K-vector Space Research Articles

A useful semistability criterion

We provide a short proof for a semistability criterion which is crucial to the construction of master spaces which has drawn interest in recent research in Geometric Invariant Theory. Among the main objects of study of Geometric Invariant Theory are representations of reductive algebraic groups. Given a reductive algebraic group G, defined over a field k, a finite dimensional k-vector space W, and a rational representation p: G GL(W), one obtains an action of G on IP(W), the space of hyperplanes in W, and a representation of G on the algebra R := Sym W. By the Hilbert-Nagata theorem, the algebra RG of G-invariant elements in R is a finitely generated graded subalgebra of R. Therefore, IP(W)//G Proj RG is a projective variety, and there is a rational map 7r: IP(W) --+ IP(W)//G. This is the GIT procedure of forming a quotient. A closed point [w] C EP(W) is called semistable, if 7r is defined in [w], i.e., if there is a non-constant homogeneous element in RG not vanishing in [w]. The set IP(W)88 of semistable points is open, and a point [w] c IP(W) is called polystable, if [w] is semistable and the G-orbit of [w] is closed in P(W)88. The map 7r identifies the closed points of IP(W)//G with the set of G-orbits of polystable points. A central task of GIT is now to identify the semistable and polystable points in IP(W). This is usually achieved by the Hilbert-Mumford criterion [1]. However, if p is the direct sum of other representations, this might become too difficult. In this note, we will provide a short and elementary proof of a theorem from [2], dealing with this situation. More precisely, let k and G be as before, and W1,...,W, finite dimensional k-vector spaces. Suppose we are given representations pi: G GL(Wi), i = 1, ..., s, with W = W1 ... ED W, and p = p, ED * * ffl For any t = (Li, ...,tt) with 0 0. In all these cases, one can safely apply the Hilbert-Mumford criterion. The outcome is that the semistable points correspond to semistable L-oriented pairs, using the formulation of semistability in terms of the parameter dependent semistability concept for Bradlow pairs. This content downloaded from 157.55.39.19 on Tue, 14 Jun 2016 05:30:35 UTC All use subject to http://about.jstor.org/terms A USEFUL SEMISTABILITY CRITERION 1925 PROOF OF THE THEOREM We first prove the semistable part. By definition, the point w is semistable if and only if, for some k > 0, there is an invariant section in H0((9(k)) = SkW which does not vanish in w. Now, SkW = Oki+*+k,=k Ski W0 (g ... S S . Ws and the representation of G on SkW respects this decomposition, i.e., (SkW)G (~kl+--+k =k(SkiWl (... (? SksWs)G. Therefore, w is semistable if and only if we find a section in some (Sk~l W,, o . Sklt Wt )G which does not vanish in w. This is precisely the assertion of the theorem. Note that we have used only the very definition of semistability. The polystable part will follow easily by means of an induction from the following ClaiR. Let w' ([w= 1,v WL2], [WL3], ..., [wjt]) be a point in P(WVV, e W142) X P(13 t. ) Then w' is semistable (polystable) w.r.t. the given linearization in ((k, k3, ..., kt), if and only if either ([WI, [wI 3 ],) ..., [wI J]) is semistable (polystable) in P(.tt) w.r.t. the linearization in O (k, k3I ...., kt) for either i = 1 (and w,2 = 0) or i = 2 (and W,, = 0), or there are positive natural numbers n, k1, and k2, such that k[ +-k2 = nk and the point (Q[W,], [W62], [WL3], ..., [wI ]W1) is semistable (polystable) in P(,1,2,,3.tt) w.r.t. the linearization in O (kj, k2, nk3, ..., nkt). The semistable part is proved as before. First, we suppose that w' is polystable. With the Hilbert-Mumford criterion [1], the claim can be rephrased as follows: There exist non-negative rational numbers K, and K2, not both zero, with fK1 + r/2= k such that for every one-parameter subgroup A of G, we have

Hochschild Cohomology of Triangular Matrix Algebras

We study the Hochschild cohomology of triangular matrix rings B=R0AMRA , where A and R are finite dimensional algebras over an algebraically closed field K and M is an A-R-bimodule. We prove the existence of two long exact sequences of K-vector spaces relating the Hochschild cohomology of A, R, and B.

