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At least 19 recordsLinked to original sources
arXiv2007

Depth three towers and Jacobson-Bourbaki correspondence

We introduce a notion of depth three tower of three rings C < B < A as a useful generalization of depth two ring extension. If A = End B_C and B | C is a Frobenius extension, this also captures the notion of depth three for a Frobenius extension in math.RA/0107064 and math.RA/0108067 such that if B | C is depth three, then A | C is depth two (cf. math.QA/0001020). If A, B and C correspond to a tower of subgroups G > H > K via the group algebra over a fixed base ring, the depth three condition is the condition that subgroup K has normal closure K^G contained in H. For a depth three tower of rings, there is a pre-Galois theory for the ring End {}_BA_C and coring (A \otimes_B A)^C involving Morita context bimodules and left coideal subrings. This is applied in the last two sections to a specialization of a Jacobson-Bourbaki correspondence theorem for augmented rings to depth two extensions with depth three intermediate division rings.

math.RA↗
arXiv2012

Chevalley Supergroups of type D(2,1;a)

I present a construction "a` la Chevalley" of affine supergroups associated with simple Lie superalgebras of (classical) type D(2,1;a), for any possible value of the parameter a - in particular, including non-integral values of a. This extends the similar work performed in [R. Fioresi, F. Gavarini, "Chevalley Supergroups", Memoirs of the AMS 215 (2012), no. 1014 - arXiv:0808.0785v8 [math.RA]], where all other simple Lie superalgebras of classical type were considered. The case of simple Lie superalgebras of Cartan type is dealt with in [F. Gavarini, "Algebraic supergroups of Cartan type", Forum Mathematicum (to appear), 92 pages - arXiv:1109.0626v5 [math.RA], so this work completes the program of constructing connected affine supergroups associated with any simple Lie superalgebra.

math.RA↗
arXiv2004

Depth Two and the Galois Coring

We study the cyclic module ${}_SR$ for a ring extension $A \| B$ with centralizer $R$ and bimodule endomorphism ring $S = End {}_BA_B$. We show that if $A \| B$ is an H-separable Hopf subalgebra, then $B$ is a normal Hopf subalgebra of $A$. We observe from math.RA/0107064 and math.RA/0108067 depth two in the role of noncommutative normality (as in field theory) in a depth two separable Frobenius characterization of irreducible semisimple-Hopf-Galois extensions. We prove that a depth two extension has a Galois $A$-coring structure on $A ø_R T$ where $T$ is the right $R$-bialgebroid dual to $S$.

math.RA↗
arXiv2004

Hopf algebroids and Galois extensions

To a finite Hopf-Galois extension $A | B$ we associate dual bialgebroids $S := \End_BA_B$ and $T := (A ø_B A)^B$ over the centralizer $R$ using the depth two theory in math.RA/0108067. First we extend results on the equivalence of certain properties of Hopf-Galois extensions with corresponding properties of the coacting Hopf algebra \cite{KT,Doi} to depth two extensions using coring theory math.RA/0002105. Next we show that $T^{\rm op}$ is a Hopf algebroid over the centralizer $R$ via Lu's theorem 5.1 in math.QA/9505024 for smash products with special modules over the Drinfel'd double, the Miyashita-Ulbrich action, the fact that $R$ is a commutative algebra in the pre-braided category of Yetter-Drinfel'd modules \cite[Schauenburg]{Sch} and the equivalence of Yetter-Drinfel'd modules with modules over Drinfel'd double \cite[Majid]{Maj}. In our last section, an exposition of results of Sugano \cite{Su82,Su87} leads us to a Galois correspondence between sub-Hopf algebroids of $S$ over simple subalgebras of the centralizer with finite projective intermediate simple subrings of a finite projective H-separable extension of simple rings $A \supseteq B$.

math.QA↗
arXiv2005

Galois theory for bialgebroids, depth two and normal Hopf subalgebras

We reduce certain proofs in math.RA/0108067, math.RA/0408155, and math.QA/0409589 to depth two quasibases from one side only, a minimalistic approach which leads to a characterization of Galois extensions for finite projective bialgebroids without the Frobenius extension property. We prove that a proper algebra extension is a left $T$-Galois extension for some right finite projective left bialgebroid over some algebra $R$ if and only if it is a left depth two and left balanced extension. Exchanging left and right in this statement, we have a characterization of right Galois extensions for left finite projective right bialgebroids. Looking to examples of depth two, we establish that a Hopf subalgebra is normal if and only if it is a Hopf-Galois extension. We characterize finite weak Hopf-Galois extensions using an alternate Galois canonical mapping with several corollaries: that these are depth two and that surjectivity of the Galois mapping implies its bijectivity.

