Balanced Deep Matrix Algebras

This paper continues the study of associative and Lie deep matrix algebras, DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) and \mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}), and their subalgebras. After a brief overview of the general construction, balanced deep matrix subalgebras, BDM(X,\mathbbK){\mathcal{BDM}}(X,{\mathbb{K}}) and \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}), are defined and studied for an infinite set X. The global structures of these two algebras are studied, devising a depth grading on both as well as determining their ideal lattices. In particular, \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) is shown to be semisimple. The Lie algebra \mathfrakbld(X,\mathbbK){\mathfrak{bld}}(X,{\mathbb{K}}) possesses a deep Cartan decomposition and is locally finite with every finite subalgebra naturally enveloped by a semi-direct product of \mathfraksl_n{\mathfrak{{sl}_n}}’s. We classify all associative bilinear forms on \mathfraksl₂\mathfrakd{\mathfrak{sl}_2\mathfrak{d}} (a natural depth analogue of \mathfraksl₂{\mathfrak{{sl}_2}}) and \mathfrakbld{\mathfrak{bld}}.     

Simple and Nearly Simple Deep Matrix Algebras

Deep matrix algebras were originally created by Cuntz (Comm. Math. Phys. 57:173–185, 1977) and McCrimmon (2006). Further study of the associative case was done by the author in Kennedy (2004) and Kennedy (Algebr. Represent. Theory 9:525–537, 2006). In this paper, the associative algebra DM(X,\mathbbK){\mathcal{DM}}(X,{\mathbb{K}}) based on a set X over a field \mathbbK{\mathbb{K}} and various of its subalgebras are studied for the purpose of determining the structure of the associated Lie algebra \mathfrakgld(X,\mathbbK){\mathfrak{gld}}(X,{\mathbb{K}}) and its subalgebras. Several key examples of deep matrix Lie algebras are constructed. These are shown to be either simple or nearly simple depending on the cardinality of the set X. Cartan subalgebras are constructed and two of the key Lie algebras are then decomposed with respect to the adjoint action of these subalgebras. In the process, an infinite dimensional analogue to \mathfraksl₂(\mathbbK)\mathfrak{sl}_2({\mathbb{K}}) is naturally realized as a key subalgebra in deep matrix Lie algebras.     

Reduced invariant sets

Let K be a compact Lie group and W a finite-dimensional real K-module. Let X be a K-stable real algebraic subset of W. Let I(X){\mathcal{I}(X)} denote the ideal of X in \mathbbR[W]{\mathbb{R}[W]} and let I_K(X){\mathcal{I}_{K}(X)} be the ideal generated by I(X)^K{\mathcal{I}(X)^{K}} . We find necessary conditions and sufficient conditions for I(X) = I_K(X){{\mathcal{I}(X) = \mathcal{I}_{K}(X)}} and for ?{I_K(X)} = I(X){{\sqrt{\mathcal{I}_{K}(X)} = \mathcal{I}(X)}} . We consider analogous questions for actions of complex reductive groups.     

Vertex algebras and the formal loop space

We construct a certain algebro-geometric version L(X)\mathcal{L}(X) of the free loop space for a complex algebraic variety X. This is an ind-scheme containing the scheme L⁰(X)\mathcal{L}^{0}(X) of formal arcs in X as studied by Kontsevich and Denef-Loeser. We describe the chiral de Rham complex of Malikov, Schechtman and Vaintrob in terms of the space of formal distributions on L(X)\mathcal{L}(X) supported in L⁰(X)\mathcal{L}^{0}(X) . We also show that L(X)\mathcal{L}(X) possesses a factorization structure: a certain non-linear version of a vertex algebra structure. This explains the heuristic principle that all linear constructions applied to the free loop space produce vertex algebras.     

