# Lie algebras, torsors and cohomological invariants (12w5008)

Arriving in Banff, Alberta Sunday, September 30 and departing Friday October 5, 2012

## Organizers

Stefan Gille (University of Alberta)

Nikita Karpenko (University of Alberta)

Arturo Pianzola (University of Alberta)

Vera Serganova (University of California, Berkeley)

Kirill Zainoulline (University of Ottawa)

## Objectives

The theory of Lie algebras and the theory of torsors are well-established areas of modern mathematics. The first deals with the study and classification of (in-)finite dimensional Lie algebras and has many applications in representation theory, combinatorics and mathematical physics. The second studies and classifies so-called twisted forms of algebraic objects (example: associative algebras that become matrix algebras after extending the base field), and has many applications in algebraic geometry and number theory. The bridge between Lie algebras and torsors is provided by the various cohomological invariants, e.g. de Rham and Galois cohomology, motives, Chow groups, K-theory, algebraic cobordism.

The last five years can be characteristed as a boom of research activity in these two areas. To support this observation we should mention the recent results by

(a) Garibald-Merkurjev-Rost-Serre-Totaro-Zainoulline which set up various connections between cohomological invariants and irreducible representations of Lie algebras, e.g. the Dynkin index;

(b) Karpenko-Merkurjev-Reichstein on essential and canonical dimensions of quadratic forms and linear algebaic groups which essentially used the representation theory.

(c) Petrov-Semenov-Zainoulline on motivic decompositions of projective homogeneous varieties, which are based on V.~Kac computations of the Chow groups of compact Lie groups.

(d) Gille-Pianzola, where the classification of multiloop Lie algebras has been related to the classification of torsors over Laurent polynomial rings.

(e) Kac-Lau-Pianzola the remarkable fact that the torsor point of view can also be used to study conformal superalgebras, a fact that the lead to the concept of differential conformal superlagebras.

Note also that two of the confirmed participants of the workshop (Karpenko and Reichstein) were invited to give sectional talks at the International Congress of Mathematicians (2010).

We would like to stress that during the last 5 years there were no common activities between researchers representing these two areas (Lie algebras and Torsors) except of a minor activity supported by the Fields Institute and the CRM at the University of Ottawa during March 26-28, 2010. Because of the great success of that mini-workshop (it has already lead to several successfull joint projects) it is reasonable to expect that a large interdisciplinary meeting of the same format can contribute a lot to the development of these areas; can bring together specialists and young researchers (PhDs and PDFs) in Lie algebras and Torsors; can help to establish new links and projects between these two subjects.

The main topics of the workshop will be the following:

I. Torsors and cohomological invariants

Cohomological invariants existed long before this terminology was introduced by Jean-Pierre Serre in the mid of 90's. For instance the (signed) determinant and the Clifford algebra of a quadratic form can be considered as a cohomological invariant with values in the Galois cohomology. To bring some order in the various existing invariants Serre developed a theory of cohomological invariants in the following abstract setting:

By a cohomological invariant one means a natural transformation from the first Galois cohomology with coefficients in an algebraic group G (the pointed set which describes all G-torsors) to a cohomology functor h(-), where h is a Galois cohomology with torsion coefficients, a Witt group, a Chow group with coefficients in a Rost cyclic module M, etc. The ideal result here would be to construct enough invariants to classify all G-torsors. This concept was developed further by Garibaldi-Merkurjev-Rost-Totaro, leading to the complete understanding of invariants in lower degrees. For instance, in degree 2 the group of invariants is generated by the classes of Tits algebras in the Brauer group and in degree 3 it is generated by the Rost invariant.

I.a) Applications to extended affine Lie algebras

The connection between the ``forms" point of view, which is related to the theory of Reductive Group Schemes developed by Demazure and Grothendieck, and Extended Affine Lie Algebras (EALAs) is one of the central themes of the proposal. P. Gille and A. Pianzola have pioneered this approach. The language and theory of G-torsors, where G is a reductive group scheme over a Laurent polynomial ring, appears then quite naturally. This point of view brings extremely powerful tools to the study of infinite dimensional Lie theory.

On the other side the study of EALAs has been greatly simplified by work of E. Neher, which reduces the classification problem to the so called cores of the EALA, which are are in fact multiloop algebras (except for a family of fully understood cases). It is not true, however, that every multiloop algebra is the centreless core of an EALA. Finding cohomological invariants that characterize the isomorphism classes of torsors corresponding to the multiloop algebras attached to EALAs is part of the proposal. This question is connected with some deep work in progress by Chernousov, Gille and Pianzola dealing with the problem of conjugacy of Cartan subalgebras of multiloop algebras.

I.b) Applications to representations and cohomology of Lie algebras and homogeneous spaces

The cohomology theory of Lie algebras on one hand is used to compute central and abelian extensions of Lie algebras. Central extensions of Lie algebras often have richer representation theories than their centerless quotients. On the other hand, cohomology of manifolds connects representation theory with geometry and physics. An example of such an interplay is a recently established cohomological interpretation of the elliptic genus, an object originally introduced in string theory.