math.QA↗
arXiv2006

Anchor maps and stable modules in depth two

An algebra extension A | B is right depth two if its tensor-square A\otimes_B A is in the Dress category Add A as A-B-bimodules. We consider necessary conditions for right, similarly left, D2 extensions in terms of partial A-invariance of two-sided ideals in A contracted to the centralizer. Finite dimensional algebras extending central simple algebras are shown to be depth two. Following P. Xu math.QA/9905192, left and right bialgebroids over a base algebra R may be defined in terms of anchor maps, or representations on R. The anchor maps for the bialgebroids S = End {}_BA_B and T = End {}_AA\otimes_BA_A over the centralizer R = C_A(B) are the modules {}_SR and R_T studied in math.RA/0505004, math.RA/0408155 and math.GR/0409346, which provide information about the bialgebroids and the extension (cf. math.QA/0409106). The anchor maps for the Hopf algebroids in math.KT/0105105 and math.QA/0508411 reverse the order of right multiplication and action by a Hopf algebra element, and lift to the isomorphism in math.QA/0508638. We sketch a theory of stable $A$-modules and their endomorphism rings and generalize the smash product decomposition in Prop. 1.1, (L. Kadison, Hopf Algebroid and H-separable extensions, Proc. A.M.S. 131 (2003), 2993-3002) to any A-module. We observe that Schneider's coGalois theory (Isr.J.Math 1990) provides examples of codepth two, such as the quotient epimorphism of a finite dimensional normal Hopf subalgebra. A homomorphism of finite dimensional coalgebras is codepth two if and only if its dual homomorphism of algebras is depth two.

math.QA↗
arXiv2006

Infinite index subalgebras of depth two

An algebra extension $A \| B$ is right depth two in this paper if its tensor-square is $A$-$B$-isomorphic to a direct summand of any (not necessarily finite) direct sum of $A$ with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite dimensional algebras are depth two in this extended sense. The added generality loses some duality results obtained in the finite theory math.RA/0108067 but extends the main theorem of depth two theory, as for example in math.RA/0107064. That is, a right depth two extension has right bialgebroid T = (A \otimes_B A)^B$ over its centralizer R = C_A(B). The main theorem: an extension A | B is right depth two and right balanced if and only if A | B is T-Galois wrt. left projective, right R-bialgebroid T.

math.QA↗
arXiv2007

Graded identities of matrix algebras and the universal graded algebra

We consider fine G-gradings on M_n(C) (i.e. gradings of the matrix algebra over the complex numbers where each component is 1 dimensional). Groups which provide such a grading are known to be solvable. We consider the T-ideal of G-graded identities and show that it is generated by a special type of binomial identities which we call elementary. In particular we show that the ideal of graded identities is finitely generated as a T-ideal. Next, given such grading we construct a universal algebra U_{G,c} in two different ways: one by means of polynomial identities and the other one by means of a generic two-cocycle (this parallels the classical constructions in the non-graded case). We show that a suitable central localization of U_{G,c} is Azumaya over its center and moreover, its homomorphic images are precisely the G-graded forms of M_n(C). Finally, we consider the ring of central quotients Q(U_{G,c}) (this is an F-central simple algebra where F=Frac(Z) and Z is the center of of U_{G,c}). Using an earlier results of the authors (see E. Aljadeff, D. Haile and M. Natapov, Projective bases of division algebras and groups of central type, Israel J. Math.146 (2005) 317-335 and M. Natapov arXiv:0710.5468v1 [math.RA]) we show that this is a division algebra for a very explicit (and short) family of nilpotent groups. As a consequence, for groups G such that Q(U_{G,c}) is not a division algebra, one can find a non identity polynomial p(x_{i,g}) such that p(x_{i,g})^r is a graded identity for some integer r. We illustrate this phenomenon with a fine G-grading of M_6(C) where G is a semidirect product of S_3 and C_6.

math.RA↗
arXiv2008

Finitely presented groups and homotopy of presentations of triangular algebras

Given any finitely presented group G we find a triangular algebra such that has two presentations, one with fundamental group G and another with trivial group. Thus proving that given a collection G1,...,Gn of finitely presented groups there exist a triangular algebra A such that all Gi appear as fundamental group of some presentation of A, extending one of the result in arXiv:math/0405127v3 [math.RA]

math.GR↗
arXiv2010

Mathematical Structures Defined by Identities II

In our paper arXiv: math.RA/0110333 v1 Oct 2001 we showed that the number of algebras defined by a binary operation satisfying a formally irreducible identity between two n-iterates is O( e^{-n/16}S_{n}^{2} for n --> infinity, S_{n} being the nth-Catalan number. This was proved by using exclusively the series of tableaux A_{n}. By using also the series of tableaux B_{n}, we now sharpen this result to O{(n+2)/n|e^{-n/16}-2/n)|S_{n}^{2}}

math.RA↗
arXiv2002

Magic Squares of Lie Algebras

An improved (streamlined and extended) version of this paper is available as math.RA/0203010, which however omits some details. We recommend the later version unless details are essential.

