Commuting Derivations and Automorphisms of Certain Nilpotent Lie Algebras Over Commutative Rings

Let L be a finite-dimensional complex simple Lie algebra, L _? be the ?-span of a Chevalley basis of L, and L _R  = R ?_? L _? be a Chevalley algebra of type L over a commutative ring R. Let 𝒩(R) be the nilpotent subalgebra of L _R spanned by the root vectors associated with positive roots. A map ? of 𝒩(R) is called commuting if [?(x), x] = 0 for all x ∈ 𝒩(R). In this article, we prove that under some conditions for R, if Φ is not of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a central derivation (resp., automorphism), and if Φ is of type A ₂, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism).     

Second Homology Groups and Universal Coverings of Steinberg Leibniz Algebras of Small Characteristic

It is known that the second Leibniz homology group HL ₂ (𝔰𝔱𝔩 _n (R)) of the Steinberg Leibniz algebra 𝔰𝔱𝔩 _n (R) is trivial for n ≥ 5. In this article, we determine HL ₂(𝔰𝔱𝔩 _n (R)) explicitly (which are shown to be not necessarily trivial) for n = 3, 4 without any assumption on the base ring.     

Coherent Continuation on the Category of (𝔤, K)-Modules

Abstract

Let 𝔤 be a complex semisimple Lie algebra. Let K be an algebraic group acting on the flag variety of 𝔤 with finitely many orbits. We give a geometric interpretation of the coherent continuation on the category of finitely generated (𝔤, K )-modules in terms of the intertwining functors on the category of K-equivariant 𝒟-modules.     

Symplectic meanders

Analogous to the 𝔰𝔩(n) case, we address the computation of the index of seaweed subalgebras of 𝔰𝔭(2n) by introducing graphical representations called symplectic meanders. Formulas for the algebra’s index may be computed by counting the connected components of its associated meander. In certain cases, formulas for the index can be given in terms of elementary functions.     

Nonlinear generalized Lie triple derivation on triangular algebras

Let ? be a commutative ring with identity and let 𝔄 = Tri(𝒜,?,?) be a triangular algebra consisting of unital algebras 𝒜,? over ? and an (𝒜,?)-bimodule ? which is faithful as a left 𝒜-module as well as a right ?-module. In this paper, we prove that under certain assumptions every nonlinear generalized Lie triple derivation G_L:𝔄→𝔄 is of the form G_L = δ+τ, where δ:𝔄→𝔄 is an additive generalized derivation on 𝔄 and τ is a mapping from 𝔄 into its center which annihilates all Lie triple products [[x,y],z].     

Restrictions of Symmetric Invariants of a Complex Semisimple Lie Algebra and Some Applications

Let 𝔤 be a finite-dimensional complex semisimple Lie algebra and σ an arbitrary semisimple automorphism of 𝔤. Let 𝔱 be a Cartan subalgebra of 𝔨 = 𝔤^σ and 𝔥 =Z _𝔤(𝔱) be the centralizer of 𝔱 in 𝔤. Then 𝔥 is a σ-invariant Cartan subalgebra of 𝔤 and 𝔱 = 𝔥^σ. Let W(𝔤, 𝔥) be the Weyl group. One knows that Δ(𝔤, 𝔱), the set of roots of 𝔤 in 𝔱, is also a root system. It is proved that the corresponding Weyl group W(𝔤, 𝔱) is isomorphic to W(𝔤, 𝔥)^σ, which is the subgroup of W(𝔤, 𝔥) consisting of those elements commuting with σ. It is also shown that the image of the restriction map S(𝔥*) ^{W(𝔤, 𝔥)} → S(𝔱*) ^{W(𝔨, 𝔱)}, where S(𝔥*) and S(𝔱*) are the polynomial algebras on 𝔥 and 𝔱, respectively, is exactly S(𝔱*) ^{W(𝔤, 𝔱)}. Based on the above result, we also get a complete classification of the pairs (𝔤, σ) such that 𝔤^σ is noncohomologous to zero in 𝔤.     

Reductivity of the Lie Algebra of a Bilinear Form

Let f: V × V → F be a totally arbitrary bilinear form defined on a finite dimensional vector space V over a field F, and let L(f) be the subalgebra of 𝔤𝔩(V) of all skew-adjoint endomorphisms relative to f. Provided F is algebraically closed of characteristic not 2, we determine all f, up to equivalence, such that L(f) is reductive. As a consequence, we find, over an arbitrary field, necessary and sufficient conditions for L(f) to be simple, semisimple or isomorphic to 𝔰𝔩(n) for some n.     