Quantum Spaces Associated to Multipermutation Solutions of Level Two

We study finite set-theoretic solutions (X,r) of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over ℂ with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra A(\mathbbC,X,r)\mathcal{A}(\mathbb{C},X,r) having a q-commutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots of a prescribed form appear as determined by the representation theory of the finite abelian group G\mathcal{G} of left actions on X. We study the structure of A(\mathbbC,X,r)\mathcal{A}(\mathbb{C},X,r) and show that they have a ∙-product form ‘quantizing’ the commutative algebra of polynomials in |X| variables. We obtain the ∙-product both as a Drinfeld cotwist for a certain canonical 2-cocycle and as a braided-opposite product for a certain crossed G\mathcal{G}-module (over any field k). We provide first steps in the noncommutative differential geometry of A(k,X,r)\mathcal{A}(k,X,r) arising from these results. As a byproduct of our work we find that every such level 2 solution (X,r) factorises as r = f ∘ τ ∘ f ^− 1 where τ is the flip map and (X,f) is another solution coming from X as a crossed G\mathcal{G}-set.     

Group measure space decomposition of II1 factors and W*-superrigidity

We prove a “unique crossed product decomposition” result for group measure space II₁ factors L ^∞(X)⋊Γ arising from arbitrary free ergodic probability measure preserving (p.m.p.) actions of groups Γ in a fairly large family G\mathcal{G}, which contains all free products of a Kazhdan group and a non-trivial group, as well as certain amalgamated free products over an amenable subgroup. We deduce that if T _n denotes the group of upper triangular matrices in PSL (n,ℤ), then any free, mixing p.m.p. action of G = \operatornamePSL(n,\mathbbZ)*_Tn\operatornamePSL(n,\mathbbZ)\Gamma=\operatorname{PSL}(n,\mathbb{Z})*_{T_{n}}\operatorname{PSL}(n,\mathbb{Z}) is W^∗-superrigid, i.e. any isomorphism between L ^∞(X)⋊Γ and an arbitrary group measure space factor L ^∞(Y)⋊Λ, comes from a conjugacy of the actions. We also prove that for many groups Γ in the family G\mathcal{G}, the Bernoulli actions of Γ are W^∗-superrigid.     

An L0（F,R）-valued Function＇s Intermediate Value Theorem and Its Applications to Random Uniform Convexity

Let (Ω , F , P ) be a probability space and L0 ( F, R ) the algebra of equivalence classes of real- valued random variables on (Ω , F , P ). When L0 ( F, R ) is endowed with the topology of convergence in probability, we prove an intermediate value theorem for a continuous local function from L0 ( F, R ) to L0 ( F, R ). As applications of this theorem, we first give several useful expressions for modulus of random convexity, then we prove that a complete random normed module ( S,|| · ||) is random uniformly convex iff Lp ( S ) is uniformly convex for each fixed positive number p such that 1 p + ∞ .     

Good fuzzy preorders on fuzzy power structures

This paper deals with good fuzzy preorders on fuzzy power structures. It is shown that a fuzzy preorder R on an algebra (X,\mathbbF){(X,\mathbb{F})} is compatible if and only if it is Hoare good, if and only if it is Smyth good.     

The 2-factoriality of the O’Grady moduli spaces

The aim of this work is to show that the moduli space M ₁₀ introduced by O’Grady is a 2-factorial variety. Namely, M ₁₀ is the moduli space of semistable sheaves with Mukai vector v: = (2, 0, −2) in H^ev(X,\mathbbZ){H^{ev}(X,\mathbb{Z})} on a projective K3 surface X. As a corollary to our construction, we show that the Donaldson morphism gives a Hodge isometry between v^{^}{v^{\perp}} (sublattice of the Mukai lattice of X) and its image in H² ([(M)\tilde]₁₀, \mathbbZ){H^{2} (\widetilde{M}_{10}, \mathbb{Z})}, lattice with respect to the Beauville form of the 10-dimensional irreducible symplectic manifold [(M)\tilde]₁₀{\widetilde{M}_{10}}, obtained as symplectic resolution of M ₁₀. Similar results are shown for the moduli space M ₆ introduced by O’Grady to produce its 6-dimensional example of irreducible symplectic variety.     

