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Tag: simples

Segal’s formal neighbourhood result

Published December 8, 2007 by lievenlb

Yesterday, Ed Segal gave a talk at the Arts. His title “Superpotential algebras from 3-fold singularities” didnt look too promising to me. And sure enough it was all there again : stringtheory, D-branes, Calabi-Yaus, superpotentials, all the pseudo-physics babble that spreads virally among the youngest generation of algebraists and geometers.

Fortunately, his talk did contain a general ringtheoretic gem. After a bit of polishing up this gem, contained in his paper The A-infinity Deformation Theory of a Point and the Derived Categories of Local Calabi-Yaus, can be stated as follows.

Let $A $ be a $\mathbb{C} $-algebra and let $M = S_1 \oplus \ldots \oplus S_k $ be a finite dimensional semi-simple representation with distinct simple components. Let $\mathfrak{m} $ be the kernel of the algebra epimorphism $A \rightarrow S $ to the semi-simple algebra $S=End(M) $. Then, the $\mathfrak{m} $-adic completion of $A $ is Morita-equivalent to the completion of a quiver-algebra with relations. The nice thing is that both the quiver and relations come in a canonical way from the $A_{\infty} $-structure on the Ext-algebra $Ext^{\bullet}_A(M,M) $. More precisely, there is an isomorphism

$\hat{A}_{\mathfrak{m}} \simeq \frac{\hat{T}_S(Ext^1_A(M,M)^{\ast})}{(Im(HMC)^{\ast})} $

where the homotopy Maurer-Cartan map comes from the $A_{\infty} $ structure maps

$HMC = \oplus_i m_i~:~T_S(Ext_A^1(M,M)) \rightarrow Ext^2_A(M,M) $

and hence the defining relations of the completion are given by the image of the dual of this map.

For ages, Ive known this result in the trivial case of formally smooth algebras (where $Ext^2_A(M,M)=0 $ and hence there are no relations to divide out) and where it is a consequence of a special case of the Cuntz-Quillen “tubular neighborhood” result. Completions of formally smooth algebras at semi-simples are Morita equivalent to completions of path algebras. This fact motivated all the local-quiver technology that was developed here in Antwerp over the last decade (see my book if you want to know the details).

Also for 3-dimensional Calabi-Yau algebras it states that the completions at semi-simples are Morita equivalent to completions of quotients of path algebras by the relations coming from a superpotential (aka a necklace) by taking partial noncommutative derivatives. Here the essential ingredient is that $Ext^2_A(M,M)^{\ast} \simeq Ext^1_A(M,M) $ in this case.

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problema bovinum

Published December 3, 2007 by lievenlb

Suppose for a moment that some librarian at the Bodleian Library announces that (s)he discovered an old encrypted book attributed to Isaac Newton. After a few months of failed attempts, the code is finally cracked and turns out to use a Public Key system based on the product of two gigantic prime numbers, $2^{32582657}-1 $ and $2^{30402457}-1 $, which were only discovered to be prime recently. Would one deduce from this that Newton invented public key cryptography and that he used alchemy to factor integers? (( Come to think of it, some probably would ))

The cynic in me would argue that it is a hell of a coincidence for this text to surface exactly at the moment in history when we are able to show these numbers to be prime and understand their cryptographic use, and conclude that the book is likely to be a fabrication. Still, stranger things have happened in the history of mathematics…

In 1773, Gotthold Ephraim Lessing at that time librarian at the Herzog-August-Bibliothek discovered and published a Greek epigram in 22 elegiac couplets. The manuscript describes a problem sent by Archimedes to the mathematicians in Alexandria.

In his beautiful book “Number Theory, an approach through history. From Hammurapi to Legendre” Andre Weil asserts (( Chapter I,IX )):

Many mathematical epigrams are known. Most of them state problems of little depth; not so Lessing’s find; there is indeed every reason to accept the attribution to Archimedes, and none for putting it into doubt.

This Problema Bovidum (the cattle problem) is a surprisingly difficult diophantine problem and the simplest complete solution consists of eigth numbers, each having about 206545 digits. As we will see later the final ingredient in the solution is the solution of Pell’s equation using continued fractions discovered by Lagrange in 1768 and published in 1769 in a long memoir. Lagrange’s solution to the Pell equation was inserted in Euler’s “Algebra” which was composed in 1771 but published only in 1773… the very same year as Lessing’s discovery! (( all dates learned from Weil’s book Chp. III,XII ))

Weil’s book doesn’t include the details of the original epigram. The (lost) archeologist in me wanted to see the original Greek 22 couplets as well as a translation. So here they are : (( thanks to the Cattle problem site ))

A PROBLEM

which Archimedes solved in epigrams, and which he communicated to students of such matters at Alexandria in a letter to Eratosthenes of Cyrene.

