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Let X denote a quasi-projective variety over a field on which a connected linear algebraic group G acts with finitely many orbits. Then, the G-orbits define a stratification of X. We establish several key properties of the category of equivariant perverse sheaves on X, which have locally constant cohomology sheaves on each of the orbits. Under the above assumptions, we show that this category comes close to being a highest weight category in the sense of Cline, Parshall and Scott and defines a quasi-hereditary algebra. We observe that the above hypotheses are satisfied by all toric varieties and by all spherical varieties associated to connected reductive groups over any algebraically closed field. Next we show that the odd dimensional intersection cohomology sheaves vanish on all spherical varieties defined over algebraically closed fields of positive characteristics, extending similar results for spherical varieties defined over the field of complex numbers by Michel Brion and the author in prior work. Assuming that the linear algebraic group G and the action of G on X are defined over a finite field F q , and where the odd dimensional intersection cohomology sheaves on the orbit closures vanish, we also establish several basic properties of the mixed category of mixed equivariant perverse sheaves so that the associated terms in the weight filtration are finite sums of the shifted equivariant intersection cohomology complexes on the orbit closures.

Let sl N ˆ ( C q ) be the core of extended affine Lie algebra of type A N − 1 coordinated by the rational quantum 2-torus C q . In this paper, we first prove that for any complex number l, the category of restricted sl N ˆ ( C q ) -modules of level l is canonically isomorphic to the category of twisted modules for the vertex algebra V C g ˆ ( l , 0 ) arising from a conformal Lie algebra C g , where C g ˆ is isomorphic to a toroidal Lie algebra. Then we prove that for any nonnegative integer l, the integrable restricted sl N ˆ ( C q ) -modules of level l are exactly the twisted modules for the quotient vertex algebra L C g ˆ ( l , 0 ) of V C g ˆ ( l , 0 ) . Finally, we classify irreducible graded twisted L C g ˆ ( l , 0 ) -modules.

The additive monoid R + ( x ) is defined as the set of all nonnegative integer linear combinations of binomial polynomials ( x n ) for n ∈ Z + . This paper is concerned with the inquiry into the structure of R + ( α ) for complex numbers α. Particularly interesting is the case of algebraic α which are not non-negative integers. This question is motivated by the study of functors between Deligne categories Rep ( S t ) (and also Rep ( GL t ) ) for t ∈ C ﹨ Z + . We prove that this object is a ring if and only if α is an algebraic number that is not a nonnegative integer. Furthermore, we show that all algebraic integers generated by α, i.e. all elements of O Q ( α ) , are also contained in this ring. We also give two explicit representations of R + ( α ) for both algebraic integers and general algebraic numbers α. One is in terms of inequalities for the valuations with respect to certain prime ideals and the other is in terms of explicitly constructed generators. We show how these results work in the context of the study of symmetric monoidal functors between Deligne categories in positive characteristic. Moreover, this leads to a particularly simple description of R + ( α ) for both quadratic algebraic numbers and roots of unity.

We show that any two planar cubic embeddings of a pseudoplanar curve of genus 1 (over complex numbers) are linearly equivalent. We describe the ‘abstract moduli’ of nonsingular affine plane cubic curves with 3 places at infinity. We note some examples of inequivalent planar embeddings of a nonsingular affine curve of genus 0 having 2 places at infinity.

Involving a symmetric Hochschild 1-cocycle condition, we equip the space of decorated planar rooted forests with a coproduct which turns the space into a dendriform-Nijenhuis bialgebra. We combine dendriform-Nijenhuis bialgebras with operated algebras and introduce the notation of an Ω-operated DN-bialgebra. Applying the universal property of the underlying operated algebras, we construct free objects in the category of Ω-cocycle DN-bialgebras. We introduce the notation of a DN-associative Yang-Baxter equation (AYBE) and show that the dendriform-Nijenhuis bialgebra offers an algebraic framework of the DN-associative Yang-Baxter equation. We construct a Leroux's TD operator from a solution of the DN-AYBE. We also give two different ways to derive Lie algebras from quasitriangular DN-bialgebras. Finally, we classify the solutions of the DN-AYBE in the unitary algebras of dimensions two and three over the field of complex numbers.

