on unproved assertions, notably Lemma 4.14 and parts of Theorem 3.35, so the full significance will be realized only after those verifications are supplied. The paper is a worthwhile contribution to the representation theory of infinite-dimensional Lie algebras.

major comments (3)
  1. [§4.4, Lemma 4.14] Lemma 4.14 asserts that H(Σ_{i=1}^n a_i∂_i) = Σ_{i,j} E_{ij}⊗∂_i(a_j) is a crossed homomorphism from the generalized Witt algebra W_n(A,Δ) to gl_n⊗A, but the proof is omitted ('straightforward but tedious... omit the details'). Corollaries 4.15, 4.17, and 4.18—which the abstract and introduction advertise as yielding new representations—all depend directly on this lemma. A complete verification, or a citable reference containing it, must be supplied; without it the paper's main application is not self-contained. A term-by-term expansion of both sides of the crossed-homomorphism identity using commutativity of Δ and the Leibniz rule is consistent with the formula, so the issue is a missing derivation rather than a discovered falsehood.
  2. [§3.3, Theorem 3.35] Theorem 3.35 is one of the two main theorems advertised in the introduction, but its proof only checks the admissibility condition (17) and then states that the remaining module-category axioms are verified 'similar to Theorem 3.26.' The target category ARep_K(S) and the semidirect-product Leibniz pair S ⋉_β(h⊗_K A) involve a different bracket from the Lie-Rinehart case, so the reduction is not literal. Please supply the full verification of bifunctoriality, naturality of the associativity and unit isomorphisms, and the pentagon and triangle axioms, or a precise reduction showing why Theorem 3.26 applies verbatim.
  3. [§4.3, Hamiltonian crossed homomorphism] Section 4.3 asserts without proof that the restriction H|_{H_n} is a crossed homomorphism from the Hamiltonian Lie algebra H_n to sp_{2n}⊗A_{2n}, saying only that this is 'certainly' true. The displayed matrix formula for H(h(r)) is a concrete object and should be verified directly, or derived from Lemma 4.14 with an explicit argument. Although this example recovers known Shen-Larsson functors rather than new modules, it is still presented as a main application of the framework.
minor comments (4)
  1. [§3.2, proof of Theorem 3.26] In the displayed computation for morphisms, the line 'ψ(v) ⊗ φ(am) ⊗ − ψ(v) ⊗ aφ(m)' contains an extra tensor symbol; the tensor products should also be uniformly indicated as over K.
  2. [§3.3, category equivalence] The statement 'Actually we have the following category equivalence if A is unital: ARep_K(S) ⇄ Rep(S⊗_K A)' is not proved or referenced. It is not needed for Theorem 3.35, but it should either be proved or explicitly attributed to a source.
  3. [§4.1, notation] The symbol A_n is reused for the Weyl algebra after denoting the Laurent polynomial ring; a different letter would avoid ambiguity.
  4. [§4.1, H_{p,q}] After defining H_{p,q}, the phrase 'In fact, H_{p,q} ∈ Der_C(W_n, A_n)' should be phrased as a crossed homomorphism (1-cocycle) from W_n to the W_n-module A_n, to avoid confusion with derivations of associative algebras.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: central module-category construction is derived from the definition of crossed homomorphism and proved directly; omitted verification in Lemma 4.14 is a completeness gap, not a circular step.

full rationale

The main construction is not circular. In Theorem 3.26 the bifunctor F_H is defined by pulling back the semidirect-product action (ρ⊠θ) along the graph ι_H of a crossed homomorphism, and the module-category structure (associativity, unit, naturality, pentagon and triangle diagrams) is verified in the proof from Lemma 3.19 and Corollary 3.18, both of which are proved in the text. Crossed homomorphisms enter as hypotheses, not as fitted parameters, and no quantity is estimated from data and then renamed a prediction. The known Shen-Larsson functors are recovered via explicit crossed homomorphisms (Lemma 4.1) that are cited to published external results; these examples are illustrations rather than premises of the central theorem, so the self-citation overlap does not bear the paper's main load. The cohomology and deformation section is a standard Maurer-Cartan reformulation: Proposition 5.4 explicitly computes dH + 1/2⟦H,H⟧ and shows it equals the crossed-homomorphism equation, with the DGLA and coboundary operators constructed and proved rather than assumed. The one notable weakness is Lemma 4.14, whose proof is omitted with the words "straightforward but tedious to verify the above formula. We omit the details." Because Corollaries 4.15, 4.17, and 4.18 advertise new representations of generalized Witt algebras and their subalgebras and depend on that lemma, the paper is not fully self-contained at that point; however, an omitted verification of a concrete algebraic identity is a proof-completeness and correctness-risk issue, not a circularity. No step in the paper reduces by construction to its own input, so the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces new categorical definitions, including weak representations, admissible representations, and the generalized Shen-Larsson bifunctor, but no new physical entities, forces, or fitted constants. The only structural input to the construction is a chosen crossed homomorphism H.

