Steinberg李代数及其中心扩张理论
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摘要
在本文中，我们主要研究Steinberg李代数st(n，R)和Steinberg李超代数st(m，n，R)的泛中心扩张问题，也就是要计算它们的二阶同调群H_2(st(n，R))以及H_2(st(m，n，R))的结构．
Steinberg李代数st(n，R)及其泛中心扩张等问题被Bloch[Bl]，Kassel-Loday[KL]，Kassel[Ka]，Faulkner[F]，Allison-Faulkner[AF]，Berman-Moody[BM]，Gao[G1，2]和Allison-Gao[AG]等人做过许多研究．这些研究与根系分次李代数以及代数K-理论等方向都有紧密的联系．在多数情形下Steinberg李代数st(n，R)都是特殊线性李代数st(n，R)的泛中心扩张，且扩张的核同构于R的第一循环同调群H_1(R)且有H_2(st(n，R))=0．在[Bl]和[KL]中，已经证明了当n≥5时，有H_2(st(n，R))=0．
在本文的第3章中，我们计算出了当n=3，4时，H_2(st(n，R))的结构，看到它不一定为零，当K的特征较小时．由此，我们得到本文的第一个主要定理．
定理1：设K是一个含么的交换环，而R是一个含么的结合的K-代数．假设R有一组K-基包含了单位元．那么有
这里，对K上结合代数R，Rm(m∈N∪{0})的定义详见第3章的第2节．
类似于Steinberg李代数st(n，R)的理论，A．V．Mikhalev和I．A．Pinchuk在[MP]中研究了Steinberg李超代数st(m，n，R)，它是特殊线性李超代数sl(m，n，R)的一个中心扩张．他们在文中证明了当m+n≥5时，st(m，n，R)是st(m，n，R)的泛中心扩张，且中心扩张的核同构于(HC1(R))_0(?)(0)_1．在本文中，我们将看到，有必要对中心扩张核的Z_(2-)分次做一下强调．
在本文的第4章中，我们研究了当m+n=3，4时的情形，这又可以分为三种情形m=2，n=1，m=3，n=1和m=2，n=2．文中分别计算了这三种情形时H_2(m、n，R))的结构，得到本文的第二个主要定理．
定理2：设K是一个含么的交换环，而R是一个含么的结合的K-代数．假设R有一组K-基包含了单位元．那么有
这里，H_2(st(m，n，R))是一个Z_(2-)分次空间．
In this thesis, we will mainly study the universal central extensions of the Steinberg Lie algebras st(n, R) and the Steinberg Lie superalgebras st(m, n, R). It is equivalent to work out the second homology group H_2(st(n, R)) and H_2(st(m, n, R)).
Steinberg Lie algebras si(n, R) and/or their universal coverings have been studied by Bloch [B1], Kassel-Loday [KL], Kassel [Ka], Faulkner [F], Allison-Faulkner [AF], Berman-Moody [BM], Gao [G1, 2] and Allison-Gao [AG], and among others. They play an important role in the study of root graded Lie algebras and the additive algebraic K-theory. In most situations, the Steinberg Lie algebra si(n, R) is the universal covering of the Lie algebra sl_n(R) whose kernel is isomorphic to the first cyclic homology group HC_1(R) of the associative algebra R and the second Lie algebra homology group H_2(st(n,R)) = 0. It was shown in [Bl] and [KL] that H_2(st(n,R)) = 0 for n≥5.
In the Chapter 3 of this thesis, we shall work out H_2(st(n, R)) explicitly for n = 3,4, which is not necessarily equal to 0 when the base commutative ring K is of small characteristic. Our first main result of this thesis is the following.
Theorem 1: let K be a unital commutative ring and R be a unital associative K-algebra. Assume that R has a K-basis containing the identity element. Thenwhere R_m(m,∈N∩{0}) is defined in the Section 2 of Chapter 3.
Similarly with the theory of the Steinberg Lie algebras st(n, R), A.V.Mikhalev and I.A.Pinchuk [MP] studied the Steinberg Lie superalgebras st(m,n,R) which are central extensions of Lie superalgebras sl(m, n, R). They showed that when m+n≥5, st(m, n, R) is the universal central extension of sl(m, n, R) whose kernel is isomorphic to (HC_1(R))_0 (?) (0)1. Here we would like to emphasize the Z_2-gradation of the kernel.