It is well known that for two univariate polynomials over the complex number field the number of their common roots is equal to the order of their resultant. In this paper, we show that this fundamental relationship still holds for the tropical polynomials under suitable adaptation of the notion of order, if the roots are simple and non-zero., Ministerio de Economía y Competitividad

The study of singularities of algebraic or analytic spaces over the field of complex numbers is a traditional subject, but, in contrast, the parallel theory over arbitrary algebraically closed fields is still poor and there are lots of interesting questions to be answered. Our main concern here is the study of the Milnor number μ of an isolated hypersurface singularity which is defined as the codimension of the ideal generated by the partial derivatives of a power series that represents locally the hypersurface. This is an important topological invariant of the singularity over the complex numbers, but its behavior changes dramatically when the base field has positive characteristic, in which case it may be infinite and depends upon the local equation of the hypersurface, not being an intrinsic invariant. In this paper we will study the variation of the Milnor number in terms of the local equations, showing that its minimum value has an intrinsic meaning and give necessary and sufficient conditions for its invariance. We also relate the smoothness of the generic fiber of an isolated hypersurface singularity deformation with the finiteness of this number, connecting it to a Bertini type result. At the end, we will show how, in arbitrary characteristic, one defines the sequence μ ⁎ of Milnor numbers of sections of a hypersurface singularity by general linear spaces of increasing codimension.

We classify finite irreducible conformal modules over a class of infinite Lie conformal algebras ${\frak {B}}(p)$ of Block type, where $p$ is a nonzero complex number. In particular, we obtain that a finite irreducible conformal module over ${\frak {B}}(p)$ may be a nontrivial extension of a finite conformal module over ${\frak {Vir}}$ if $p=-1$, where ${\frak {Vir}}$ is a Virasoro conformal subalgebra of ${\frak {B}}(p)$. As a byproduct, we also obtain the classification of finite irreducible conformal modules over a series of finite Lie conformal algebras ${\frak b}(n)$ for $n\ge1$., 15 pages

In this paper, we study the representation theory of the Fuss–Catalan algebras, FC n ( a , b ) . We prove that this algebra is cellular with a cellular basis and forms a tower of recollement, as defined by Cox, Martin, Parker, and Xi [7] , and hence, it is quasi-hereditary algebra if a , b are non-zero complex numbers.

Simple finite dimensional Kantor triple systems over the complex numbers are classified in terms of Satake diagrams. We prove that every simple and linearly compact Kantor triple system has finite dimension and give an explicit presentation of all the classical and exceptional systems., Comment: 46 pages, 3 tables; v2: Major revision of the introduction; v3: Final version to appear in Journal of Algebra

Starting with a highest weight representation of a Kac–Moody group over the complex numbers, we construct a monoid whose unit group is the image of the Kac–Moody group under the representation, multiplied by the nonzero complex numbers. We show that this monoid has similar properties to those of a J -irreducible reductive linear algebraic monoid. In particular, the monoid is unit regular and has a Bruhat decomposition, and the idempotent lattice of the generalized Renner monoid of the Bruhat decomposition is isomorphic to the face lattice of the convex hull of the Weyl group orbit of the highest weight.

In this paper, we constructed a new family of Virasoro modules Ω ( λ , α , h ) parameterized by complex numbers λ ≠ 0 , α ∈ C and a polynomial h ( t ) . They are a special type of modules on which the non-central Cartan part acts freely of infinite rank. These modules are natural generalizations of the modules Ω ( λ , α ) defined and studied in [5] , [11] and [19] . We determine the sufficient and necessary conditions for them to be irreducible. When they are reducible, we also determine all their maximal submodules and corresponding simple quotients explicitly. For some special cases, we can determine all submodules.

We consider the finite $W$-superalgebra $U(\mathfrak{g_\bbf},e)$ for a basic Lie superalgebra ${\ggg}_\bbf=(\ggg_\bbf)_\bz+(\ggg_\bbf)_\bo$ associated with a nilpotent element $e\in (\ggg_\bbf)_{\bar0}$ both over the field of complex numbers $\bbf=\mathbb{C}$ and over $\bbf={\bbk}$ an algebraically closed field of positive characteristic. In this paper, we mainly present the PBW theorem for $U({\ggg}_\bbf,e)$. Then the construction of $U({\ggg}_\bbf,e)$ can be understood well, which in contrast with finite $W$-algebras, is divided into two cases in virtue of the parity of $\text{dim}\,\mathfrak{g_\bbf}(-1)_{\bar1}$. This observation will be a basis of our sequent work on the dimensional lower bounds in the super Kac-Weisfeiler property of modular representations of basic Lie superalgebras (cf. \cite[\S7-\S9]{ZS})., Comment: 42 pages. This version is revised from the first 6 chapters of the manuscript "Finite W-superalgebras for basic classical Lie superalgebras" (arXiv:1404.1150 [math.RT])

