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## Über dieses Buch

This book provides explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic codes, combinatorics and algebraic varieties, summarizing the author’s works and his joint works with his former students. Further, it presents various oscillator generalizations of the classical representation theorem on harmonic polynomials, and highlights new functors from the representation category of a simple Lie algebra to that of another simple Lie algebra.

Partial differential equations play a key role in solving certain representation problems. The weight matrices of the minimal and adjoint representations over the simple Lie algebras of types E and F are proved to generate ternary orthogonal linear codes with large minimal distances. New multi-variable hypergeometric functions related to the root systems of simple Lie algebras are introduced in connection with quantum many-body systems in one dimension. In addition, the book identifies certain equivalent combinatorial properties on representation formulas, and the irreducibility of representations is proved directly related to algebraic varieties. The book offers a valuable reference guide for mathematicians and scientists alike. As it is largely self-contained – readers need only a minimal background in calculus and linear algebra – it can also be used as a textbook.

## Inhaltsverzeichnis

### Chapter 1. Preliminary of Lie Algebras

We give basic concepts and examples of Lie algebras. Moreover, Engel’s theorem on nilpotent Lie algebras and Lie’s theorem on solvable Lie algebras are proved. Furthermore, we derive the Jordan-Chevalley decomposition of a linear transformation and use it to show Cartan’s criterion on the solvability.
Xiaoping Xu

### Chapter 2. Semisimple Lie Algebras

We use the Killing form to derive the decomposition of a finite-dimensional semisimple Lie algebra over $$\mathbb {C}$$ into a direct sum of simple ideals. Moreover, we prove the Weyl’s theorem of complete reducibility, and the equivalence of the complete reducibility of real and complex modules is also given. Cartan’s root-space decomposition of a finite-dimensional semisimple Lie algebra over $$\mathbb {C}$$ is derived.
Xiaoping Xu

### Chapter 3. Root Systems

We start with the axiom of root system and give the root systems of special linear algebras, orthogonal Lie algebras and symplectic Lie algebras. Then we derive some basic properties of root systems. As finite symmetries of root systems, the Weyl groups are introduced and studied in detail. The classification and explicit constructions of root systems are presented. The automorphism groups of root systems are determined. As a preparation for later representation theory of Lie algebras, the corresponding weight lattices and their saturated subsets are investigated.
Xiaoping Xu

### Chapter 4. Isomorphisms, Conjugacy and Exceptional Types

We show that the structure of a finite-dimensional semisimple Lie algebra over $$\mathbb {C}$$ is completely determined by its root system. Moreover, we prove that any two Cartan subalgebras of such a Lie algebra $${\mathscr {G}}$$ are conjugated under the group of inner automorphisms of $${\mathscr {G}}$$. In particular, the automorphism group of $${\mathscr {G}}$$ is determined when it is simple. Furthermore, we give explicit constructions of the simple Lie algebras of exceptional types.
Xiaoping Xu

### Chapter 5. Highest-Weight Representation Theory

We study finite-dimensional irreducible representations of a finite-dimensional semisimple Lie algebra over $$\mathbb {C}$$. First we introduce the universal enveloping algebra of a Lie algebra and prove the Poincaré-Birkhoff-Witt (PBW) Theorem. Then we use the universal enveloping algebra o to construct Verma modules. Moreover, we prove that any finite-dimensional-module is the quotient of a Verma module modulo its maximal proper submodule, whose generators are explicitly given. Furthermore, the Weyl’s character formula is derived and the dimensional formula is determined.
Xiaoping Xu

### Chapter 6. Representations of Special Linear Algebras

First we present a fundamental lemma of solving flag partial differential equations for polynomial solutions. Then we present the canonical bosonic and fermionic oscillator representations of type A over their minimal natural modules and minimal orthogonal modules. Moreover, we determine the structure of the noncanonical oscillator representations obtained from the canonical bosonic oscillator representations over their minimal natural modules and minimal orthogonal modules by partially swapping differential operators and multiplication operators. Furthermore we construct a functor from the category of $$A_{n-1}$$-modules to the category of $$A_n$$-modules, which is related to n-dimensional projective transformations. Finally, we present multi-parameter families of irreducible projective oscillator representations of the algebras.
Xiaoping Xu

### Chapter 7. Representations of Even Orthogonal Lie Algebras

First we present the canonical bosonic and fermionic oscillator representations over their minimal natural modules of type D. Then we determine the structure of the noncanonical oscillator representations obtained from the above bosonic representations by partially swapping differential operators and multiplication operators. Furthermore, we speak about a functor from the category of $$D_n$$-modules to the category of $$D_{n+1}$$-modules, which is related to 2n-dimensional conformal transformations. In addition, we present multi-parameter families of irreducible conformal oscillator representations of the algebras and some of them are related to an explicitly given algebraic variety.
Xiaoping Xu