We shall work out H_2(st(m, n, R)) explicitly for m + n = 3,4 in Chapter 4. We will treat for m = 2, n = 1, m = 3, n = 1 and m = 2, n = 2 case by case, and obtain the second main result of this thesis,
Theorem 2: let K be a unital commutative ring and R be a unital associative K-algebra. Assume that R has a K-basis containing the identity element. Thenwhich are Z_2-graded spaces.
Steinberg李代数st(n，R)及其泛中心扩张等问题被Bloch[Bl]，Kassel-Loday[KL]，Kassel[Ka]，Faulkner[F]，Allison-Faulkner[AF]，Berman-Moody[BM]，Gao[G1，2]和Allison-Gao[AG]等人做过许多研究．这些研究与根系分次李代数以及代数K-理论等方向都有紧密的联系．在多数情形下Steinberg李代数st(n，R)都是特殊线性李代数st(n，R)的泛中心扩张，且扩张的核同构于R的第一循环同调群H_1(R)且有H_2(st(n，R))=0．在[Bl]和[KL]中，已经证明了当n≥5时，有H_2(st(n，R))=0．
在本文的第3章中，我们计算出了当n=3，4时，H_2(st(n，R))的结构，看到它不一定为零，当K的特征较小时．由此，我们得到本文的第一个主要定理．
定理1：设K是一个含么的交换环，而R是一个含么的结合的K-代数．假设R有一组K-基包含了单位元．那么有
这里，对K上结合代数R，Rm(m∈N∪{0})的定义详见第3章的第2节．
类似于Steinberg李代数st(n，R)的理论，A．V．Mikhalev和I．A．Pinchuk在[MP]中研究了Steinberg李超代数st(m，n，R)，它是特殊线性李超代数sl(m，n，R)的一个中心扩张．他们在文中证明了当m+n≥5时，st(m，n，R)是st(m，n，R)的泛中心扩张，且中心扩张的核同构于(HC1(R))_0(?)(0)_1．在本文中，我们将看到，有必要对中心扩张核的Z_(2-)分次做一下强调．
在本文的第4章中，我们研究了当m+n=3，4时的情形，这又可以分为三种情形m=2，n=1，m=3，n=1和m=2，n=2．文中分别计算了这三种情形时H_2(m、n，R))的结构，得到本文的第二个主要定理．
定理2：设K是一个含么的交换环，而R是一个含么的结合的K-代数．假设R有一组K-基包含了单位元．那么有
这里，H_2(st(m，n，R))是一个Z_(2-)分次空间．
In this thesis, we will mainly study the universal central extensions of the Steinberg Lie algebras st(n, R) and the Steinberg Lie superalgebras st(m, n, R). It is equivalent to work out the second homology group H_2(st(n, R)) and H_2(st(m, n, R)).
Steinberg Lie algebras si(n, R) and/or their universal coverings have been studied by Bloch [B1], Kassel-Loday [KL], Kassel [Ka], Faulkner [F], Allison-Faulkner [AF], Berman-Moody [BM], Gao [G1, 2] and Allison-Gao [AG], and among others. They play an important role in the study of root graded Lie algebras and the additive algebraic K-theory. In most situations, the Steinberg Lie algebra si(n, R) is the universal covering of the Lie algebra sl_n(R) whose kernel is isomorphic to the first cyclic homology group HC_1(R) of the associative algebra R and the second Lie algebra homology group H_2(st(n,R)) = 0. It was shown in [Bl] and [KL] that H_2(st(n,R)) = 0 for n≥5.
In the Chapter 3 of this thesis, we shall work out H_2(st(n, R)) explicitly for n = 3,4, which is not necessarily equal to 0 when the base commutative ring K is of small characteristic. Our first main result of this thesis is the following.
Theorem 1: let K be a unital commutative ring and R be a unital associative K-algebra. Assume that R has a K-basis containing the identity element. Thenwhere R_m(m,∈N∩{0}) is defined in the Section 2 of Chapter 3.
Similarly with the theory of the Steinberg Lie algebras st(n, R), A.V.Mikhalev and I.A.Pinchuk [MP] studied the Steinberg Lie superalgebras st(m,n,R) which are central extensions of Lie superalgebras sl(m, n, R). They showed that when m+n≥5, st(m, n, R) is the universal central extension of sl(m, n, R) whose kernel is isomorphic to (HC_1(R))_0 (?) (0)1. Here we would like to emphasize the Z_2-gradation of the kernel.