All Lie algebras and representations will be assumed to be finite dimensional over the complex numbers. Let V ( m ) be the irreducible sl ( 2 ) -module with highest weight m ⩾ 1 and consider the perfect Lie algebra g = sl ( 2 ) ⋉ V ( m ) . Recall that a g -module is uniserial when its submodules form a chain. In this paper we classify all uniserial g -modules. The main family of uniserial g -modules is actually constructed in greater generality for the perfect Lie algebra g = s ⋉ V ( μ ) , where s is a semisimple Lie algebra and V ( μ ) is the irreducible s -module with highest weight μ ≠ 0 . The fact that the members of this family are, but for a few exceptions of lengths 2, 3 and 4, the only uniserial sl ( 2 ) ⋉ V ( m ) -modules depends in an essential manner on the determination of certain non-trivial zeros of Racah–Wigner 6j-symbol.

After the fundamental theorem for nonregular reductive prehomogeneous vector spaces was proved by A. Gyoja (Tsukuba J. Math.14 (1990), 437–457), the construction of the theory of nonregular prehomogeneous vector spaces became an interesting problem. However, only (Spn×GL2,Λ1⊗2Λ1,M(2n,3)) is known as an interesting example of nonregular reductive prehomogeneous vector spaces. In this paper, we give some other interesting examples by constructing all relative invariants of nonregular 2-simple prehomogeneous vector spaces of type I. We consider everything over the complex number field C.

We show that almost all fields of characteristic 0 carry a directed partial order. Especially, the field of complex numbers C can be made into a directed field, which answers an open question in [Y. Yang, On the existence of directed rings and algebras with negative squares, J. Algebra 295 (2006) 452–457].

It is proved that any vertex operator algebra for which the image of the Virasoro element in Zhu's algebra is algebraic over complex numbers is finitely generated. In particular, any vertex operator algebra with a finite dimensional Zhu's algebra is finitely generated. As a result, any rational vertex operator algebra is finitely generated., 6 pages

Let H C (B n ) (respectively H C (D n ) ) be the Hecke algebra of type Bn (respectively of type Dn) over the complex numbers field C . Let ζ be a primitive 2lth root of unity in C . For any Kleshchev bipartition (with respect to (ζ,1,−1)) λ=(λ(1),λ(2)) of n, let D λ be the corresponding irreducible H C (B n ) -module. In the present paper we explicitly determine which D λ split and which D λ remains irreducible when restricts to H C (D n ) . This yields a complete classification of all the simple modules for Hecke algebra H C (D n ) . Our proof makes use of the crystal bases theory for the Fock representation of the quantum affine algebra U q ( sl 2l ) and deep result of Ariki's proof of LLT's conjecture [J. Math. Kyoto Univ. 36 (1996) 789–808].

F is a differential field of characteristic zero with algebraically closed field of constants C . A Picard–Vessiot antiderivative closure of F is a differential field extension E ⊃ F which is a union of Picard–Vessiot extensions of F , each obtained by iterated adjunction of antiderivatives, and such that every such Picard–Vessiot extension of F has an isomorphic copy in E . The group G of differential automorphisms of E over F is shown to be prounipotent. When C is the complex numbers and F = C ( t ) the rational functions in one variable, G is shown to be free prounipotent.

In this paper we present a complete classification, up to isomorphism, of affine and projective fully commutative (m + k, m)-groups, defined on subsets of the set of complex numbers C. The classification is via characteristic vectors and their associated complex curves (real surfaces).

Suppose that Ω = {1, 2,…, ab } for some non-negative integers a and b . Denote by P ( a , b ) the set of unordered partitions of Ω into a parts of cardinality b . In this paper we study the decomposition of the permutation module C P ( a , b ) where C is the field of complex numbers. In particular, we show that C P (3, b ) is isomorphic to a submodule of C P ( b , 3) for b ≥ 3. This settles the next unproven case of a conjecture of Foulkes.

For two different integral formsKof the exceptional Jordan algebra we show that AutKis generated by octave reflections. These provide “geometric” examples of discrete reflection groups acting with finite covolume on the octave (or Cayley) hyperbolic plane OH2, the exceptional rank one symmetric space. (The isometry group of the plane is the exceptional Lie groupF4(−20).) Our groups are defined in terms of Coxeter's discrete subring K of the nonassociative division algebra O and we interpret them as the symmetry groups of “Lorentzian lattices” over K. We also show that the reflection group of the “hyperbolic cell” over K is the rotation subgroup of a particularrealreflection group acting onH8≅OH1. Part of our approach is the treatment of the Jordan algebra of matrices that are Hermitian with respect to any real symmetric matrix.