Theta correspondence for unitary groups

In this paper we study the theta correspondence for Unitary groups ofthe same size over local and global fields. This correspondence has been studied in many cases by several authors. We are able to unify and generalise all these known results in terms oftwo conjectures, one local and the other global. These conjectures are in terms ofthe parametrisation ofirreducible admissible representations ofgroups over local fields which are formulated by David Vogan refining Langlands parametrization, and which are now called Vogan parameters. In turn, the simple form of the conjecture here, gives support to the importance ofVogan’s refinement of Langlands parametrisation. 1. Generalities. Let K be a quadratic extension of a local field k, and let V and W be two finite dimensional Hermitian spaces over K. Let δ ∈ K ∗ be fixed with tr(δ) = 0. Multiplication by δ turns the Hermitian space W into a skewHermitian space, and therefore we get a symplectic structure on the kvector space V ⊗K W , making (U (V ) ,U (W )) into a dual reductive pair in Sp(V ⊗K W ). In the paper [HKS], Harris, Kudla and Sweet have made a detailed studyof the C ∗ -metaplectic covering of Sp(V ⊗K W ) restricted to U (V ) × U (W ). Theyprove that the C ∗ -metaplectic cover of Sp(V ⊗K W ) splits over U (V ) × U (W ) for non-Archimedean local fields, and that the splittings over U (V ) × U (W ) can be parametrised bypairs of characters

Simple Lie Algebras of Special Type

Let K be a field, let A be an associative, commutative K-algebra, and let Δ be a nonzero K-vector space of commuting K-derivations of A. Then, with a rather natural definition, W(A,Δ)=A⊗KΔ=AΔ becomes a Lie algebra, a Witt type algebra. In addition, there is a map div: W(A,Δ)→A called the divergence and its kernel S=S(A,Δ) is a Lie subalgebra, a special type algebra. In this paper, we study S from a ring theoretic point of view, obtaining sufficient conditions for the Lie simplicity of [S,S]. While the main result here is somewhat cumbersome to state, it does handle a number of examples in a fairly efficient manner. Furthermore, some of the preliminary lemmas are of interest in their own right and may, in time, lead to a more satisfactory answer.

Partial Representations and Partial Group Algebras

The partial group algebra of a group G over a field K, denoted by Kpar(G), is the algebra whose representations correspond to the partial representations of G over K-vector spaces. In this paper we study the structure of the partial group algebra Kpar(G), where G is a finite group. In particular, given two finite abelian groups G1 and G2, we prove that if the characteristic of K does not divide the order of G1, then Kpar(G1) is isomorphic to Kpar(G2) if and only if G1 is isomorphic to G2.

A Five-term Exact Sequence for Opext of Crossed Modules in Lie Algebras

Extensions of crossed modules in Lie algebras with abelian kernel are studied, particularly backward and forward induced extensions and related properties. The set Opext((U, Q, ɛ), (R, K, ∂)) of congruence classes of extensions of (R, K, ∂) by (U, Q, ɛ) is endowed with a K-vector space structure. This K-vector space appears in a five-term natural and exact sequence associated with an extension of crossed modules.

A Five-term Exact Sequence for Opext of Crossed Modules in Lie Algebras

Extensions of crossed modules in Lie algebras with abelian kernel are studied, particularly backward and forward induced extensions and related properties. The set Opext((U, Q, ɛ), (R, K, ∂)) of congruence classes of extensions of (R, K, ∂) by (U, Q, ɛ) is endowed with a K-vector space structure. This K-vector space appears in a five-term natural and exact sequence associated with an extension of crossed modules.

Tree structure on the set of multiplicative seminorms of Krasner

Let K be an algebraically closed field, complete for an ultra- metric absolute value, let D be an infinite subset of K and let H(D) be the set of analytic elements on D. We denote by Mult(H(D), UD) the set of semi-norms P of the K-vector space H(D) which are continuous with respect to the topology of uniform convergence on D and wich satisfy further P(f g)=P(j) P(g) whenever f,g concern H(D) such that fg concern H(D). This set is provided with the topology of simple convergence. By the way of a metric topology thinner than the simple convergence, we establish the equivalence between the connectedness of Mult(H(D), UD), the arc-connectedness of Mult(H(D), UD) and the infraconnectedness of D. This generalizes a result of Berkovich given on affinoid algebras. Next, we study the filter of neighbourhoods of an element of Mult(H(D), UD), and we give a condition on the field K such that this filter admits a countable basis. We also prove the local arc-connectedness of Mult(H(D), UD) when D is infraconnected. Finally, we study the metrizability of the topology of simple convergence on Mult(H(D), UD) and we give some conditions to have an equivalence with the metric topology defined above. The fundamental tool in this survey consists of circular filters.

