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Dimitry Leites - One of the best experts on this subject based on the ideXlab platform.
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duflo serganova homology for exceptional modular lie superalgebras with Cartan Matrix
arXiv: Representation Theory, 2020Co-Authors: Andrey Krutov, Dimitry Leites, Jin ShangAbstract:For the exceptional finite-dimensional modular Lie superalgebras $\mathfrak{g}(A)$ with indecomposable Cartan Matrix $A$, and their simple subquotients, we computed non-isomorphic Lie superalgebras constituting the homologies of the odd elements with zero square. These homologies are~key ingredients in the Duflo--Serganova approach to the representation theory. There were two definitions of defect of Lie superalgebras in the literature with different ranges of application. We suggest a third definition and an easy-to-use way to find its value. In positive characteristic, we found out one more reason to consider the space of roots over reals, unlike the space of weights, which should be considered over the ground field. We proved that the rank of the homological element (decisive in calculating the defect of a given Lie superalgebra) should be considered in the adjoint module, not the irreducible module of least dimension (although the latter is sometimes possible to consider, e.g., for $p=0$). We also computed the above homology for the only case of simple Lie superalgebras with symmetric root system not considered so far over the field of complex numbers, and its modular versions: $\mathfrak{psl}(a|a+pk)$ for $a$ and $k$ small, and $p=2, 3, 5$.
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The Roots of Exceptional Modular Lie Superalgebras with Cartan Matrix
Arnold Mathematical Journal, 2020Co-Authors: Sofiane Bouarroudj, Dimitry Leites, Olexander Lozhechnyk, Jin ShangAbstract:For each of the exceptional (not entering infinite series) finite-dimensional modular Lie superalgebras with indecomposable Cartan Matrix, we give the explicit list of its roots, and the corresponding Chevalley basis, for one of its inequivalent Cartan matrices, namely the one corresponding to the greatest number of mutually orthogonal isotropic odd simple roots (this number, called the defect of the Lie superalgebra, is important in the representation theory). Our main tools: Grozman’s Mathematica-based code SuperLie, Python, and A. Lebedev’s help.
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the roots of exceptional modular lie superalgebras with Cartan Matrix
arXiv: Representation Theory, 2019Co-Authors: Sofiane Bouarroudj, Dimitry Leites, Alexander Lozhechnyk, Jin ShangAbstract:For each of the exceptional Lie superalgebras with indecomposable Cartan Matrix, we give the explicit list of its roots of and the corresponding Chevalley basis for one of the inequivalent Cartan matrices, the one corresponding to the greatest number of mutually orthogonal isotropic odd simple roots. Our main tools: Grozman's Mathematica-based code SuperLie, and Python.
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simple prolongs of the non positive parts of graded lie algebras with Cartan Matrix in characteristic 2
arXiv: Representation Theory, 2013Co-Authors: Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites, Alexei Lebedev, Irina ShchepochkinaAbstract:Over an algebraically closed fields, an alternative to the method due to Kostrikin and Shafarevich was recently suggested. It produces all known simple finite dimensional Lie algebras in characteristic p > 2. For p = 2, we investigate one of the steps of this method, interpret several other simple Lie algebras, previously known only as sums of their components, as Lie algebras of vector fields. One new series of exceptional simple Lie algebras is discovered, together with its "hidden supersymmetries". In characteristic 2, certain simple Lie algebras are "desuperizations" of simple Lie superal- gebras. Several simple Lie algebras we describe as results of generalized Cartan prolongation of the non-positive parts, relative a simplest (by declaring degree of just one pair of root vec- tors corresponding to opposite simple roots nonzero) grading by integers, of Lie algebras with Cartan Matrix are "desuperizations" of characteristic 2 versions of complex simple exceptional vectorial Lie superalgebras. We list the Lie superalgebras (some of them new) obtained from the Lie algebras considered by declaring certain generators odd. One of the simple Lie algebras obtained is the prolong relative to a non-simplest grading, so the classification to be obtained might be more involved than we previously thought.
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divided power co homology presentations of simple finite dimensional modular lie superalgebras with Cartan Matrix
arXiv: Representation Theory, 2009Co-Authors: Sofiane Bouarroudj, Pavel Grozman, Alexei Lebedev, Dimitry LeitesAbstract:For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras with indecomposable Cartan Matrix in characteristic 2 (and in other characteristics for completeness of the picture). We correct the currently available in the literature notions of Chevalley generators and Cartan Matrix in the modular and super cases, and an auxiliary notion of the Dynkin diagram. In characteristic 2, the defining relations of simple classical Lie algebras of the A, D, E types are not only Serre ones; these non-Serre relations are same for Lie superalgebras with the same Cartan Matrix and any distribution of parities of the generators. Presentations of simple orthogonal Lie algebras having no Cartan Matrix are also given..