We give a short proof that the Diophantine theory of the complex function field C ( t 1 , t 2 ) where t i are algebraically independent over the complex numbers, in the language {+, ×, t 1 , t 2 , 0, 1} is undecidable. We construct a model of ( Z , +, ×) represented by isogenies of an elliptic curve.

The purpose of this paper is to give an explicit procedure for the computation of the equivariant Poincart series of a complete symmetric variety, in terms of associated combinatorial objects. We recall that given G, a semisimple algebraic group of adjoint type over the complex numbers, 0: G + G an order two automorphism, and letting H = G” = (g E G 1 g(g) = g), by a wonderful symmetric variety we mean a pointed smooth G-variety (Y, p) such that

In a study [6] of the equivalence of forms, Jordan derived the finiteness of the orthogonal group of a form defined over the complex number field whose degree is at least three and whose discriminant is not zero. Herein we demonstrate the validity of Jordan’s result for any field, with the usual restriction that the characteristic of the field, if not zero, dominate the degree of the form. Jordan’s proof rests on an ingenious inequality argument applied to the coefficients of the form. Our argument begins with a proposition of Bott and Tate [3] which focuses attention on unipotent isometries. The proof provided here (due to Harrison [Sj) is regardless of the characteristic of the underlying field, as is the subsequent elimination of unipotent isometries. The argument is completed by showing the group under consideration to be periodic of bounded period with exponent relatively prime to the characteristic of the field, if it is nonzero. Finiteness then follows from a theorem of Burnside. Rather than a theorem of forms we adopt throughout the equivalent setting of symmetric multilinear maps on a vector space in the hope of achieving greater clarity of exposition and with the risk of obscuring the elementary nature of the argument. Let 0 denote a symmetric multilinear map of degree Y 2 3 on a vector space V of dimension n over a field F whose characteristic p, if nonzero, is greater than r. Let 0 = o(8) denote the orthogonal group of 0 in GL(n,F). Employing the standard extension of the scalar field to its algebraic closure, we may assume that F is algebraically closed. Recall that B is singular if and only if for some 0 # z E V, 0(X,..., z, v) = 0 for all 0 E V.

In this paper, we classify all homogeneous spaces ρ ( H ) \ GL ( V ) with an open dense P-orbit, except when ( GL 1 × H , Λ 1 ⊗ ρ , V ( 1 ) ⊗ V ) is a trivial P.V., where P is a proper parabolic subgroup of GL ( V ) and ρ : H → GL ( V ) is a finite-dimensional irreducible representation of a semisimple algebraic group H. When P is of type P ( e 1 , e 2 , e 3 ) , we classify all the cases up to equivalence. We consider everything over the complex number field C .

In this note we consider a denumerable hilbertian field K equipped with an absolute value v which is either the usual absolute value induced by that of the complex numbers or a non-archimedean valuation with values in a commutative ordered group r. The absolute value v is assumed to have been extended in some fixed way to the algebraic closure k? of K. The purpose of this note is to strengthen Theorem A for such K in the following way.

R”G (and RSG) on the one side, and R;G (and RiG) on the other side, are defined as subgroups of RG and R,G, respectively, having as generators the G-modules which admit an orthogonal (a symplectic), G-invariant bilinear form over the field C of complex numbers, respectively over K. RFG is the subgroup of R,G that is spanned by the G-modules which are equipped with an orthogonal, G-invariant bilinear form over K to which there exists an orthonormal K-basis. RYG then is the subgroup of R, G that is generated by the PZ 9 K, where wz itself possesses an orthogonal, G-invariant bilinear form over D to which there exists an orthonormal D-basis. The following diagram roughly shows how all these groups are correlated.

Let V 1 , …, V m be inner product spaces and A a linear operator on V 1 ⊗ ··· ⊗ V m . Suppose that an equation involving A holds for all tensors of a given rank. Does it follow that the equation holds for all tensors ? We answer this question for some equations involving the inner production V 1 ⊗ ··· ⊗ V m . For example, it is shown that if the field is the complex numbers and ( At , t ) = 0, for all decomposable tensors t , then ( At , t ) = 0, for all tensors t . Thus, A = 0.

Let s and u be nonconjugate elements of a finite group and let a , b , and c be complex numbers, ab ≠ 0. Groups satisfying aX ( s ) + bX ( u ) = c for every non-principal irreducible character X are completely classified. They are the PSL (2, 2 n ), n ⩾ 2 and some solvable group of derived length at most 3.

