A SURVEY OF THE ADDITIVE EIGENVALUE PROBLEM (WITH APPENDIX BY M. KAPOVICH)

The classical Hermitian eigenvalue problem addresses the following question: What are the possible eigenvalues of the sum A + B of two Hermitian matrices A and B, provided we fix the eigenvalues of A and B. A systematic study of this problem was initiated by H. Weyl (1912). By virtue of contributions from a long list of mathematicians, notably Weyl (1912), Horn (1962), Klyachko (1998) and Knutson–Tao (1999), the problem is finally settled. The solution asserts that the eigenvalues of A + B are given in terms of certain system of linear inequalities in the eigenvalues of A and B. These inequalities (called the Hom inequalities) are given explicitly in terms of certain triples of Schubert classes in the singular cohomology of Grassmannians and the standard cup product. Belkale (2001) gave a smaller set of inequalities for the problem in this case (which was shown to be optimal by Knutson–Tao–Woodward). The Hermitian eigenvalue problem has been extended by Berenstein–Sjamaar (2000) and Kapovich–Leeb–Millson (2009) for any semisimple complex algebraic group G. Their solution is again in terms of a system of linear inequalities obtained from certain triples of Schubert classes in the singular cohomology of the partial ag varieties G/P (P being a maximal parabolic subgroup) and the standard cup product. However, their solution is far from being optimal. In a joint work with P. Belkale, we define a deformation of the cup product in the cohomology of G/P and use this new product to generate our system of inequalities which solves the problem for any G optimally (as shown by Ressayre). This article is a survey (with more or less complete proofs) of this additive eigenvalue problem. The eigenvalue problem is equivalent to the saturated tensor product problem. We also give an extension of the saturated tensor product problem to the saturated restriction problem for any pair G ? ? of connected reductive algebraic groups. In the appendix by M. Kapovich, a connection between metric geometry and the representation theory of complex semisimple algebraic groups is explained. The connection runs through the theory of buildings. This connection is exploited to give a uniform (though not optimal) saturation factor for any G.     

Hilbert's Theorem 90 for Hopf Galois Extensions

For a faithfully flat extension A/B and a right A-module M, we give a new characterization of the set of descent data on M. Assuming that B is a simple Artinian ring and A/B is H-Galois, for a certain finite dimensional Hopf algebra H, we prove that Sweedler's noncommutative cohomology H ¹(H?, A) is trivial as a pointed set.     

Quantum homogeneous spaces with faithfully flat module structures

LetA be a Hopf algebra with bijective antipode andB⊃A a right coideal subalgebra ofA. Formally, the inclusionB⊃A defines a quotient mapG→X whereG is a quantum group andX a right homogeneousG-space. From an algebraic point of view theG-spaceX only has good properties ifA is left (or right) faithfully flat as a module overB. In the last few years many interesting examples of quantumG-spaces for concrete quantum groupsG have been constructured by Podleś, Noumi, Dijkhuizen and others (as analogs of classical compact symmetric spaces). In these examplesB consists of infinitesimal invariants of the function algebraA of the quantum group. As a consequence of a general theorem we show that in all these casesA as a left or rightB-module is faithfully flat. Moreover, the coalgebraA/AB ⁺ is cosemisimple.     

The Nodal Cubic is a Quantum Homogeneous Space

The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring B can be embedded as a right coideal subalgebra into a Hopf algebra A such that A is faithfully flat as a B-module. In the present article such a Hopf algebra A is constructed for the coordinate ring B of the nodal cubic, thus further motivating the question which affine varieties are quantum homogeneous spaces.     

Quantization via mirror symmetry

When combined with mirror symmetry, the A-model approach to quantization leads to a fairly simple and tractable problem. The most interesting part of the problem then becomes finding the mirror of the coisotropic brane. We illustrate how it can be addressed in a number of interesting examples related to representation theory and gauge theory, in which mirror geometry is naturally associated with the Langlands dual group. Hyperholomorphic sheaves and (B, B, B) branes play an important role in the B-model approach to quantization.     

Newton-Like Dynamics and Forward-Backward Methods for Structured Monotone Inclusions in Hilbert Spaces

In a Hilbert space setting we introduce dynamical systems, which are linked to Newton and Levenberg–Marquardt methods. They are intended to solve, by splitting methods, inclusions governed by structured monotone operators M=A+B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy–Lipschitz theorem, and involve separately B and the resolvents of A. In the convex subdifferential case, by using Lyapunov asymptotic analysis, we prove a descent minimizing property and weak convergence to equilibria of the trajectories. Time discretization of these dynamics gives algorithms combining Newton’s method and forward-backward methods for solving structured monotone inclusions.     