If thou art diligent and wise, O stranger, compute the number of cattle of the Sun, who once upon a time grazed on the fields of the Thrinacian isle of Sicily, divided into four herds of different colours, one milk white, another a glossy black, a third yellow and the last dappled. In each herd were bulls, mighty in number according to these proportions: Understand, stranger, that the white bulls were equal to a half and a third of the black together with the whole of the yellow, while the black were equal to the fourth part of the dappled and a fifth, together with, once more, the whole of the yellow. Observe further that the remaining bulls, the dappled, were equal to a sixth part of the white and a seventh, together with all of the yellow. These were the proportions of the cows: The white were precisely equal to the third part and a fourth of the whole herd of the black; while the black were equal to the fourth part once more of the dappled and with it a fifth part, when all, including the bulls, went to pasture together. Now the dappled in four parts were equal in number to a fifth part and a sixth of the yellow herd. Finally the yellow were in number equal to a sixth part and a seventh of the white herd. If thou canst accurately tell, O stranger, the number of cattle of the Sun, giving separately the number of well-fed bulls and again the number of females according to each colour, thou wouldst not be called unskilled or ignorant of numbers, but not yet shalt thou be numbered among the wise.

But come, understand also all these conditions regarding the cattle of the Sun. When the white bulls mingled their number with the black, they stood firm, equal in depth and breadth, and the plains of Thrinacia, stretching far in all ways, were filled with their multitude. Again, when the yellow and the dappled bulls were gathered into one herd they stood in such a manner that their number, beginning from one, grew slowly greater till it completed a triangular figure, there being no bulls of other colours in their midst nor none of them lacking. If thou art able, O stranger, to find out all these things and gather them together in your mind, giving all the relations, thou shalt depart crowned with glory and knowing that thou hast been adjudged perfect in this species of wisdom.

The Lessing epigram may very well be an extremely laborious hoax but it is still worth spending a couple of posts on it. It gives us the opportunity to retell the amazing history of Pell’s problem rangingfrom the ancient Greeks and Indians, over Fermat and his correspondents, to Euler and Lagrange (with a couple of recent heroes entering the story). And, on top of this, the modular group is all the time just around the corner…

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M-geometry (3)

Published September 18, 2007 by lievenlb

For any finite dimensional A-representation S we defined before a character $\chi(S) $ which is an linear functional on the noncommutative functions $\mathfrak{g}_A = A/[A,A]_{vect} $ and defined via

$\chi_a(S) = Tr(a | S) $ for all $a \in A $

We would like to have enough such characters to separate simples, that is we would like to have an embedding

$\mathbf{simp}~A \hookrightarrow \mathfrak{g}_A^* $

from the set of all finite dimensional simple A-representations $\mathbf{simp}~A $ into the linear dual of $\mathfrak{g}_A^* $. This is a consequence of the celebrated Artin-Procesi theorem.

Michael Artin was the first person to approach representation theory via algebraic geometry and geometric invariant theory. In his 1969 classical paper “On Azumaya algebras and finite dimensional representations of rings” he introduced the affine scheme $\mathbf{rep}_n~A $ of all n-dimensional representations of A on which the group $GL_n $ acts via basechange, the orbits of which are exactly the isomorphism classes of representations. He went on to use the Hilbert criterium in invariant theory to prove that the closed orbits for this action are exactly the isomorphism classes of semi-simple -dimensional representations. Invariant theory tells us that there are enough invariant polynomials to separate closed orbits, so we would be done if the caracters would generate the ring of invariant polynmials, a statement first conjectured in this paper.

Claudio Procesi was able to prove this conjecture in his 1976 paper “The invariant theory of $n \times n $ matrices” in which he reformulated the fundamental theorems on $GL_n $-invariants to show that the ring of invariant polynomials of m $n \times n $ matrices under simultaneous conjugation is generated by traces of words in the matrices (and even managed to limit the number of letters in the words required to $n^2+1 $). Using the properties of the Reynolds operator in invariant theory it then follows that the same applies to the $GL_n $-action on the representation schemes $\mathbf{rep}_n~A $.

So, let us reformulate their result a bit. Assume the affine $\mathbb{C} $-algebra A is generated by the elements $a_1,\ldots,a_m $ then we define a necklace to be an equivalence class of words in the $a_i $, where two words are equivalent iff they are the same upto cyclic permutation of letters. For example $a_1a_2^2a_1a_3 $ and $a_2a_1a_3a_1a_2 $ determine the same necklace. Remark that traces of different words corresponding to the same necklace have the same value and that the noncommutative functions $\mathfrak{g}_A $ are spanned by necklaces.

The Artin-Procesi theorem then asserts that if S and T are non-isomorphic simple A-representations, then $\chi(S) \not= \chi(T) $ as elements of $\mathfrak{g}_A^* $ and even that they differ on a necklace in the generators of A of length at most $n^2+1 $. Phrased differently, the array of characters of simples evaluated at necklaces is a substitute for the clasical character-table in finite group theory.

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