Let V be an n-dimensional vector space over complex numbers C. Let W be the mth tensor product of V. If T C Homc(V, V), let ?m T EHomC(W, W) be the mth tensor product of T. The homomorphism T -4 (m T is a representa- tion of the full linear group GL (C). If H is a subgroup of the symmetric group SMI and X a linear character on H, let VX(G) be the subspace of W consisting of all tensors symmetric with respect to H and x. Then Vm(H) is invariant under gm T. Let K(T) be the restriction of (m T to Vxm(H). For n large compared with m and for H transitive, we determine all cases when the representation T - K(T) is irreducible.

The study of the algebraic theory of characters of commutative semigroups was initiated by Schwarz [8, 9, IO] and Hewitt and Zuckerman [#, 51. The theory has been further developed and expanded by Clifford and Preston [I] and several other authors [3,6, 7, II]. A character of a semigroup S is simply a homomorphism of S into the multiplicative semigroup of complex numbers. The collection of all characters of a semigroup S is a semigroup with respect to pointwise multiplication. We denote the semigroup of characters of S by V(S) and we define

The classical Frobenius-Schur indicators for finite groups are character sums defined for any representation and any integer m greater or equal to 2. In the familiar case m=2, the Frobenius-Schur indicator partitions the irreducible representations over the complex numbers into real, complex, and quaternionic representations. In recent years, several generalizations of these invariants have been introduced. Bump and Ginzburg, building on earlier work of Mackey, have defined versions of these indicators which are twisted by an automorphism of the group. In another direction, Linchenko and Montgomery have defined Frobenius-Schur indicators for semisimple Hopf algebras. In this paper, the authors construct twisted Frobenius-Schur indicators for semisimple Hopf algebras; these include all of the above indicators as special cases and have similar properties., Comment: 12 pages. Minor revisions

For a simple Lie algebra, L, over the complex numbers we study generalized Verma modules induced from modules which are torsion-free over a parabolic sub-algebra and the irreducible quotients of such modules. We construct a new category O α which generalizes the well-known category O .

(1) f: R” + R” is a C” map. Here k is the real numbers R and R is the ring of C” maps from R” to R. (2) f: C” + C” is an analytic map. Here k is the complex numbers C and R is the ring of analytic maps from C” to C. (3) f: k” -+ k” is a polynomial map. Here k is a commutative ring and R is the polynomial ring kc”‘. (4) f:k[["I',k[[nll is a k-algebra homomorphism. Here k is a commutative ring and R is the formal power series ring kccnl’.

In this paper, we study stability properties of norms on the complex numbers and on the quaternions. Our main findings are that these norms are stable if and only if they majorize the modulus function and that not all stable norms are strongly stable. Part of the paper is devoted to the standard matrix representations of the above number systems, where we show that norms on the corresponding matrix algebras are stable if and only if they are spectrally dominant. We conclude by considering proper seminorms, observing that none are stable on the complex numbers or on the quaternions.

We give a proof of complex Bott periodicity that uses some of the general ideas of Segal and Quillen on classifying spaces [I, 2, 41. We attempt to understand which essential properties of the complex numbers and the unitary group are responsible for the periodicity, as contrasted with general facts on classifying spaces and linear groups over general topological rings (i.e., we compare complex K-theory and general algebraic K-theory). From this point of view the only essential properties of the complex numbers are: the “spectral theorem” (n x n unitary matrices can be diagonalized) and the fact that the Stiefel manifold of k-frames in Cd” is highly connected for large IV.

Regular actions of the additive group of complex numbers on complex surfaces and on complex affine space are considered. A proper action on an affine surface admits a geometric quotient which is an affine curve. A proper action on a normal quasiaffine surface is equivariantly trivial. New criteria for local and “global” triviality of proper actions on a complex affine space of arbitrary dimension are presented.

This paper continues the search for bounds an the order of the Sylow p-subgroup P of a primitive complex projective linear group of degree less than 3(p -- 1)/2. We assume that all groups are finite and all representations, linear groups, or projective groups are over the complex numbers. The following is a consequence of our results.

Let X be a scheme of finite type over the field of complex numbers and let x be the generic point of the center of a permissible monoidal transformation X′ → X. If the generic point y of the projectivization of the strict tangent space TxX is an infinitely near H-fold point to X at x (i.e., the Hilbert-Samuel function of X′ at y “coincides” with the Hilbert-Samuel function of X at x) then the length of the Samuel stratum of X passing through x strictly decreases whenever it is finite.