A Class of Nongraded Simple Lie Algebras

Abstract

Nongraded simple Lie algebras appear naturally in mathematical physics. In this paper, a new class of nongraded simple Lie algebras are presented based on the pairs (𝒜, 𝒟) consisting of a commutative associative unital algebra 𝒜 and a finite dimensional commutative derivation subalgebra 𝒟 such that 𝒜 is 𝒟-simple. The isomorphism classes of these nongraded Lie algebras are also determined and the structure space of these algebras is given explicitly.     

Irreducible representations over the diamond Lie algebra

In this paper, we use Block’s results to classify irreducible modules over the diamond Lie algebra 𝔇. As a corollary, we also give a classification of irreducible modules over the Euclidean algebra 𝔢(2).     

On the Comparison of Rees's Valuations Over a Quasi-Unmixed Local Ring

In this paper, we will show that if (R, 𝔪) is a quasi-unmixed local ring, I an 𝔪-primary ideal of R and ?𝒱(I) is the set of Rees valuations of I, then the number of minimal prime ideals in the 𝔪-adic completion of R equals exactly the number of equivalence classes on the set ?𝒱(I) under the equivalence relation ～defined by: ν₁ ～ ν₂ if there exist a constant c ≥ 1 such that for all x ∈ R, ν₁(x) ≤ cν₂(x) and ν₂(x) ≤ cν₁(x).     

Formal Expansion of Harish-Chandra Homomorphism

Abstract

Let 𝔤 be a complex semisimple Lie algebra with adjoint group G and let 𝔥 be a Cartan subalgebra of 𝔤. Let Â(𝔤) and Â(𝔥) denote the algebra of differential operators with formal power series coefficients on 𝔤 and 𝔥 respectively. We construct a subalgebra A _𝔤 of Â(𝔤) containing all the pull-backs of the differential operators in G attached to any element x in 𝔤. We also consider the projection P: A _𝔤 → Â _𝔥. Then, we calculate explicity the pull-back of the differential operator in G attached to an element h in 𝔥 modulo Ker P.     

The Lasker-Noether Theorem for Unstable Modules over the Steenrod Algebra

Abstract

We show that a 𝒫* submodule of an unstable Noetherian R-module over a Noetherian ring R has a 𝒫*-invariant primary decomposition.     

Verbal Subgroups of T r (∞, K)

In this note we investigate the verbal subgroups of T _r (∞, K), the group of upper triangular, infinite, row-finite matrices over a field K containing more than two elements. The main result of the work states that every fully invariant subgroup of T _r (∞, K), which is contained in its unitriangular subgroup, coincides with some term of the lower central series of T _r (∞, K). Using this fact, we describe the verbal subgroups of this group. Moreover, we prove some facts concerning the verbal width of T _r (∞, K). In particular, we state that this width is finite.     

Non-weight representations of Cartan type S Lie algebras

For the two Cartan type S subalgebras of the Witt algebra 𝒲_n, called Lie algebras of divergence-zero vector fields, we determine all module structures on the universal enveloping algebra of their Cartan subalgebra 𝔥_n. We also give all submodules of these modules.     

A Family of Complex Irreducible Characters Possessed by Unitary and Special Unitary Groups Defined over Local Rings

Abstract

Let 𝒪 be a discrete valuation ring whose residue field 𝒪/𝔭 is finite and has odd characteristic. Let l be a positive integer. Set R = 𝒪/𝔭 ^l and let R = R[θ] be the ring obtained by adjoining to R a square root of a non-square unit. Consider the involution σ of R that fixes R elementwise and sends θ to ? θ. Let V be a free R-module of rank n > 0 endowed with a non-degenerate hermitian form ( , ) relative to σ. Let U _n (R) be the subgroup of GL(V) that preserves ( , ). Let SU _n (R) be the subgroup of all g ∈ U _n (R) whose determinant is equal to one. Let Ψ be the Weil character of U _n (R).

All irreducible constituents of Ψ are determined. An explicit character formula is given for each of them. In particular, all character degrees are computed. For n > 2 the corresponding results are also obtained for the restriction of Ψ to SU _n (R).     

Ideals and simplicity of unitary Lie algebras

Double graded ideals and simplicity of elementary unitary Lie algebra eu _n (R,⁻, γ) and Steinberg unitary Lie algebra stu _n (R,⁻, γ) are characterized, where R is a unital involutory associative algebra over a field F of characteristic zero, n ⩾ 5.     

Tangent Cone of Numerical Semigroup Rings of Embedding Dimension Three

In this article, we give new characterizations of the Buchsbaum and Cohen–Macaulay properties of the tangent cone gr _𝔪 (R), where (R, 𝔪) is a numerical semigroup ring of embedding dimension 3. In particular, we confirm the conjectures raised by Sapko on the Buchsbaumness of gr _𝔪 (R).