Author: Peter Niemann
Publisher: American Mathematical Soc.
ISBN: 0821828886
Category : Mathematics
Languages : en
Pages : 137
Book Description
Starting from Borcherds' fake monster Lie algebra, this text construct a sequence of six generalized Kac-Moody algebras whose denominator formulas, root systems and all root multiplicities can be described explicitly. The root systems decompose space into convex holes, of finite and affine type, similar to the situation in the case of the Leech lattice. As a corollary, we obtain strong upper bounds for the root multiplicities of a number of hyperbolic Lie algebras, including $AE_3$.
Some Generalized Kac-Moody Algebras with Known Root Multiplicities
Author: Peter Niemann
Publisher: American Mathematical Soc.
ISBN: 0821828886
Category : Mathematics
Languages : en
Pages : 137
Book Description
Starting from Borcherds' fake monster Lie algebra, this text construct a sequence of six generalized Kac-Moody algebras whose denominator formulas, root systems and all root multiplicities can be described explicitly. The root systems decompose space into convex holes, of finite and affine type, similar to the situation in the case of the Leech lattice. As a corollary, we obtain strong upper bounds for the root multiplicities of a number of hyperbolic Lie algebras, including $AE_3$.
Publisher: American Mathematical Soc.
ISBN: 0821828886
Category : Mathematics
Languages : en
Pages : 137
Book Description
Starting from Borcherds' fake monster Lie algebra, this text construct a sequence of six generalized Kac-Moody algebras whose denominator formulas, root systems and all root multiplicities can be described explicitly. The root systems decompose space into convex holes, of finite and affine type, similar to the situation in the case of the Leech lattice. As a corollary, we obtain strong upper bounds for the root multiplicities of a number of hyperbolic Lie algebras, including $AE_3$.
Some Generalized Kac-Moody Algebras with Known Root Multiplicities
Author: Nanhua XI
Publisher:
ISBN: 9781470403393
Category : Electronic books
Languages : en
Pages : 119
Book Description
Introduction Generalized Kac-Moody algebras Modular forms Lattices and their Theta-functions The proof of Theorem 1.7 The real simple roots Hyperbolic Lie algebras Appendix A Appendix B Bibliography Notation
Publisher:
ISBN: 9781470403393
Category : Electronic books
Languages : en
Pages : 119
Book Description
Introduction Generalized Kac-Moody algebras Modular forms Lattices and their Theta-functions The proof of Theorem 1.7 The real simple roots Hyperbolic Lie algebras Appendix A Appendix B Bibliography Notation
Root Multiplicities of the Indefinite Type Kac-Moody Algebras HC[subscript N](1)
Author:
Publisher:
ISBN:
Category :
Languages : en
Pages :
Book Description
Victor Kac and Robert Moody independently introduced Kac-Moody algebras around 1968. These Lie algebras have numerous applications in physics and mathematics and thus have been the subject of much study over the last three decades. Kac-Moody algebras are classified as finite, affine, or indefinite type. A basic problem concerning these algebras is finding their root multiplicities. The root multiplicities of finite and affine type Kac-Moody algebras are well known. However, determining the root multiplicities of indefinite type Kac-Moody algebras is an open problem. In this thesis we determine the multiplicities of some roots of the indefinite type Kac-Moody algebras HC[subscript n](1). A well known construction allows us to view HC[subscript n](1) as the minimal graded Lie algebra with local part V direct sum g0 direct sum V', where g0 is the affine Kac-Moody algebra C[subscript n](1). and V, V' are suitable g0-modules. From this viewpoint, root spaces of HC[subscript n](1) become weight spaces of certain C[subscript n](1)-modules. Using a multiplicity formula due to Kang we reduce our problem to finding weight multiplicities in certain irreducible highest weight C[subscript n](1)-modules. We then use crystal basis theory for the affine Kac-Moody algebras C[subscript n](1) to find these weight multiplicities. With this strategy we calculate the multiplicities of some roots of HC[subscript n](1). In particular, we determine the multiplicities of the level two roots -2(alpha1)-k(delta) of HC[subscript n](1) for 1 less than or equal to k less than or equal to 10. We also show that the multiplicities of the roots of HC[subscript n](1) of the form -l(alpha−1) -k(delta) are n for l equal to k and 0 for l greater than k. In the process, we observe that Frenkel's c.