Another interesting direction is to study maps between cohomologies of vector bundles on homogeneous varieties arising from homomorphisms of the underlying homogeneous varieties. So far the best understood case is the case of varieties of Borel subgroups. In particular, Dimitrov-Roth have showed that the diagonal embedding leads to a natural geometric construction of extreme components of the tensor product of irreducible representations.

II. Torsors and Motives

Following the general philosophy of Grothendieck one can introduce a universal cohomological invariant which takes values in the category of motives. The link between the world of motives and torsors is provided by the celebrated Rost Nilpotence Theorem (RNT) which can be viewed as a generalized Galois descent property. In [Duke, 2003] Chernousov-Gille-Merkurjev proved the RNT for arbitrary projective homogeneous varieties over semisimple algebraic groups, hence, opening the door to the study of motives of projective homogeneous varieties.

II.a) Applications to linear algebraic groups

Based on this result and computations of Victor Kac of the Chow ring of G Petrov-Semenov-Zainoulline [Ann. Sci. ENS, 2008] computed the motive of generically split projective homogeneous varieties in terms of generalized Rost motives introduced by Voevodsky. They also showed that the motivic behavior of such varieties can be described by a certain discrete numerical invariant, the J-invariant. As an application of the motivic J-invariant Petrov-Semenov [Duke, 2010] classified all generically split homogeneous varieties of linear algebraic groups. As another application, Semenov (2010) has given a construction of a cohomological invariant of degree 5 of an exceptional group of type $E_8$, hence, proving a conjecture by J.-P.Serre. This invariant has tremendous applications to the study of subgroups of compact Lie groups of type $E_8$.

II.b) Applications to quadratic forms and algebras with involutions

Quadratic forms and central simple algebras with involutions provide classical examples of torsors for a (projective) orthogonal group. Its cohomological invariants and motives have been extensively studied during the last decade. We should mention here the works of Karpenko and Vishik who use motives to investigate the splitting behavior of quadratic forms and algebras with involutions, e.g. the Vishik's construction of fields with u-invariant $2^r+1$, $r>3$, his motivic decomposition type theory; Karpenko's result on the first Witt indices of quadratic forms and the proof of the Hoffmann's conjecture, his recent proofs of hyperbolicity and isotropy conjectures for algebras with involutions.

In the proof of the hyperbolicity conjecture Karpenko (2010) uses the theory of upper motives. Observe that for a projective homogeneous variety the dimension of its upper motive measures its canonical p-dimension, hence, providing a new approach to study the canonical and essential dimensions of algebraic groups.

II.c) Applications to Del Pezzo surfaces and representations of exceptional algebraic groups.

In 1990 Batyrev conjectured that universal torsors over Del Pezzo surfaces can be embedded into homogeneous spaces of exceptional algebraic groups. The particular cases of this conjecture were proven by Popov and Derenthal. Serganova and Skorobogatov recently suggested a universal proof of the Batyrev conjecture which covers a new case of $E_8$. Very recently S. Gille [Invent. Math., 2010] has proven the RNT for del Pezzo surfaces, hence, providing a new approach to study their geometry, motives and Chow groups.

The last five years can be characteristed as a boom of research activity in these two areas. To support this observation we should mention the recent results by

(a) Garibald-Merkurjev-Rost-Serre-Totaro-Zainoulline which set up various connections between cohomological invariants and irreducible representations of Lie algebras, e.g. the Dynkin index;

(b) Karpenko-Merkurjev-Reichstein on essential and canonical dimensions of quadratic forms and linear algebaic groups which essentially used the representation theory.

(c) Petrov-Semenov-Zainoulline on motivic decompositions of projective homogeneous varieties, which are based on V.~Kac computations of the Chow groups of compact Lie groups.

(d) Gille-Pianzola, where the classification of multiloop Lie algebras has been related to the classification of torsors over Laurent polynomial rings.

(e) Kac-Lau-Pianzola the remarkable fact that the torsor point of view can also be used to study conformal superalgebras, a fact that the lead to the concept of differential conformal superlagebras.

Note also that two of the confirmed participants of the workshop (Karpenko and Reichstein) were invited to give sectional talks at the International Congress of Mathematicians (2010).

We would like to stress that during the last 5 years there were no common activities between researchers representing these two areas (Lie algebras and Torsors) except of a minor activity supported by the Fields Institute and the CRM at the University of Ottawa during March 26-28, 2010. Because of the great success of that mini-workshop (it has already lead to several successfull joint projects) it is reasonable to expect that a large interdisciplinary meeting of the same format can contribute a lot to the development of these areas; can bring together specialists and young researchers (PhDs and PDFs) in Lie algebras and Torsors; can help to establish new links and projects between these two subjects.

