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Tag: necklace

Segal’s formal neighbourhood result

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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M-geometry (3)

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.