CHAO MA

Semisimplicity and Jacobson Radicals of Reduced Enveloping Algebras of Cartan Type

Let k be an algebraically closed field of characteristic p > 3, let L be a restricted simple Lie algebra of Cartan type, and let u_chi(L) be the reduced enveloping algebra attached to chi in L*. We determine the semisimple locus and give a pointwise formula for the Jacobson radic...

Cartan Invariants of Reduced Enveloping Algebras of Cartan Type

Let k be an algebraically closed field of characteristic p>3 and let g be a restricted simple Lie algebra of Cartan type W, S, H, or K. We determine the complete labelled Cartan matrix of u_chi(g) for every p-character chi. At zero character the Witt, Special, Hamiltonian, and st...

Representation Type and Stable Auslander-Reiten Components for Reduced Enveloping Algebras of Cartan Type

On Problem 8 of Benkart and Feldvoss: the representation type of the blocks of u_chi(g) for g a restricted simple Lie algebra of Cartan type, and the shapes of the stable Auslander-Reiten components. For terminal blocks with p >= 7 every nonperiodic stable component is one of ZA_...

Augmentation Realization of N-Series and Finite-Quotient Descent

Let G = H1 >= H2 >= ... be a prescribed N-series and let A be its weighted augmentation filtration. In every degree we identify the kernel of the canonical map Hn/Hn+1 -> An/An+1 with the cokernel of an explicit map between normalized bar groups. When the series is finite and sep...

Augmentation-Ideal Realization of Special Nottingham Congruence Series and Cartier Indecomposables

Let k = F_p with p >= 5, let A = k[[x_1,...,x_n]] with n >= 3, and let m be its maximal ideal. Let G be the group of volume-preserving formal automorphisms of A tangent to the identity, let G_s consist of those g in G such that g(x_i) - x_i lies in m^s for every i, and let T_N =...

Cartier Relation Symbols and Frattini Transgression in Special Nottingham Groups

Let k = F_p, let V = k^n with n >= 3, and let T_N be a finite congruence quotient of the tangent-to-identity Jacobian-one automorphism group of k[[x_1,...,x_n]]. Put d_r = (p - 1)(n - 1) + pr, and let L_s be the coefficient-degree-s divergence-free polynomial vector fields. Carti...

The Critical Cartier Diagonal in the Lower p-Central Series of Jacobian-One Formal Automorphisms

Let k = F_p with p ≥ 5, let V = k^n with n ≥ 3, and let T(N) be a finite congruence quotient of the tangent-to-the-identity Jacobian-one formal automorphism group in n variables. Write P(1) = T(N), P(i + 1) = P(i)^p[P(i), T(N)], and d(r) = (p − 1)(n − 1) + pr. For every r ≥ 2 and...

The First Frattini–Commutator Layer of Jacobian-One Formal Automorphisms

Let k = F_p with p >= 5, let V = k^n with n >= 3, and let G_1 be the tangent-to-the-identity Jacobian-one formal automorphism group, with congruence quotients T_N and Frattini subgroups Phi_N. We determine the first Frattini-commutator layer J_N = [T_N,Phi_N] / ([T_N,[T_N,Phi_N]]...

The Critical Cartier Diagonal in the Lower p-Central Series of Jacobian-One Formal Automorphisms

Let k = F_p with p ≥ 5, let V = k^n with n ≥ 3, and let T(N) be a finite congruence quotient of the tangent-to-the-identity Jacobian-one formal automorphism group in n variables. Write P(1) = T(N), P(i + 1) = P(i)^p[P(i), T(N)], and d(r) = (p − 1)(n − 1) + pr. For every r ≥ 2 and...

The First Frattini–Commutator Layer of Jacobian-One Formal Automorphisms

Let k = F_p with p >= 5, let V = k^n with n >= 3, and let G_1 be the tangent-to-the-identity Jacobian-one formal automorphism group, with congruence quotients T_N and Frattini subgroups Phi_N. We determine the first Frattini-commutator layer J_N = [T_N,Phi_N] / ([T_N,[T_N,Phi_N]]...

Cartier Relation Symbols and Frattini Transgression in Special Nottingham Groups

Let k = F_p, let V = k^n with n >= 3, and let T_N be a finite congruence quotient of the tangent-to-identity Jacobian-one automorphism group of k[[x_1,...,x_n]]. Put d_r = (p - 1)(n - 1) + pr, and let L_s be the coefficient-degree-s divergence-free polynomial vector fields. Carti...