Ricci-flat Kähler metrics on crepant resolutions of Kähler cones

We prove that a crepant resolution π : Y → X of a Ricci-flat Kähler cone X admits a complete Ricci-flat Kähler metric asymptotic to the cone metric in every Kähler class in ${H^2_c(Y,\mathbb{R})}We prove that a crepant resolution π : Y → X of a Ricci-flat K?hler cone X admits a complete Ricci-flat K?hler metric asymptotic to the cone metric in every K?hler class in H²_c(Y,\mathbbR){H^2_c(Y,\mathbb{R})}. A K?hler cone (X,[`(g)]){(X,\bar{g})} is a metric cone over a Sasaki manifold (S, g), i.e. ${X=C(S):=S\times\mathbb{R}_{ >0 }}${X=C(S):=S\times\mathbb{R}_{ >0 }} with [`(g)]=dr² +r² g{\bar{g}=dr^2 +r^2 g}, and (X,[`(g)]){(X,\bar{g})} is Ricci-flat precisely when (S, g) Einstein of positive scalar curvature. This result contains as a subset the existence of ALE Ricci-flat K?hler metrics on crepant resolutions p:Y? X=\mathbbC<sup>n</sup> /G{\pi:Y\rightarrow X=\mathbb{C}^n /\Gamma}, with G ì SL(n,\mathbbC){\Gamma\subset SL(n,\mathbb{C})}, due to P. Kronheimer (n = 2) and D. Joyce (n > 2). We then consider the case when X = C(S) is toric. It is a result of A. Futaki, H. Ono, and G. Wang that any Gorenstein toric K?hler cone admits a Ricci-flat K?hler cone metric. It follows that if a toric K?hler cone X = C(S) admits a crepant resolution π : Y → X, then Y admits a T <sup>n</sup> -invariant Ricci-flat K?hler metric asymptotic to the cone metric (X,[`(g)]){(X,\bar{g})} in every K?hler class in H²_c(Y,\mathbbR){H^2_c(Y,\mathbb{R})}. A crepant resolution, in this context, is a simplicial fan refining the convex polyhedral cone defining X. We then list some examples which are easy to construct using toric geometry.     

Symplectic automorphisms on Kummer surfaces

Nikulin proved that the isometries induced on the second cohomology group of a K3 surface X by a finite abelian group G of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of H²(X,\mathbbZ){H^2(X,\mathbb{Z})} which is fixed by the isometries induced by G. However for certain groups these discriminants are not the same as those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.     

Differential operators and BV structures in noncommutative geometry

We introduce a new formalism of differential operators for a general associative algebra A. It replaces Grothendieck’s notion of differential operators on a commutative algebra in such a way that derivations of the commutative algebra are replaced by \mathbbDer(A){\mathbb{D}{\rm er}(A)}, the bimodule of double derivations. Our differential operators act not on the algebra A itself but rather on F(A){\mathcal{F}(A)}, a certain ‘Fock space’ associated to any noncommutative algebra A in a functorial way. The corresponding algebra D(F(A)){\mathcal{D}(\mathcal{F}(A))} of differential operators is filtered and gr D(F(A)){\mathcal{D}(\mathcal{F}(A))}, the associated graded algebra, is commutative in some ‘wheeled’ sense. The resulting ‘wheeled’ Poisson structure on gr D(F(A)){\mathcal{D}(\mathcal{F}(A))} is closely related to the double Poisson structure on T_A \mathbbDer(A){T_{A} \mathbb{D}{\rm er}(A)} introduced by Van den Bergh. Specifically, we prove that gr D(F(A)) @ F(T_A(\mathbbDer(A)),{\mathcal{D}(\mathcal{F}(A))\cong\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)),} provided the algebra A is smooth. Our construction is based on replacing vector spaces by the new symmetric monoidal category of wheelspaces. The Fock space F(A){\mathcal{F}(A)} is a commutative algebra in this category (a “commutative wheelgebra”) which is a structure closely related to the notion of wheeled PROP. Similarly, we have Lie, Poisson, etc., wheelgebras. In this language, D(F(A)){\mathcal{D}(\mathcal{F}(A))} becomes the universal enveloping wheelgebra of a Lie wheelgebroid of double derivations. In the second part of the paper, we show, extending a classical construction of Koszul to the noncommutative setting, that any Ricci-flat, torsion-free bimodule connection on \mathbbDer(A){\mathbb{D}{\rm er}(A)} gives rise to a second-order (wheeled) differential operator, a noncommutative analogue of the Batalin-Vilkovisky (BV) operator, that makes F(T_A(\mathbbDer(A))){\mathcal{F}(T_{A}(\mathbb{D}{\rm er}(A)))} a BV wheelgebra. In the final section, we explain how the wheeled differential operators D(F(A)){\mathcal{D}(\mathcal{F}(A))} produce ordinary differential operators on the varieties of n-dimensional representations of A for all n ≥ 1.     