Publisher:
ISBN:
Category :
Languages : en
Pages :
Book Description
Victor Kac and Robert Moody independently introduced Kac-Moody algebras around 1968. These Lie algebras have numerous applications in physics and mathematics and thus have been the subject of much study over the last three decades. Kac-Moody algebras are classified as finite, affine, or indefinite type. A basic problem concerning these algebras is finding their root multiplicities. The root multiplicities of finite and affine type Kac-Moody algebras are well known. However, determining the root multiplicities of indefinite type Kac-Moody algebras is an open problem. In this thesis we determine the multiplicities of some roots of the indefinite type Kac-Moody algebras HC[subscript n](1). A well known construction allows us to view HC[subscript n](1) as the minimal graded Lie algebra with local part V direct sum g0 direct sum V', where g0 is the affine Kac-Moody algebra C[subscript n](1). and V, V' are suitable g0-modules. From this viewpoint, root spaces of HC[subscript n](1) become weight spaces of certain C[subscript n](1)-modules. Using a multiplicity formula due to Kang we reduce our problem to finding weight multiplicities in certain irreducible highest weight C[subscript n](1)-modules. We then use crystal basis theory for the affine Kac-Moody algebras C[subscript n](1) to find these weight multiplicities. With this strategy we calculate the multiplicities of some roots of HC[subscript n](1). In particular, we determine the multiplicities of the level two roots -2(alpha1)-k(delta) of HC[subscript n](1) for 1 less than or equal to k less than or equal to 10. We also show that the multiplicities of the roots of HC[subscript n](1) of the form -l(alpha−1) -k(delta) are n for l equal to k and 0 for l greater than k. In the process, we observe that Frenkel's c.
Recent Advances in Representation Theory, Quantum Groups, Algebraic Geometry, and Related Topics
Author: Pramod M. Achar
Publisher: American Mathematical Society
ISBN: 0821898523
Category : Mathematics
Languages : en
Pages : 296
Book Description
This volume contains the proceedings of two AMS Special Sessions "Geometric and Algebraic Aspects of Representation Theory" and "Quantum Groups and Noncommutative Algebraic Geometry" held October 13–14, 2012, at Tulane University, New Orleans, Louisiana. Included in this volume are original research and some survey articles on various aspects of representations of algebras including Kac—Moody algebras, Lie superalgebras, quantum groups, toroidal algebras, Leibniz algebras and their connections with other areas of mathematics and mathematical physics.
Publisher: American Mathematical Society
ISBN: 0821898523
Category : Mathematics
Languages : en
Pages : 296
Book Description
This volume contains the proceedings of two AMS Special Sessions "Geometric and Algebraic Aspects of Representation Theory" and "Quantum Groups and Noncommutative Algebraic Geometry" held October 13–14, 2012, at Tulane University, New Orleans, Louisiana. Included in this volume are original research and some survey articles on various aspects of representations of algebras including Kac—Moody algebras, Lie superalgebras, quantum groups, toroidal algebras, Leibniz algebras and their connections with other areas of mathematics and mathematical physics.
Automorphic Forms and Lie Superalgebras
Author: Urmie Ray
Publisher: Springer Science & Business Media
ISBN: 1402050100
Category : Mathematics
Languages : en
Pages : 293
Book Description
This book provides the reader with the tools to understand the ongoing classification and construction project of Lie superalgebras. It presents the material in as simple terms as possible. Coverage specifically details Borcherds-Kac-Moody superalgebras. The book examines the link between the above class of Lie superalgebras and automorphic form and explains their construction from lattice vertex algebras. It also includes all necessary background information.
Publisher: Springer Science & Business Media
ISBN: 1402050100
Category : Mathematics
Languages : en
Pages : 293
Book Description
This book provides the reader with the tools to understand the ongoing classification and construction project of Lie superalgebras. It presents the material in as simple terms as possible. Coverage specifically details Borcherds-Kac-Moody superalgebras. The book examines the link between the above class of Lie superalgebras and automorphic form and explains their construction from lattice vertex algebras. It also includes all necessary background information.
Root Multiplicities of Some Kac-Moody Lie Algebras of Indefinite Type
Author: Jennifer Mae Hontz
Publisher:
ISBN:
Category : Kac-Moody algebras
Languages : en
Pages : 284
Book Description
Publisher:
ISBN:
Category : Kac-Moody algebras
Languages : en
Pages : 284
Book Description
Root Multiplicities of Some Hyperbolic Kac-Moody Lie Algebras
Author: Michael Aaron Baker
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Book Description
Publisher:
ISBN:
Category :
Languages : en
Pages : 0
Book Description
Dualities on Generalized Koszul Algebras
Author: Edward L. Green
Publisher: American Mathematical Soc.