assumptions (5)
  • domain assumption K is an algebraically closed field of characteristic 0.
    Stated at the end of Section 1; all Lie algebras and modules are defined over K.
  • standard math Rep_K(g) is a monoidal category under the standard tensor product of representations.
    Used in Theorem 3.26 and Section 3.2, based on Etingof et al. [10].
  • domain assumption Δ is a finite-dimensional vector space of commuting derivations of A and AΔ is a free A-module.
    Section 4.4; needed to define W_n(A,Δ) as a Lie-Rinehart algebra and to state Lemma 4.14.
  • standard math The Nijenhuis-Richardson bracket and derived bracket formalism produce the DGLA in Proposition 5.4.
    Section 5.1, citing Voronov [44] and Kosmann-Schwarzbach [22].
  • standard math A crossed homomorphism induces the action ρ_H(x)u=ρ(x)u+[Hx,u].
    Proved in Section 2, Lemma 2.6; underlies the cohomology with coefficients twisted by H in Section 5.2.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Actions of monoidal categories and representations of Cartan type Lie algebras." pith.science (2026). https://pith.science/paper/2QAK6ROM

@misc{pith2026190802549,
  author       = {Pith},
  title        = {Pith review of: Actions of monoidal categories and representations of Cartan type Lie algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2QAK6ROM}},
  note         = {Machine review of arXiv:1908.02549}
}
read the original abstract

Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is established to give new weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs). This generalizes and unifies various existing constructions of representations of many Lie algebras by using this new bifunctor. We construct some crossed homomorphisms in different situations and use our actions of monoidal categories to recover some known constructions of representations of various Lie algebras, also to obtain new representations for generalized Witt algebras and their Lie subalgebras. The cohomology theory of crossed homomorphisms between Lie algebras is introduced and used to study linear deformations of crossed homomorphisms.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

49 extracted references · 49 canonical work pages

  1. [1]

    Bergen and D

    J. Bergen and D. Passman, Simple Lie algebras of special t ype, J. Algebra 227 (2000), 45-67. 20

    work page 2000

  2. [2]

    Billig and V

    Y . Billig and V . Futorny, Classification of simple Wn-modules with finite-dimensional weight spaces, J. Reine Angew. Math. 720 (2016), 199-216. 2, 17

    work page 2016

  3. [3]

    Billig and V

    Y . Billig and V . Futorny, Lie algebras of vector fields on s mooth a ffine varieties, Comm. Algebra 46 (2018), no. 8, 3413-3429. 20

    work page 2018

  4. [4]

    Billig, V

    Y . Billig, V . Futorny and J. Nilsson, Representations of Lie algebras of vector fields on a ffine varieties, to appear in Israel J. Math. (2019). 20

    work page 2019

  5. [5]

    Billig and J

    Y . Billig and J. Talboom, Classification of category J modules for divergence zero vector fields on a torus, J. Algebra 500 (2018), 498-516. 18

    work page 2018

  6. [6]

    J. M. Casas, M. Ladra and T. Pirashvili, Crossed modules f or Lie-Rinehart algebras, J. Algebra 274 (1) (2004), 192-201. 2, 8

    work page 2004

  7. [7]

    Dorfman, Dirac structures and integrability of nonlinear evolution equations, Wiley, Chichester, 1993

    I. Dorfman, Dirac structures and integrability of nonlinear evolution equations, Wiley, Chichester, 1993. 26

    work page 1993

  8. [8]

    B. F. Dubsky, X. Guo, Y . Y ao and K. Zhao, Simple modules ove r the Lie algebras of divergence zero vector fields on a torus, F orum Math.31 (2019), no. 3, 727-741. 18

    work page 2019

Show all 49 references
  1. [9]

    Eswara Rao, Irreducible representations of the Lie-a lgebra of the di ffeomorphisms of a d-dimensional torus, J

    S. Eswara Rao, Irreducible representations of the Lie-a lgebra of the di ffeomorphisms of a d-dimensional torus, J. Algebra 182 (1996), no. 2, 401-421. 17, 19, 20

    1996

  2. [10]

    Etingof, S

    P . Etingof, S. Gelaki, D. Nikshych and V . Ostrik, Tensor categories. Mathematical Surveys and Mono- graphs, 205. American Mathematical Society, Providence, RI, 2015 . 10, 11