Sofiane Bouarroudj - One of the best experts on this subject based on the ideXlab platform.
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The Roots of Exceptional Modular Lie Superalgebras with Cartan Matrix
Arnold Mathematical Journal, 2020Co-Authors: Sofiane Bouarroudj, Dimitry Leites, Olexander Lozhechnyk, Jin ShangAbstract:For each of the exceptional (not entering infinite series) finite-dimensional modular Lie superalgebras with indecomposable Cartan Matrix, we give the explicit list of its roots, and the corresponding Chevalley basis, for one of its inequivalent Cartan matrices, namely the one corresponding to the greatest number of mutually orthogonal isotropic odd simple roots (this number, called the defect of the Lie superalgebra, is important in the representation theory). Our main tools: Grozman’s Mathematica-based code SuperLie, Python, and A. Lebedev’s help.
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the roots of exceptional modular lie superalgebras with Cartan Matrix
arXiv: Representation Theory, 2019Co-Authors: Sofiane Bouarroudj, Dimitry Leites, Alexander Lozhechnyk, Jin ShangAbstract:For each of the exceptional Lie superalgebras with indecomposable Cartan Matrix, we give the explicit list of its roots of and the corresponding Chevalley basis for one of the inequivalent Cartan matrices, the one corresponding to the greatest number of mutually orthogonal isotropic odd simple roots. Our main tools: Grozman's Mathematica-based code SuperLie, and Python.
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double extensions of lie superalgebras in characteristic 2 with nondegenerate invariant supersymmetric bilinear form
Journal of Algebra, 2018Co-Authors: Said Benayadi, Sofiane BouarroudjAbstract:Abstract A Lie (super)algebra with a non-degenerate invariant symmetric bilinear form will be called a NIS-Lie (super)algebra. The double extension of a NIS-Lie (super)algebra is the result of simultaneously adding to it a central element and an outer derivation so that the larger algebra has also a NIS. Affine loop algebras, Lie (super)algebras with symmetrizable Cartan Matrix over any field, Manin triples, symplectic reflection (super)algebras are among the Lie (super)algebras suitable to be doubly extended. We consider double extensions of Lie superalgebras in characteristic 2, and concentrate on peculiarities of these notions related with the possibility for the bilinear form, the center, and the derivation to be odd. Two Lie superalgebras we discovered by this method are indigenous to the characteristic 2.
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simple prolongs of the non positive parts of graded lie algebras with Cartan Matrix in characteristic 2
arXiv: Representation Theory, 2013Co-Authors: Sofiane Bouarroudj, Pavel Grozman, Dimitry Leites, Alexei Lebedev, Irina ShchepochkinaAbstract:Over an algebraically closed fields, an alternative to the method due to Kostrikin and Shafarevich was recently suggested. It produces all known simple finite dimensional Lie algebras in characteristic p > 2. For p = 2, we investigate one of the steps of this method, interpret several other simple Lie algebras, previously known only as sums of their components, as Lie algebras of vector fields. One new series of exceptional simple Lie algebras is discovered, together with its "hidden supersymmetries". In characteristic 2, certain simple Lie algebras are "desuperizations" of simple Lie superal- gebras. Several simple Lie algebras we describe as results of generalized Cartan prolongation of the non-positive parts, relative a simplest (by declaring degree of just one pair of root vec- tors corresponding to opposite simple roots nonzero) grading by integers, of Lie algebras with Cartan Matrix are "desuperizations" of characteristic 2 versions of complex simple exceptional vectorial Lie superalgebras. We list the Lie superalgebras (some of them new) obtained from the Lie algebras considered by declaring certain generators odd. One of the simple Lie algebras obtained is the prolong relative to a non-simplest grading, so the classification to be obtained might be more involved than we previously thought.
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divided power co homology presentations of simple finite dimensional modular lie superalgebras with Cartan Matrix
arXiv: Representation Theory, 2009Co-Authors: Sofiane Bouarroudj, Pavel Grozman, Alexei Lebedev, Dimitry LeitesAbstract:For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras with indecomposable Cartan Matrix in characteristic 2 (and in other characteristics for completeness of the picture). We correct the currently available in the literature notions of Chevalley generators and Cartan Matrix in the modular and super cases, and an auxiliary notion of the Dynkin diagram. In characteristic 2, the defining relations of simple classical Lie algebras of the A, D, E types are not only Serre ones; these non-Serre relations are same for Lie superalgebras with the same Cartan Matrix and any distribution of parities of the generators. Presentations of simple orthogonal Lie algebras having no Cartan Matrix are also given..