Character formulas for Feigin–Stoyanovsky’s type subspaces of standard $\mathfrak{sl}(3, \mathbb{C})^{\widetilde{}}$-modules

Exact sequences of Feigin–Stoyanovsky’s type subspaces for affine Lie algebra \mathfraksl(l+1,\mathbbC)^[\tilde]\mathfrak{sl}(l+1,\mathbb{C})^{\widetilde{}} lead to systems of recurrence relations for formal characters of those subspaces. By solving the corresponding system for \mathfraksl(3,\mathbbC)^[\tilde]\mathfrak{sl}(3,\mathbb{C})^{\widetilde{}}, we obtain a new family of character formulas for all Feigin–Stoyanovsky’s type subspaces at the general level.     

On Energy, Discrepancy and Group Invariant Measures on Measurable Subsets of Euclidean Space

Given $\mathcal{X}Given X\mathcal{X}, some measurable subset of Euclidean space, one sometimes wants to construct a finite set of points, P ì X\mathcal{P}\subset\mathcal {X}, called a design, with a small energy or discrepancy. Here it is shown that these two measures of design quality are equivalent when they are defined via positive definite kernels K:X²(=X×X)?\mathbbRK:\mathcal{X}^{2}(=\mathcal{X}\times\mathcal {X})\to\mathbb{R}. The error of approximating the integral ò_Xf(x) dm(x)\int_{\mathcal{X}}f(\boldsymbol{x})\,\mathrm{d}\mu(\boldsymbol{x}) by the sample average of f over P\mathcal{P} has a tight upper bound in terms of the energy or discrepancy of P\mathcal{P}. The tightness of this error bound follows by requiring f to lie in the Hilbert space with reproducing kernel K. The theory presented here provides an interpretation of the best design for numerical integration as one with minimum energy, provided that the measure μ defining the integration problem is the equilibrium measure or charge distribution corresponding to the energy kernel, K.     

On support points and continuous extensions

A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by ${\mathcal{BCC}(X)}A selection theorem concerning support points of convex sets in a Banach space is proved. As a corollary we obtain the following result. Denote by BCC(X){\mathcal{BCC}(X)} the metric space of all nonempty bounded closed convex sets in a Banach space X. Then there exists a continuous mapping S : BCC(X) ? X{S : \mathcal{BCC}(X) \rightarrow X} such that S(K) is a support point of K for each K ? BCC(X){K \in \mathcal{BCC}(X)}. Moreover, it is possible to prescribe the values of S on a closed discrete subset of BCC(X){\mathcal{BCC}(X)}.     

The Crossed Burnside Ring,the Drinfel’d Double,and the Dade Group of a <Emphasis Type="Italic">p</Emphasis>-Group

Let p be a prime number. This paper solves the question of the difference between the rank of the crossed Burnside ring B ^c (P) of a finite p-group P and of the rational representation ring R_\mathbbQ (D(P))R_{{\mathbb{Q}}} (\mathcal{D}(P)) of the Drinfel’d double D(P)\mathcal{D}(P) of the group algebra ℚ P. The difference is represented by using the Dade groups of certain subgroups of P.     

The $${{\rm GL}(2,\mathbb{C})}$$ McKay correspondence

In this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of GL(2,\mathbbC){{\rm GL}(2,\mathbb{C})}, it means that the endomorphism ring of the special CM \mathbbC{\mathbb{C}} [[x, y]] ^G -modules can be used to build the dual graph of the minimal resolution of \mathbbC²/G{\mathbb{C}^{2}/G}, extending McKay’s observation (McKay, Proc Symp Pure Math, 37:183–186, 1980) for finite subgroups of SL(2,\mathbbC){{\rm SL}(2,\mathbb{C})} to all finite subgroups of GL(2,\mathbbC){{\rm GL}(2,\mathbb{C})}.