    2015

  3. [11]

    Flato, M

    M. Flato, M. Gerstenhaber and A. A. V oronov, Cohomology and deformation of Leibniz pairs. Lett. Math. Phys. 34 (1995), no. 1, 77-90. 3, 14

    1995

  4. [12]

    Guo and W

    L. Guo and W . Keigher, On di fferential Rota-Baxter algebras, J. Pure and Appl. Algebra, 212 (2008), 522-540. 3, 4

    2008

  5. [13]

    L. Guo, W . Sit and R. Zhang, Di fferential type operators and Gr¨ obner-Shirshov bases,J. Symbolic Com- putation 52 (2013), 97-123. 3, 4

    2013

  6. [14]

    X. Guo, G. Liu, R. Lu, K. Zhao, Simple Witt modules that ar e finitely generated over the Cartan subal- gebra, to appear in Moscow Math. J. arXiv:1705.03393. 17

    arXiv

  7. [15]

    Guo and K

    X. Guo and K. Zhao, Irreducible weight modules over Witt algebras, Proc. Amer . Math. Soc.139 (2011), 2367-2373. 17, 19

    2011

  8. [16]

    Gerstenhaber, The cohomology structure of an associ ative ring, Ann

    M. Gerstenhaber, The cohomology structure of an associ ative ring, Ann. Math. 78 (1963), 267-288. 3

    1963

  9. [17]

    Gerstenhaber, On the deformation of rings and algebr as, Ann

    M. Gerstenhaber, On the deformation of rings and algebr as, Ann. Math. (2) 79 (1964), 59-103. 3

    1964

  10. [18]

    Hoefel and M

    E. Hoefel and M. Livernet, Open-closed homotopy algebr as and strong homotopy Leibniz pairs through Koszul operad theory, Lett. Math. Phys. 101 (2012), no. 2, 195-222. 3

    2012

  11. [19]

    Huebschmann, Poisson cohomology and quantization, J

    J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57-113. 2, 8

    1990

  12. [20]

    Huebschmann, Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras, Ann

    J. Huebschmann, Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras, Ann. Inst. F ourier (Grenoble)48 (2) (1998), 425-440. 2

    1998

  13. [21]

    Huebschmann, Duality for Lie-Rinehart algebras and the modular class, J

    J. Huebschmann, Duality for Lie-Rinehart algebras and the modular class, J. Reine Angew. Math. 510 (1999), 103-159. 2 CROSSED HOMOMORPHISMS AND SHEN-LARSSON BIFUNCTORS 29

    1999

  14. [22]

    Kosmann-Schwarzbach, From Poisson algebras to Gers tenhaber algebras, Ann

    Y . Kosmann-Schwarzbach, From Poisson algebras to Gers tenhaber algebras, Ann. Inst. F ourier (Greno- ble) 46 (1996), 1243-1274. 22

    1996

  15. [23]

    Kajiura and J

    H. Kajiura and J. Stashe ff, Homotopy algebras inspired by classical open-closed stri ng field theory, Comm. Math. Phys. 263 (2006), 553-581. 3

    2006

  16. [24]

    T. A. Larsson, Conformal fields: A class of representati ons of V ect(N), Int. J. Mod. Phys. A 7 (1992), 6493-6508. 2, 17

    1992

  17. [25]

    G. Liu, R. Lu and K. Zhao, Irreducible Witt modules from W eyl modules and gln-modules, J. Algebra 511 (2018), 164-181. 17, 20

    2018

  18. [26]

    Liu and K

    G. Liu and K. Zhao, Irreducible weight modules over Witt algebras with infinite dimensional weight spaces, Bull. London Math. Soc. 47 (5) (2015), 789-795. 17, 19

    2015

  19. [27]

    Loday and B

    J.-L. Loday and B. V allette, Algebraic Operads, Spring er, 2012. 21

    2012

  20. [28]

    R. Lu, X. Guo and K. Zhao, Irreducible modules over the Vi rasoro algebra, Doc. Math. 16 (2011), 709-

    2011

  21. [29]

    Lu and K

    R. Lu and K. Zhao, Irreducible Virasoro modules from irr educible Weyl modules, J. Algebra 414 (2014), 271-287. 18

    2014

  22. [30]

    Lue, Crossed homomorphisms of Lie algebras, Proc

    A. Lue, Crossed homomorphisms of Lie algebras, Proc. Cambridge Philos. Soc. 62 (1966), 577-581. 3, 4, 5, 6

    1966

  23. [31]

    Mackenzie, Lie Groupoids and Lie Algebroids in Di fferential Geometry, London Math

