On the global dimension of an algebra

Let algebra R = Λ/P, where Λ is a free algebra over a field w. gl. dim R: = {min n ¦? R-modules X, Y, Tor _n+1 ^R (X, Y)=0}. In order that w. gl. dim R≤2n (w. gl. dim R≤2n+1), it is necessary and sufficient that, for any two ideals of algebra Λ, a left ideal A and a right ideal B, containing ideal P, the following equation holds: $$AP^n \cap P^n B = AP^n B + P^{n + 1} (AP^n B \cap P^{n + 1} = AP^{n + 1} + P^{n + 1} B).$$      

The Krull-Gabriel dimension of the representation theory of a tame hereditary Artin algebra and applications to the structure of exact sequences

For an Artin Algebra of finite representation type, the category -mod, considered as a ring with several objects, has Krull dimension zero. Contrary, for a wild hereditary Artin Algebra this dimension does not exist. In this paper we show that the Krull dimension of -mod for an Artin Algebra of tame representation type is two. The corresponding Krull-Gabriel filtration by Serre subcategories of the category F of finitely presented contravariant functors on -mod leads to a hierarchy of exact sequences in -mod. Influenced by the functorial approach to almost split sequences by M. Auslander and I. Reiten, we investigate the exact sequences whose corresponding functors become simple in one of the successive quotient categories of the Krull-Gabriel filtration of F.     

On the dimension of the algebra generated by a boolean matrix

We investigate the subspace of the space of all n × n Boolean (0,1)-matrices, spanned by the powers of an arbitrary matrix. We estimate the maximum dimension of such spaces as a function of n and show that their bases consists of consecutive integer powers of the matrix, starting at I. We also determine the maximum dimension of the space spanned by the powers of as symmetric matrix and characterise the matrices achieving that maximum.     

On the dimension of the algebra generated by a boolean matrix

We investigate the subspace of the space of all n × n Boolean (0,1)-matrices, spanned by the powers of an arbitrary matrix. We estimate the maximum dimension of such spaces as a function of n and show that their bases consists of consecutive integer powers of the matrix, starting at I. We also determine the maximum dimension of the space spanned by the powers of as symmetric matrix and characterise the matrices achieving that maximum.     

Krull dimension of the enveloping algebra of a semisimple Lie algebra

Let be a complex semisimple Lie algebra and be its enveloping algebra. We deduce from the work of R. Bezrukavnikov, A. Braverman and L. Positselskii that the Krull-Gabriel-Rentschler dimension of is equal to the dimension of a Borel subalgebra of .

     

On the Lie algebra of skew-symmetric elements of an enveloping algebra

Let L be a restricted Lie algebra over a field of characteristic p>2 and denote by u(L) its restricted enveloping algebra. We establish when the Lie algebra of skew-symmetric elements of u(L) under the principal involution is solvable, nilpotent, or satisfies an Engel condition.     

On the symmetric algebra of an ideal

The symmetric algebra of an ideal I may be compared to the Rees algebra via the canonical epimorphism :Sym(I)R(I). A necessary and sufficient criterion is given for a to be an isomorphism, and sequential conditions on the symmetric algebra are studied. Some applications are given to Proj :ProjR(I)Pro'j Sym(I) and to the theory of approximation complexes.The material presented in this paper constitutes part of the author's thesis submitted to Universität Essen     

On the socle of an endomorphism algebra

The socle of an endomorphism algebra of a finite dimensional module of a finite dimensional algebra is described. The results are applied to the modular Hecke algebra of a finite group with a cyclic Sylow subgroup.     

Finite Schur filtration dimension for modules over an algebra with Schur filtration

Let G = GL _N or SL _N as reductive linear algebraic group over a field k of characteristic p > 0. We prove several results that were previously established only when N ⩽ 5 or p > 2  ^N : Let G act rationally on a finitely generated commutative k-algebra A and let grA be the Grosshans graded ring. We show that the cohomology algebra H ^*(G, grA) is finitely generated over k. If moreover A has a good filtration and M is a Noetherian A-module with compatible G action, then M has finite good filtration dimension and the H ⁱ (G, M) are Noetherian A ^G -modules. To obtain results in this generality, we employ functorial resolution of the ideal of the diagonal in a product of Grassmannians.     

On the first Hochschild cohomology group of an algebra

Let A≡KΔ /I be a factor of a path algebra. We develop a strategy to compute dim H ¹(A), the dimension of the first Hochschild cohomology group of A, using combinatorial data from (Δ,I). That allows us to connect dim H ¹(A) with the rank and p-rank of the fundamental group π₁(Δ,I) of (Δ,I). We get explicit formulae for dim H ¹(A), when every path in Δ parallel to an arrow belongs to I or when I is homogeneous. Received: 12 April 1999 / Revised version: 9 October 2000     

The dimension of the tensor algebra of a projective bimodule

It is proved that the dimension of the tensor algebra of a bimodule which is projective with one of the sides differs from the dimension of the ring over which this bimodule is considered by not more than one. A criterion for these dimensions to coincide is given.     

GK dimension of the relatively free algebra for sl_2

Let \(sl_2(K)\) be the Lie algebra of the \(2\times 2\) traceless matrices over an infinite field \(K\) of characteristic different from 2, denote by \(R_m= R_m(sl_2(K))\) the relatively free (also called universal) algebra of rank \(m\) in the variety of Lie algebras generated by \(sl_2(K).\) In this paper we compute the Gelfand–Kirillov dimension of the Lie algebra \(R_m(sl_2(K)).\) It turns out that whenever \(m\ge 2\) one has \(\mathrm{GK}\dim R_m = 3(m-1).\) In order to compute it we use the explicit form of the Hilbert series of \(R_m\) described by Drensky. This result is new for \(m>2\) ; the case \(m=2\) was dealt with by Bahturin in 1979.     

On the normalizers of subalgebras in an infinite Lie algebra

An algorithm for calculating the normalizer of subalgebra in an infinite Lie symmetry algebra is proposed. The classification problem for a subalgebra spanned by generators that depend on arbitrary functions is formulated. This problem lies in finding the specifications of arbitrary functions and calculating the normalizers of the subalgebras so obtained. As an example, we consider the Lie symmetry algebra L admitted by the thermal diffusion equations. The first-order optimal system of subalgebras Θ₁L is constructed and the normalizers of finite subalgebras from this system are found. The classification of subalgebras depending on arbitrary functions is made.