### Chapter 8. Representations of Odd Orthogonal Lie Algebras

We give the canonical bosonic and fermionic oscillator representations over their minimal natural modules of type B. Moreover, we determine the structure of the noncanonical oscillator representations obtained from the above bosonic representations by partially swapping differential operators and multiplication operators. Furthermore, we present a functor from the category of $$B_{n}$$-modules to the category of $$B_{n+1}$$-modules, which is related to $$(2n + 1)$$-dimensional conformal transformations. Besides, we present multi-parameter families of irreducible conformal oscillator representations of the algebras and some of them are related to an explicitly given algebraic variety.
Xiaoping Xu

### Chapter 9. Representations of Symplectic Lie Algebras

We determin the structure of the canonical bosonic and fermionic oscillator representations of type C over their minimal natural modules. Moreover, we study the noncanonical oscillator representations obtained from the above bosonic representations by partially swapping differential operators and multiplication operators. Finally, we present multi parameter families of irreducible projective oscillator representations of the algebras.
Xiaoping Xu

### Chapter 10. Representations of and

We determine the structure of the canonical bosonic and fermionic oscillator representations of the simple Lie algebra of type $$G_2$$ over its 7-dimensional module. Moreover, we present a one-parameter family of conformal oscillator representations of $$G_2$$ derived from those of the simple Lie algebra of type $$B_3$$ and determine their irreducibility. Furthermore, we use partial differential equations to find the explicit irreducible decomposition of the space of polynomial functions on 26-dimensional basic irreducible module of the simple Lie algebra of type $$F_4$$.
Xiaoping Xu

### Chapter 11. Representations of

This chapter studies explicit representations of the simple Lie algebra of type $$E_6$$. First we prove the cubic $$E_6$$-generalization of the classical theorem on harmonic polynomials. Then we study the functor from the module category of $$D_5$$ to the module category of $$E_6$$. Finally, we give a family of inhomogeneous oscillator representations of the simple Lie algebra of type $$E_6$$ on a space of exponential-polynomial functions and prove that their irreducibility is related to an explicit given algebraic variety.
Xiaoping Xu

### Chapter 12. Representations of

Explicit representations of the simple Lie algebra of type $$E_7$$ are given . By solving certain partial differential equations, we find the explicit decomposition of the polynomial algebra over the 56-dimensional basic irreducible module of the simple Lie algebra $$E_7$$ into a sum of irreducible submodules. Then we study the functor from the module category of $$E_6$$ to the module category of $$E_7$$ developed. Moreover, we construct a family of irreducible inhomogeneous oscillator representations of the simple Lie algebra of type $$E_7$$ on a space of exponential-polynomial functions, related to an explicitly given algebraic variety.
Xiaoping Xu

### Chapter 13. Oscillator Representations of gl(n|m) and

We first establish two-parameter $$\mathbb {Z}^2$$-graded supersymmetric oscillator generalizations of the classical theorem on harmonic polynomials for the general linear Lie superalgebra. Then we extend the result to two-parameter $$\mathbb {Z}$$-graded supersymmetric oscillator generalizations of the classical theorem on harmonic polynomials for the ortho-symplectic Lie superalgebras.
Xiaoping Xu

### Chapter 14. Representation Theoretic Codes

We study the binary and ternary orthogonal codes generated by the weight matrices of finite-dimensional modules of simple Lie algebras. The Weyl groups of the Lie algebras act on these codes isometrically. It turns out that certain weight matrices of the simple Lie algebras of types A and D generate doubly-even binary orthogonal codes and ternary orthogonal codes with large minimal distances. Moreover, we prove that the weight matrices of $$F_4$$, $$E_6$$, $$E_7$$ and $$E_8$$ on their minimal irreducible modules and adjoint modules all generate ternary orthogonal codes with large minimal distances.
Xiaoping Xu

### Chapter 15. Path Hypergeometric Functions

We prove that certain variations of the classical Weyl functions are solutions of the Calogero-Sutherland model and its generalizations—the Olshanestsky-Perelomov model in various cases. New multi-variable hypergeometric functions related to the root systems of classical simple Lie algebras are introduced. In particular, those of type A give rise to solutions of the Calogero-Sutherland model and those of type C yield solutions of the Olshanestsky-Perelomov model of type. The differential properties and multi-variable hypergeometric equations for these multi-variable hypergeometric functions are given. The Euler integral representations of the type-A functions are found.
Xiaoping Xu

### Backmatter

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