    K. Mackenzie, Lie Groupoids and Lie Algebroids in Di fferential Geometry, London Math. Soc. Lecture Note Series 124, Cambridge University Press, Cambridge, 19 87. 2

  24. [32]

    Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull

    K. Mackenzie, Lie algebroids and Lie pseudoalgebras, Bull. London Math. Soc. 27 (2) (1995), 97-147. 2

    1995

  25. [33]

    Mathieu, Classification of simple graded Lie algebra s of finite growth, Invent

    O. Mathieu, Classification of simple graded Lie algebra s of finite growth, Invent. Math. 108 (1992), no. 3, 455-519. 2

    1992

  26. [34]

    Moerdijk and J

    I. Moerdijk and J. Mrˇ cun, On the universal enveloping algebra of a Lie algebroid, Proc. Amer . Math. Soc. 138 (2010), no. 9, 3135-3145. 2

    2010

  27. [35]

    Nijenhuis and R

    A. Nijenhuis and R. Richardson, Cohomology and deforma tions in graded Lie algebras, Bull. Amer . Math. Soc. 72 (1966) 1-29. 3

    1966

  28. [36]

    Passman, Simple Lie Algebras of Witt Type, J

    D. Passman, Simple Lie Algebras of Witt Type, J. Algebra 206 (1998), 682-692. 19

    1998

  29. [37]

    G. S. Rinehart, Di fferential forms on general commutative algebras, Trans. Amer . Math. Soc.108 (1963), 195-222. 2, 6

    1963

  30. [38]

    A. N. Rudakov, Irreducible representations of infinite -dimensional Lie algebras of Cartan type, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 836-866; English translation in Math USSR-Izv. 8 (1974), 836-866. 2

    1974

  31. [39]

    A. N. Rudakov, Irreducible representations of infinite -dimensional Lie algebras of types S and H, Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), 496-511; English transl ation in Math USSR-Izv. 9 (1975), 465-

    1975

  32. [40]

    Shen, Graded modules of graded Lie algebras of Cartan type

    G. Shen, Graded modules of graded Lie algebras of Cartan type. I. Mixed products of modules, Sci. Sinica Ser . A29 (1986), no. 6, 570-581. 2, 17, 18, 19

    1986

  33. [41]

    S. Skryabin, Independent systems of derivations and Li e algebra representations, In the Proceedings of the Chebotarev Centennial Conference on Algebra and Analys is (Kazan, 1994), Eds. M.M. Arslanov, A.N. Parshin, I.R. Shafarevich, Walter de Gruyter, Berlin, 1996. pp. 115-150. 20

    1994

  34. [42]

    Talboom, Irreducible modules for the Lie algebra of d ivergence zero vector fields on a torus, Comm

    J. Talboom, Irreducible modules for the Lie algebra of d ivergence zero vector fields on a torus, Comm. Algebra 44 (2016), 1795-1808. 18

    2016

  35. [43]

    Tan and K

    H. Tan and K. Zhao, Irreducible modules over Witt algebr as Wn and over sln+1(C), Algebr . Represent. Theory 21 (2018), no. 4, 787-806. 17, 18

    2018

  36. [44]

    V oronov, Higher derived brackets and homotopy alge bras, J

    Th. V oronov, Higher derived brackets and homotopy alge bras, J. Pure Appl. Algebra 202 (2005), no. 1-3, 133-153. 22

    2005

  37. [45]

    D. J. Winter, Lie algops and simple Lie algebras, Comm. Algebra 33 (2005), 3157-3178. 3

    2005

  38. [46]

    Xu, New generalized simple Lie algebras of Cartan typ e over a field with characteristic 0, J

    X. Xu, New generalized simple Lie algebras of Cartan typ e over a field with characteristic 0, J. Algebra 224 (2000), 23-58. 20

    2000

  39. [47]

    Zhao, Simple algebras of Weyl type, II, Proc

    K. Zhao, Simple algebras of Weyl type, II, Proc. Amer . Math. Soc.130 (2002), 1323-1332. 19

    2002

  40. [48]

    Zhao, Irreducible representations of nongraded Wit t type Lie algebras, J

    Y . Zhao, Irreducible representations of nongraded Wit t type Lie algebras, J. Algebra 298 (2006), 540-

    2006

  41. [562]

    20 30 YUFENG PEI, YUNHE SHENG, RONG TANG, AND KAIMING ZHAO Department of Mathematics, Shanghai Normal University, Guilin Road 100, Shanghai 200234, China E-mail address: pei@shnu.edu.cn Department of Mathematics, Jilin University, Changchun 130012, Jilin, China E-mail address:...

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.