math.RA↗
arXiv2001

Functional Equations for Lexicographic Products

We generalize the main result of math.RA/9608214 concerning the convex embeddings of a chain Gamma in a lexicographic power Delta^Gamma. For a fixed nonempty chain Delta, we derive necessary and sufficient conditions for the existence of nonempty solutions Gamma to each of the lexicographic functional equations (Delta^Gamma)^{<=0} simeq Gamma, (Delta^Gamma) simeq Gamma, and (Delta^Gamma)^{<0} simeq Gamma.

math.LO↗
arXiv2004

Rational Cherednik algebras and Hilbert schemes II: representations and sheaves

Let H_c be the rational Cherednik algebra of type A_{n-1} with spherical subalgebra U_c=eH_ce. Then U_c is filtered by order of differential operators with associated graded ring gr U_c=C[h + h*]^W, where W is the n-th symmetric group. Using the Z-algebra construction from our earlier paper (math.RA/0407516) it is also possible to associate to a filtered H_c- or U_c-module M a coherent sheaf on the Hilbert scheme Hilb(n). Using this technique, we study the representation theory of U_c and H_c, and relate it to Hilb(n) and to the resolution of singularities from Hilb(n) to h+h*/W. For example, we prove: (1) If c=1/n, so that L_c(triv) is the unique one-dimensional simple H_c-module, then L_c(triv) corresponds to the structure sheaf of the punctual Hilbert scheme. (2) If c=1/n+k (for k some natural number) then, under a canonical filtration on the finite dimensional module L_c(triv), gr eL_{c}(triv) has a natural bigraded structure which coincides with that on the global sections of certain ample line bundles on the punctual Hilbert scheme; this confirms conjectures of Berest, Etingof and Ginzburg, and relates representations of H_c and U_c with Haiman's combinatorial work on the Hilbert scheme. (3) Under mild restrictions on c, the characteristic cycle of the standard H_c-modules are described in terms of certain irreducible subvarieties of the Hilbert scheme (appearing originally in work of Grojnowski) with multiplicities given by Kostka numbers.

math.RT↗
arXiv2006

Equivalences between categories of modules and categories of comodules

We show the close connection between appearingly different Galois theories for comodules introduced recently in [J. Gómez-Torrecillas and J. Vercruysse, Comatrix corings and Galois Comodules over firm rings, arXiv:math.RA/0509106.] and [R. Wisbauer, On Galois comodules (2004), to appear in Comm. Algebra.]. Furthermore we study equivalences between categories of comodules over a coring and modules over a firm ring. We show that these equivalences are related to Galois theory for comodules.

math.RA↗
arXiv2007

Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie

We use a version of the method of Deligne-Illusie to prove that the Hodge-to-de Rham, a.k.a. Hochschild-to-cyclic spectral sequence degenerates for a large class of associative, not necessariyl commutative DG algebras. This proves, under some assumption, a conjecture by Kontsevich and Soibelman made in math.RA/0606241. The approach is similar to my earlier paper math.AG/0511665, but the proof is more straightforward, and the underlying algebraic topology notions are explicitly described. The paper is independent of math.AG/0511665 and in a sense, supercedes it.

math.KT↗
arXiv2021

Matrix Algebras with a Certain Compression Property II

A subalgebra $\mathcal{A}$ of $\mathbb{M}_n(\mathbb{C})$ is said to be projection compressible if $P\mathcal{A}P$ is an algebra for all orthogonal projections $P\in\mathbb{M}_n(\mathbb{C})$. Likewise, $\mathcal{A}$ is said to be idempotent compressible if $E\mathcal{A}E$ is an algebra for all idempotents $E\in\mathbb{M}_n(\mathbb{C})$. In this paper, a case-by-case analysis is used to classify the unital projection compressible subalgebras of $\mathbb{M}_n(\mathbb{C})$, $n\geq 4$, up to transposition and unitary equivalence. It is observed that every algebra shown to admit the projection compression property is, in fact, idempotent compressible. We therefore extend the findings of Cramer, Marcoux, and Radjavi (arXiv:1904.06803 [math.RA]) in the setting of $\mathbb{M}_3(\mathbb{C})$, proving that the two notions of compressibility agree for all unital matrix algebras.

math.RA↗