Projective Schur Algebras of Nilpotent Type Are Brauer Equivalent to Radical Algebras

Let k be a field and G a finite group . For any element α ∈ H(G, k∗) (k∗ denotes the units of k , here as a trivial G-module) we can form the twisted group algebra kG. It is isomorphic to the group algebra as a vector space and its multiplication satisfies the rule xuσyuτ = xyf(σ, τ)uστ where f : G × G → k∗ is a 2-cocycle representing α. It is easily checked that the 2-cocycle condition satisfied by f insures the associativity of the multiplication in kG. A projective Schur algebra over k is by definition a k-central simple algebra which is a homomorphic image of a twisted group algebra kG for some finite group G and some α ∈ H(G, k∗). Equivalently, a k− central simple algebra A is a projective Schur algebra (over k) if it contains a subgroup Γ of the units of A, which is finite modulo k∗ and spans A as k-vector space. We often use the notation k(Γ). Clearly a projective Schur algebra over k determines an element in the Brauer group Br(k) of the field k . The subgroup in Br(k) generated by (and in fact consisting of) such elements is called the projective Schur group of k and denoted by PS(k). The notions of projective Schur algebra and projective Schur group were introduced in [6] by Lorenz and Opolka. This construction can be viewed as the projective analogue of Schur algebras and the Schur group (denoted by S(k)) of the field k. (See [10]) It is natural to restrict attention to special types of groups G. Our interest here is in k-central simple algebras that are homomorphic images of twisted group algebras kG with G nilpotent (resp. abelian) (see [6]). We call these algebras “projective Schur algebras of nilpotent type (resp. abelian type)”. Clearly, as above, these algebras represent classes in Br(k) and we denote by PNil(k) ( resp. PAb(k)) the subgroup they generate. In [6] it is shown that if k is a number field and n is the number of roots of unity in k then PAb(k) = nBr(k), where nB denotes the n-torsion subgroup of B. More generally, since a symbol algebra of degree n over k is a homomorphic

Toric degenerations and vector bundles

There are many affine subalgebras of polynomial rings with highly non-trivial projective modules, whose initial algebras (toric degenerations) are still finitely generated and have all projective modules free. Let k[X1, . . . , Xn] be a polynomial algebra (k a field, n ∈ N) and A an affine k-subalgebra. Let ≺ denote a term order on the multiplicative semigroup of monomials in the Xi and let in≺(A) denote the monomial subalgebra of k[X1, . . . , Xn], generated by the leading monomials of elements f ∈ A with respect to ≺. In case the initial algebra in≺(A) is finitely generated, one can obtain many properties of A by checking them for in≺(A) (called sometimes a toric degeneration of A) (see [CHV], [RS]). However, this is not the case for the property ‘all projective modules are free’ – thanks to Bernd Sturmfels for asking me this question. Theorem 1. Let A = k[X, Y, Z, Z − XY Z] and ≺ be the lexicographic term order corresponding to Z ≺ Y ≺ X. Then SK0(A) = Ker(K0(A) det → Pic(A)) is not trivial (equivalently, there are projective A-modules which are not even stably of type free⊕rank 1), while in≺(A) is finitely generated and all projective in≺(A)-modules are free. (Here ‘projective’ includes ‘finitely generated’.) Proof. Since (Z2−XY )k[X, Y, Z] ⊂ A, the following diagram with the upper horizontal identity embedding is a pull-back diagram (all the letters refer to variables) A −−−−→ k[X, Y, Z] y y k[U, V ] −−−−→ k[S, ST, T ] , where X 7→ S, Y 7→ T , Z 7→ ST , U 7→ S, V 7→ T . It is similarly easy to show that in≺(A) = k[X, Y, Z, XY Z] – a seminormal monomial algebra (as a kvector space it is generated by the normal monomial subalgebras k[X, Y ], k[X, Z], k[Y, Z] and k[{XaY bZc|a > 0, b > 0, c > 0}]). So by [Gu1] projective modules over in≺(A) are free, while the Mayer-Vietoris sequence (see [Bass], p.490), applied to the diagram above, implies SK0(A) = SK1(k[S, ST, T ]). Hence, by [Gu2] SK0(A) 6= 0. Received by the editors February 20, 1998. 1991 Mathematics Subject Classification. Primary 13D15, 19A49. This research was supported in part by the Alexander von Humboldt Foundation and CRDF grant #GM1-115. c ©1999 American Mathematical Society