On the efficiency of a class of a-stable methods

The numerical solution of systems of differential equations of the formB dx/dt=σ(t)Ax(t)+f(t),x(0) given, whereB andA (withB and —(A+A ^T) positive definite) are supposed to be large sparse matrices, is considered.A-stable methods like the Implicit Runge-Kutta methods based on Radau quadrature are combined with iterative methods for the solution of the algebraic systems of equations.     

Noncommutative algebras related with Schubert calculus on Coxeter groups

For any finite Coxeter system (W,S) we construct a certain noncommutative algebra, the so-called bracket algebra, together with a family of commuting elements, the so-called Dunkl elements. The Dunkl elements conjecturally generate an algebra which is canonically isomorphic to the coinvariant algebra of the Coxeter group W. We prove this conjecture for classical Coxeter groups and I₂(m). We define a “quantization” and a multiparameter deformation of our construction and show that for Lie groups of classical type and G₂, the algebra generated by Dunkl’s elements in the quantized bracket algebra is canonically isomorphic to the small quantum cohomology ring of the corresponding flag variety, as described by B. Kim. For crystallographic Coxeter systems we define the so-called quantum Bruhat representation of the corresponding bracket algebra. We study in more detail the structure of the relations in B_n-, D_n- and G₂-bracket algebras, and as an application, discover a Pieri-type formula in the B_n-bracket algebra. As a corollary, we obtain a Pieri-type formula for multiplication of an arbitrary B_n-Schubert class by some special ones. Our Pieri-type formula is a generalization of Pieri’s formulas obtained by A. Lascoux and M.-P. Schützenberger for flag varieties of type A. We also introduce a super-version of the bracket algebra together with a family of pairwise anticommutative elements, the so-called flat connections with constant coefficients, which describes “a noncommutative differential geometry on a finite Coxeter group” in the sense of S. Majid.     

The inertia of a Hermitian pencil

Let A and B be Hermitian matrices and P = λA + μB where (λ,μ)?$\text{R}$². Using parametric dependence of the eigenvalues, we study the inertia of P under variation of (λ,μ) and under small Hermitian perturbations. In particular, we give semicontinuous dependence results for the set of (λ,μ) where inertia of P is discontinuous.     

Second cohomology groups for Frobenius kernels and related structures

Let G be a simple simply connected affine algebraic group over an algebraically closed field k of characteristic p for an odd prime p. Let B be a Borel subgroup of G and U be its unipotent radical. In this paper, we determine the second cohomology groups of B and its Frobenius kernels for all simple B-modules. We also consider the standard induced modules obtained by inducing a simple B-module to G and compute all second cohomology groups of the Frobenius kernels of G for these induced modules. Also included is a calculation of the second ordinary Lie algebra cohomology group of Lie(U) with coefficients in k.     

A note on the perturbations of compact quantum metric spaces

In this short note,we consider the perturbation of compact quantum metric spaces.We first show that for two compact quantum metric spaces(A,P) and(B,Q) for which A and B are subspaces of an order-unit space C and P and Q are Lip-norms on A and B respectively,the quantum Gromov–Hausdorff distance between(A,P) and(B,Q) is small under certain conditions.Then some other perturbation results on compact quantum metric spaces derived from spectral triples are also given.     

On the finitistic dimension conjecture II: Related to finite global dimension

In this paper, we study the finitistic dimensions of artin algebras by establishing a relationship between the global dimensions of the given algebras, on the one hand, and the finitistic dimensions of their subalgebras, on the other hand. This is a continuation of the project in [J. Pure Appl. Algebra 193 (2004) 287-305]. For an artin algebra A we denote by gl.dim(A), fin.dim(A) and rep.dim(A) the global dimension, finitistic dimension and representation dimension of A, respectively. The Jacobson radical of A is denoted by rad(A). The main results in the paper are as follows: Let B be a subalgebra of an artin algebra A such that rad(B) is a left ideal in A. Then (1) if gl.dim(A)?4 and rad(A)=rad(B)A, then fin.dim(B)<∞. (2) If rep.dim(A)?3, then fin.dim(B)<∞. The results are applied to pullbacks of algebras over semi-simple algebras. Moreover, we have also the following dual statement: (3) Let ?:B?A be a surjective homomorphism between two algebras B and A. Suppose that the kernel of ? is contained in the socle of the right B-module B_B. If gl.dim(A)?4, or rep.dim(A)?3, then fin.dim(B)<∞. Finally, we provide a class of algebras with representation dimension at most three: (4) If A is stably hereditary and rad(B) is an ideal in A, then rep.dim(B)?3.     

On the Cohomology Sequence in a Semiabelian Category

In a semiabelian category, a strictly exact sequence 0→A→B→C→0 of cochain complexes gives rise to the cohomology sequence ...→H ⁿ(A) →H ⁿ(B)→ H ⁿ(C)→ H ⁿ⁺¹ (A) →.... We study conditions for exactness of the homology sequence at a given term.     