Euiyong Park - One of the best experts on this subject based on the ideXlab platform.
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geometric realization of khovanov lauda rouquier algebras associated with borcherds Cartan data
Proceedings of The London Mathematical Society, 2013Co-Authors: Seokjin Kang, Masaki Kashiwara, Euiyong ParkAbstract:We construct a geometric realization of the Khovanov-Lauda-Rouquier algebra R associated with a symmetric Borcherds-Cartan Matrix A = (aij)i,j∈I via quiver varieties. As an application, if aii 6= 0 for any i ∈ I , we prove that there exists a 1-1 correspondence between Kashiwara’s lower global basis (or Lusztig’s canonical basis) of U A (g) (resp. VA(λ)) and the set of isomorphism classes of indecomposable projective graded modules over R (resp. R).
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geometric realization of khovanov lauda rouquier algebras associated with borcherds Cartan data
arXiv: Representation Theory, 2012Co-Authors: Seokjin Kang, Masaki Kashiwara, Euiyong ParkAbstract:We construct a geometric realization of the Khovanov-Lauda-Rouquier algebra $R$ associated with a symmetric Borcherds-Cartan Matrix $A=(a_{ij})_{i,j\in I}$ via quiver varieties. As an application, if $a_{ii} \ne 0$ for any $i\in I$, we prove that there exists a 1-1 correspondence between Kashiwara's lower global basis (or Lusztig's canonical basis) of $U_\A^-(\g)$ (resp.\ $V_\A(\lambda)$) and the set of isomorphism classes of indecomposable projective graded modules over $R$ (resp.\ $R^\lambda$).
David I Olive - One of the best experts on this subject based on the ideXlab platform.
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the principal so 1 2 subalgebra of a hyperbolic kac moody algebra
arXiv: High Energy Physics - Theory, 2001Co-Authors: Hermann Nicolai, David I OliveAbstract:The analog of the principal SO(3) subalgebra of a finite dimensional simple Lie algebra can be defined for any hyperbolic Kac Moody algebra g(A) associated with a symmetrizable Cartan Matrix A, and coincides with the non-compact group SO(1,2). We exhibit the decomposition of g(A) into representations of SO(1,2); with the exception of the adjoint SO(1,2) algebra itself, all of these representations are unitary. We compute the Casimir eigenvalues; the associated ``exponents'' are complex and non-integer.
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the principal so 1 2 subalgebra of a hyperbolic kac moody algebra
Letters in Mathematical Physics, 2001Co-Authors: Hermann Nicolai, David I OliveAbstract:The analog of the principal SO(3) subalgebra of a finite-dimensional simple Lie algebra can be defined for any hyperbolic Kac–Moody algebra g(A) associated with a symmetrizable Cartan Matrix A, and coincides with the non-compact algebra SO(1,2). We exhibit the decomposition of g(A) into representations of SO(1,2). With the exception of the adjoint SO(1,2) algebra itself, all of these representations are unitary. We compute the Casimir eigenvalues; the associated ‘exponents’ are complex and noninteger.
Seokjin Kang - One of the best experts on this subject based on the ideXlab platform.
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geometric realization of khovanov lauda rouquier algebras associated with borcherds Cartan data
Proceedings of The London Mathematical Society, 2013Co-Authors: Seokjin Kang, Masaki Kashiwara, Euiyong ParkAbstract:We construct a geometric realization of the Khovanov-Lauda-Rouquier algebra R associated with a symmetric Borcherds-Cartan Matrix A = (aij)i,j∈I via quiver varieties. As an application, if aii 6= 0 for any i ∈ I , we prove that there exists a 1-1 correspondence between Kashiwara’s lower global basis (or Lusztig’s canonical basis) of U A (g) (resp. VA(λ)) and the set of isomorphism classes of indecomposable projective graded modules over R (resp. R).
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geometric realization of khovanov lauda rouquier algebras associated with borcherds Cartan data
arXiv: Representation Theory, 2012Co-Authors: Seokjin Kang, Masaki Kashiwara, Euiyong ParkAbstract:We construct a geometric realization of the Khovanov-Lauda-Rouquier algebra $R$ associated with a symmetric Borcherds-Cartan Matrix $A=(a_{ij})_{i,j\in I}$ via quiver varieties. As an application, if $a_{ii} \ne 0$ for any $i\in I$, we prove that there exists a 1-1 correspondence between Kashiwara's lower global basis (or Lusztig's canonical basis) of $U_\A^-(\g)$ (resp.\ $V_\A(\lambda)$) and the set of isomorphism classes of indecomposable projective graded modules over $R$ (resp.\ $R^\lambda$).