Frobenius Partitions in Extraspecial Groups

Let K be a field and G be the group of the upper unitriangular (n + 2) × ( n + 2) K-matrices with nonzero entries only in the first row and in the last column. Then G has a normal subgroup N with a complement H which are K-vector spaces respectively of dimensions n + 1 and n. In the present paper we show that the orbit of H under a group of automorphisms of G together with N, forms a partition of G, provided that there exists a commutative (possibly nonassociative) division algebra of dimension n + 1 over K. This algebra exists when K is a finite field.

Hochschild cohomology of truncated quiver algebras*

Consider a truncated quiver algebra , where Γ is a finite quiverJ is the bilateral ideal of kΓ generated by the arrows, and k, is a field with characteristic zero. We give another description of a minimal projective resolution of Λ as a Λe-module given in [2] and [8], and using this description we compute the dimensions, as a vector space, of each Hochschild cohomology group of the truncated quiver algebra, in terms of its quiver Γ. We also prove that the Hochschild cohomology ring of Λ is finite dimensional as a k-vector space if and only if Γ has no oriented cycles.

Jacobson–Bourbaki Theory with Parameters

In this paper we generalize the classical Jacobson–Bourbaki correspondence for field extensions to K-vector space extensions (K a field). Namely, for such extensions, we state and prove a correspondence theorem – the Jacobson–Bourbaki correspondence with parameters. Some examples are provided.

Simple Lie Color Algebras of Witt Type

LetKbe a field and let ε: Γ×Γ→K•be a bicharacter defined on the multiplicative group Γ. We suppose thatAis a Γ-graded, associativeK-algebra that is color commutative with respect to ε. Furthermore, let Δ be a nonzero Γ-graded,K-vector space of color derivations ofAand suppose that Δ is also color commutative with respect to the bicharacter ε. Then, with a rather natural definition,A⊗KΔ=AΔ becomes a Lie color algebra, and we obtain necessary and sufficient conditions here for this Lie color algebra to be simple. With two minor exceptions when dimKΔ=1, simplicity occurs if and only ifAis graded Δ-simple andAΔ⊗Δ=AΔΔ acts faithfully as color derivations onA.

New types of bialgebras arising from the hopf equation

Let M be a k-vector space and R ∈ End k (M ⊗ M)In [13] we introduced and studied what we called the Hopf equation:R 12 R 23 = R 23 R 13 R 12.By means of a FRT type theorem, we have proven that the category of H-Hopf modules is deeply involved in solving this equation. In the present paper, we continue to study the Hopf equation from another perspective:having in mind the quantum Yang-Baxter equation, in the solution of which the co-quasitriangular (or braided) bialgebras play an important role (see [8]), we introduce and study what we call bialgebras with Hopf functions. The main theorem of this paper shows that, if M is finite dimensional, any solution R of the Hopf equation has the form R = Rσ. where M is a right comodule over a bialgebra with a Hopf function (B{R),C,σ) and R σ is the special map .

Linear ultrafilters

Let X be a k-vector space, and U a maximal proper filter of subspaces of X. Then the ring of endomorphisms of X that are “continuous” with respect to U modulo the ideal of those that are “trivial” with respect to U forms a division ring E(U). (These division rings can also be described as the endomorphism rings of the simple left End(X)-modules.) We study this and the dual construction, based on maximal cofiIters of subspaces of X, in particular, the relation between the constructed division rings and the original field or division ring k. We end by examining a more general construction in which X is a module over a general ring, given with both a filter and a cofilter of submodules.

On the codimension of the image of a derivation in a communicative field

Let δ be a derivation in a field K of characteristic 0, with fixed field k. We show that, if K/k is finitely generated, then the codimension, as k-vector space, of K δ + k in K is infinite, while in the infinitely generated case, the said codimension can be any finite non-negative integer.