Nondegenerate cohomology pairing for transitive Lie algebroids,characterization

The Evens-Lu-Weinstein representation (Q _A , D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to (Q _A ^or , D^or) by tensoring by orientation flat line bundle, Q _A ^or =Q_A⊗or (M) and D ^or=D⊗∂ _A ^or . It is shown that the induced cohomology pairing is nondegenerate and that the representation (Q _A ^or , D^or) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂ _A ^or ) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid cohomology with trivial coefficients (or with coefficients in the orientation flat line bundle) gives Poincaré duality. In proofs of these theorems for Lie algebroids it is used the Hochschild-Serre spectral sequence and it is shown the general fact concerning pairings between graded filtered differential ℝ-vector spaces: assuming that the second terms live in the finite rectangular, nondegeneration of the pairing for the second terms (which can be infinite dimensional) implies the same for cohomology spaces.     

Secondary Hochschild Cohomology

For a B-algebra A we introduce a Hochschild-like cohomology and use it to describe simultaneous deformations of the product and of the B-algebra structure on A[[t]]. These deformations have the property that the natural projection A[[t]]→A is a morphism of B-algebras.     

A general Trotter-Kato formula for a class of evolution operators

In this article we prove new results concerning the existence and various properties of an evolution system UA+B(t,s)_0?s?t?T generated by the sum −(A(t)+B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express UA+B(t,s)_0?s?t?T as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by −A(t) and −B(t), respectively, thereby proving a product formula of the Trotter-Kato type under very general conditions which allow the domain D(A(t)+B(t)) to evolve with time provided there exists a fixed set D⊂_?t∈[0,T]D(A(t)+B(t)) everywhere dense in B. We obtain a special case of our formula when B(t)=0, which, in effect, allows us to reconstruct UA(t,s)_0?s?t?T very simply in terms of the semigroup generated by −A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial-boundary value problems, including one related to the theory of time-dependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics.     

Singular Hochschild cohomology via the singularity category

We show that the singular Hochschild cohomology (= Tate–Hochschild cohomology) of an algebra A is isomorphic, as a graded algebra, to the Hochschild cohomology of the differential graded enhancement of the singularity category of A. The existence of such an isomorphism is suggested by recent work by Zhengfang Wang.     

Representations and cohomology for Frobenius-Lusztig kernels

Let U_ζ be the quantum group (Lusztig form) associated to the simple Lie algebra g, with parameter ζ specialized to an ?-th root of unity in a field of characteristic p>0. In this paper we study certain finite-dimensional normal Hopf subalgebras U_ζ(G_r) of U_ζ, called Frobenius-Lusztig kernels, which generalize the Frobenius kernels G_r of an algebraic group G. When r=0, the algebras studied here reduce to the small quantum group introduced by Lusztig. We classify the irreducible U_ζ(G_r)-modules and discuss their characters. We then study the cohomology rings for the Frobenius-Lusztig kernels and for certain nilpotent and Borel subalgebras corresponding to unipotent and Borel subgroups of G. We prove that the cohomology ring for the first Frobenius-Lusztig kernel is finitely-generated when g has type A or D, and that the cohomology rings for the nilpotent and Borel subalgebras are finitely-generated in general.     

Labeled reaction matrices,a histocompatibility model

A labeled reaction (or recognition) matrix is a triple (A, ø, B) where A and B are zero-one (0–1) matrices and φ is a certain relation between the rows and columns of A and B. In the application in this paper B defines the antigens and antibodies that play a role in some histocompatibility experiment by listing specificities. A represents data observed from testing cells against sera in the experiment. Antigens or antibodies (not necessarily monospecific) whose action in the experiment can be isolated are called monic. (A, φ, B) is called monic if all antigens and antibodies are monic and B is reduced. A partial order ⩽ is put on the collection of 0–1 matrices and it is shown that if (A, φ, B) is any labeled reaction matrix, then A_ω ⩽ B where A_ω is the reduction of A. An algorithm for obtaining A_ω (that gives a labeling of A) is provided. If (A, φ, B) is monic, then A_ω and B are identical (up to a permutation of rows and columns) and the labeling of cells and sera is essentially unique.     

Hochschild cohomology and stratifying ideals

Suppose B is an algebra with a stratifying ideal BeB generated by an idempotent e. We will establish long exact sequences relating the Hochschild cohomology groups of the three algebras B, B/BeB and eBe. This provides a common generalization of various known results, all of which extend Happel’s long exact sequence for one-point extensions. Applying one of these sequences to Hochschild cohomology algebras modulo the ideal generated by homogeneous nilpotent elements shows, in some cases, that these algebras are finitely generated.