The main topics of the workshop will be the following:

I. Torsors and cohomological invariants

Cohomological invariants existed long before this terminology was introduced by Jean-Pierre Serre in the mid of 90's. For instance the (signed) determinant and the Clifford algebra of a quadratic form can be considered as a cohomological invariant with values in the Galois cohomology. To bring some order in the various existing invariants Serre developed a theory of cohomological invariants in the following abstract setting:

By a cohomological invariant one means a natural transformation from the first Galois cohomology with coefficients in an algebraic group G (the pointed set which describes all G-torsors) to a cohomology functor h(-), where h is a Galois cohomology with torsion coefficients, a Witt group, a Chow group with coefficients in a Rost cyclic module M, etc. The ideal result here would be to construct enough invariants to classify all G-torsors. This concept was developed further by Garibaldi-Merkurjev-Rost-Totaro, leading to the complete understanding of invariants in lower degrees. For instance, in degree 2 the group of invariants is generated by the classes of Tits algebras in the Brauer group and in degree 3 it is generated by the Rost invariant.

I.a) Applications to extended affine Lie algebras

The connection between the ``forms" point of view, which is related to the theory of Reductive Group Schemes developed by Demazure and Grothendieck, and Extended Affine Lie Algebras (EALAs) is one of the central themes of the proposal. P. Gille and A. Pianzola have pioneered this approach. The language and theory of G-torsors, where G is a reductive group scheme over a Laurent polynomial ring, appears then quite naturally. This point of view brings extremely powerful tools to the study of infinite dimensional Lie theory.

On the other side the study of EALAs has been greatly simplified by work of E. Neher, which reduces the classification problem to the so called cores of the EALA, which are are in fact multiloop algebras (except for a family of fully understood cases). It is not true, however, that every multiloop algebra is the centreless core of an EALA. Finding cohomological invariants that characterize the isomorphism classes of torsors corresponding to the multiloop algebras attached to EALAs is part of the proposal. This question is connected with some deep work in progress by Chernousov, Gille and Pianzola dealing with the problem of conjugacy of Cartan subalgebras of multiloop algebras.

I.b) Applications to representations and cohomology of Lie algebras and homogeneous spaces

The cohomology theory of Lie algebras on one hand is used to compute central and abelian extensions of Lie algebras. Central extensions of Lie algebras often have richer representation theories than their centerless quotients. On the other hand, cohomology of manifolds connects representation theory with geometry and physics. An example of such an interplay is a recently established cohomological interpretation of the elliptic genus, an object originally introduced in string theory.

Another interesting direction is to study maps between cohomologies of vector bundles on homogeneous varieties arising from homomorphisms of the underlying homogeneous varieties. So far the best understood case is the case of varieties of Borel subgroups. In particular, Dimitrov-Roth have showed that the diagonal embedding leads to a natural geometric construction of extreme components of the tensor product of irreducible representations.

II. Torsors and Motives

Following the general philosophy of Grothendieck one can introduce a universal cohomological invariant which takes values in the category of motives. The link between the world of motives and torsors is provided by the celebrated Rost Nilpotence Theorem (RNT) which can be viewed as a generalized Galois descent property. In [Duke, 2003] Chernousov-Gille-Merkurjev proved the RNT for arbitrary projective homogeneous varieties over semisimple algebraic groups, hence, opening the door to the study of motives of projective homogeneous varieties.

II.a) Applications to linear algebraic groups

Based on this result and computations of Victor Kac of the Chow ring of G Petrov-Semenov-Zainoulline [Ann. Sci. ENS, 2008] computed the motive of generically split projective homogeneous varieties in terms of generalized Rost motives introduced by Voevodsky. They also showed that the motivic behavior of such varieties can be described by a certain discrete numerical invariant, the J-invariant. As an application of the motivic J-invariant Petrov-Semenov [Duke, 2010] classified all generically split homogeneous varieties of linear algebraic groups. As another application, Semenov (2010) has given a construction of a cohomological invariant of degree 5 of an exceptional group of type $E_8$, hence, proving a conjecture by J.-P.Serre. This invariant has tremendous applications to the study of subgroups of compact Lie groups of type $E_8$.

II.b) Applications to quadratic forms and algebras with involutions

Quadratic forms and central simple algebras with involutions provide classical examples of torsors for a (projective) orthogonal group. Its cohomological invariants and motives have been extensively studied during the last decade. We should mention here the works of Karpenko and Vishik who use motives to investigate the splitting behavior of quadratic forms and algebras with involutions, e.g. the Vishik's construction of fields with u-invariant $2^r+1$, $r>3$, his motivic decomposition type theory; Karpenko's result on the first Witt indices of quadratic forms and the proof of the Hoffmann's conjecture, his recent proofs of hyperbolicity and isotropy conjectures for algebras with involutions.

In the proof of the hyperbolicity conjecture Karpenko (2010) uses the theory of upper motives. Observe that for a projective homogeneous variety the dimension of its upper motive measures its canonical p-dimension, hence, providing a new approach to study the canonical and essential dimensions of algebraic groups.

II.c) Applications to Del Pezzo surfaces and representations of exceptional algebraic groups.

In 1990 Batyrev conjectured that universal torsors over Del Pezzo surfaces can be embedded into homogeneous spaces of exceptional algebraic groups. The particular cases of this conjecture were proven by Popov and Derenthal. Serganova and Skorobogatov recently suggested a universal proof of the Batyrev conjecture which covers a new case of $E_8$. Very recently S. Gille [Invent. Math., 2010] has proven the RNT for del Pezzo surfaces, hence, providing a new approach to study their geometry, motives and Chow groups.