ISBN: 0821829343
Category : Mathematics
Languages : en
Pages : 90
Book Description
Koszul rings are graded rings which have played an important role in algebraic topology, noncommutative algebraic geometry and in the theory of quantum groups. One aspect of the theory is to compare the module theory for a Koszul ring and its Koszul dual. There are dualities between subcategories of graded modules; the Koszul modules.
Publisher: American Mathematical Soc.
ISBN: 0821829343
Category : Mathematics
Languages : en
Pages : 90
Book Description
Koszul rings are graded rings which have played an important role in algebraic topology, noncommutative algebraic geometry and in the theory of quantum groups. One aspect of the theory is to compare the module theory for a Koszul ring and its Koszul dual. There are dualities between subcategories of graded modules; the Koszul modules.
Kac Algebras Arising from Composition of Subfactors: General Theory and Classification
Author: Masaki Izumi
Publisher: American Mathematical Soc.
ISBN: 0821829351
Category : Mathematics
Languages : en
Pages : 215
Book Description
This title deals with a map $\alpha$ from a finite group $G$ into the automorphism group $Aut({\mathcal L})$ of a factor ${\mathcal L}$ satisfying (i) $G=N \rtimes H$ is a semi-direct product, (ii) the induced map $g \in G \to [\alpha_g] \in Out({\mathcal L})=Aut({\mathcal L})/Int({\mathcal L})$ is an injective homomorphism, and (iii) the restrictions $\alpha \! \! \mid_N, \alpha \! \! \mid_H$ are genuine actions of the subgroups on the factor ${\mathcal L}$. The pair ${\mathcal M}={\mathcal L} \rtimes_{\alpha} H \supseteq {\mathcal N}={\mathcal L} DEGREES{\alpha\mid_N}$ (of the crossed product ${\mathcal L} \rtimes_{\alpha} H$ and the fixed-point algebra ${\mathcal L} DEGREES{\alpha\mid_N}$) gives an irreducible inclusion of factors with Jones index $\# G$. The inclusion ${\mathcal M} \supseteq {\mathcal N}$ is of depth $2$ and hence known to correspond to a Kac algebra of dim
Publisher: American Mathematical Soc.
ISBN: 0821829351
Category : Mathematics
Languages : en
Pages : 215
Book Description
This title deals with a map $\alpha$ from a finite group $G$ into the automorphism group $Aut({\mathcal L})$ of a factor ${\mathcal L}$ satisfying (i) $G=N \rtimes H$ is a semi-direct product, (ii) the induced map $g \in G \to [\alpha_g] \in Out({\mathcal L})=Aut({\mathcal L})/Int({\mathcal L})$ is an injective homomorphism, and (iii) the restrictions $\alpha \! \! \mid_N, \alpha \! \! \mid_H$ are genuine actions of the subgroups on the factor ${\mathcal L}$. The pair ${\mathcal M}={\mathcal L} \rtimes_{\alpha} H \supseteq {\mathcal N}={\mathcal L} DEGREES{\alpha\mid_N}$ (of the crossed product ${\mathcal L} \rtimes_{\alpha} H$ and the fixed-point algebra ${\mathcal L} DEGREES{\alpha\mid_N}$) gives an irreducible inclusion of factors with Jones index $\# G$. The inclusion ${\mathcal M} \supseteq {\mathcal N}$ is of depth $2$ and hence known to correspond to a Kac algebra of dim
Kac-Moody Lie Algebras and Related Topics
Author: Neelacanta Sthanumoorthy
Publisher: American Mathematical Soc.
ISBN: 0821833375
Category : Mathematics
Languages : en
Pages : 384
Book Description
This volume is the proceedings of the Ramanujan International Symposium on Kac-Moody Lie algebras and their applications. The symposium provided researchers in mathematics and physics with the opportunity to discuss new developments in this rapidly-growing area of research. The book contains several excellent articles with new and significant results. It is suitable for graduate students and researchers working in Kac-Moody Lie algebras, their applications, and related areas of research.
Publisher: American Mathematical Soc.
ISBN: 0821833375
Category : Mathematics
Languages : en
Pages : 384
Book Description
This volume is the proceedings of the Ramanujan International Symposium on Kac-Moody Lie algebras and their applications. The symposium provided researchers in mathematics and physics with the opportunity to discuss new developments in this rapidly-growing area of research. The book contains several excellent articles with new and significant results. It is suitable for graduate students and researchers working in Kac-Moody Lie algebras, their applications, and related areas of research.