Bialgebras Over Noncommutative Rings and a Structure Theorem for Hopf Bimodules

It is a key property of bialgebras that their modules have a natural tensor product. More precisely, a bialgebra over k can be characterized as an algebra H whose category of modules is a monoidal category in such a way that the underlying functor to the category of k-vector spaces is monoidal (i.e. preserves tensor products in a coherent way). In the present paper we study a class of algebras whose module categories are also monoidal categories; however, the underlying functor to the category of k-vector spaces fails to be monoidal. Instead, there is a suitable underlying functor to the category of B-bimodules over a k-algebra B which is monoidal with respect to the tensor product over B. In other words, we study algebras L such that for two L-modules V and W there is a natural tensor product, which is the tensor product V⊗BW over another k-algebra B, equipped with an L-module structure defined via some kind of comultiplication of L. We show that this property is characteristic for ×B-bialgebras as studied by Sweedler (for commutative B) and Takeuchi. Our motivating example arises when H is a Hopf algebra and A an H-Galois extension of B. In this situation, one can construct an algebra L:=L(A,H), which was previously shown to be a Hopf algebra if B=k. We show that there is a structure theorem for relative Hopf bimodules in the form of a category equivalence \(_A M_A^H \cong {\text{ }}_L M\). The category on the left hand side has a natural structure of monoidal category (with the tensor product over A) which induces the structure of a monoidal category on the right hand side. The ×B-bialgebra structure of L that corresponds to this monoidal structure generalizes the Hopf algebra structure on L(A,H) known for B=k. We prove several other structure theorems involving L=L(A,H) in the form of category equivalences \(^L M \cong ^H M,{\text{ }}_A^L M \cong _k M{\text{, }}_A^L M_A \cong M_H ,{\text{ }}_A^L M^H \cong {\text{ }}M^H {\text{ and }}_A^L M_A^H \cong {\text{ }}YD_H^H \).

K-Theory of Endomorphism Rings and of Rings of Invariants

This paper gives the following description ofK0of the endomorphism ring of a finitely generated projective module.Theorem.Let T be a ring and P a finitely generated,projective T-module. Let I be the trace ideal of P. Then K0(EndPT)is isomorphic to a subgroup of K0(T,I).If,further,the natural map K1(T)→K1(T/I)is surjective then K0(EndPT)is isomorphic to the subgroup of K0(T)generated by the direct summands of Pn,for n∈N.As a corollary we can determineK0of the ring of invariants for many free linear actions. In particular, the following result is proved.Theorem.Let V be a fixed-point-free linear representation of a finite group G over a field k of characteristic zero and let S(V)be the symmetric algebra of V.Let K be any finite-dimensional k-vector space. Then K0(S(V)G⊗kS(K))=〈[S(V)G⊗kS(K)]〉.Similar results are given for suitable noncommutative versions ofS(V).

Quasitilted algebras admit a preprojective component

Quasitilted algebras are generalizations of tilted algebras. As a main result we show here that the Auslander-Reiten quiver of such an algebra has a preprojective component Let k be an algebraically closed field and A be a finite-dimensional k-algebra. We denote by mod A the category of finitely generated left A-modules. By 1A we denote the Auslander-Reiten quiver of A. Recall that the vertices of 1A correspond to the isomorphism classes of indecomposable finitely generated A-modules. The number of arrows from an indecomposable A-module X to an indecomposable Amodule Y is the dimension of the k-vector space rad(X,Y)/rad2(X,Y), where rad(-, -) denotes the Jacobson radical of mod A. We denote by TX = DTr X the Auslander-Reiten translate of the indecomposable A-module X. This is defined for each indecomposable module, and in case X is non-projective the translate TX will be indecomposable and non-injective. Dually there is defined T-X = TrDX. A connected component P of 1A is called a preprojective component if P does not contain an oriented cycle and each X c P is of the form T-rp for some r c N and an indecomposable projective A-module P. For details see [ARS]. The existence of preprojective components has been established for various classes of algebras such as tilted algebras [St] or algebras satisfying the separation condition [B]. It is well known that tilted algebras may have several preprojective components. One of the important features of an indecomposable module X lying in a preprojective component is that X is homologically trivial, i.e. ExtX(X, X) = 0 for i > 0 and EndAX = k and that its isomorphism class is uniquely determined by the composition factors. Also these modules can be constructed easily by the socalled knitting procedure. Quasitilted algebras have been introduced and investigated in [HRS2]. Recall that a finite-dimensional k-algebra A is called a quasitilted algebra if there exist a hereditary abelian k-category XH and a tilting object T c XH such that A = End'HT. In this article we will not work with this definition but rather with the homological characterization established in [HRS2]. We will use the following notation. For Received by the editors March 31, 1995 and, in revised form, November 10, 1995. 1991 Mathematics Subject Classification. Primary 16G10, 16E10.