We shall work out H_2(st(m, n, R)) explicitly for m + n = 3,4 in Chapter 4. We will treat for m = 2, n = 1, m = 3, n = 1 and m = 2, n = 2 case by case, and obtain the second main result of this thesis,
Theorem 2: let K be a unital commutative ring and R be a unital associative K-algebra. Assume that R has a K-basis containing the identity element. Thenwhich are Z_2-graded spaces.
引文
[AABGP] B. N. Allison, S. Azam, S. Berman, Y .Gao and A. Pianzola, Extended affine Lie algebras and their root systems, Mem. Amer. Math. Soc. (605) 126 (1997)
[ABG1 ] B. N. Allison, G. M. Benkart, Y. Gao, Central extensions of Lie algebras graded by finite root systems, Math. Ann. 316 (2000) 499-527
[ABG2] B. N. Allison, G. M. Benkart, Y. Gao, Lie algebra graded by the root systems BC_r, r ≥ 2, Mem. Amer. Math. Soc. (751) 158 (2002)
[AF] B. N. Allison and J. R. Faulkner, Nonassociative coefficient algebras for Steinberg unitary Lie algebras, J. Algebra 161 (1993) 1-19.
[AG] B. N. Allison and Y. Gao, Central quotients and corverings of Steinberg unitary Algebras, Canad. J. Math. 17 (1996),261-304.
[BA1] G. Benkart and E. Alberto, Lie superalgebras graded by the root system A(m,n), J. Lie Theory 12 (2003), no. 2, 387-400
[BA2] G. Benkart and E. Alberto, Lie superalgebras graded by the root system B(m,n), Slecta Math. (N.S.) 9 (2003), no. 3, 313-360
[BA3] G. Benkart and E. Alberto, Lie superalgebras graded by the root systems C(n), D(m, n), D(2,1, α), F(4), C(3), Canad. Math. Bull. 45 (2002),no. 4,509-524
[BAM] G. Benkart, E. Alberto and C. Martinec, A(m,n)-graded Lie superalgebras, J. Reine. Angew Math. 573 (2004), 139-156
[BeM] G. M. Benkart and R. V. Moody, Derivations, central extensions and affine Lie algebras, Algebras, Groups and Geometries 3 (1986) 456-492.
[BGK] S. Berman, Y. Gao and Y. S. Krylyuk, Quantum tori and structure of elliptic quasi-simple Lie algebras, J. Func. Anal. 135 (1996), no. 2, 339-389
[BK] S. Berman and Y. S. Krylyuk, Universal central extensions of twisted and untwisted Lie algebras extended over commutative rings, J. Algebra 173 (1995), no. 2, 302-347
[BM] S. Berman and R. V. Moody, Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy, Invent. Math. 108 (1992) 323-347.
[B1] S. Bloch, The dilogarithm and extensions of Lie algebras, Alg. K-theory, Evanston 1980, Springer Lecture Notes in Math 854 (1981) 1-23.
[Bou] N. Bourbaki, Groupes et algebres de Lie, Chap. IV, V, VI. Paris: Hermann 1968
[BS] S. Berman and J. Szmigielski, Principal realization for the extended affine Lie algebra of type sl_2 with coordinates in a simple quantum torus with two generators, Contemporary Mathematics 248 (1999),39-67
[BeS] G. Benkart and E. Smirnov, Lie algebras graded by the root system BC_1, J. Lie Theory 23 (2003), no. 1,91-132
[BeZ1] G. Benkart and E. Zelmanov, Lie algebras graded by the root systems, Proc. the Third Inter. Conf. on Nonassoc. alg. and appl.(S. Gonzalez, ed.), Kluwer Publ., 1994, 31-38
[BeZ2] G. Benkart and E. Zelmanov, Lie algebras graded by the finite root systems and intersection martrix algebras, Invent. Math. 126 (1996), 1-45
[C] R. W. Carter, Simple groups of Lie type, London: Wiley 1972
[CG] H. Chen and Y. Gao, BC_N-graded Lie algebras arising from fermionic representations, J. Algebra 311(2007), no. 1,216-230
[CGS] H. Chen, Y. Gao and S. Shang, BC(0, N)-graded Lie superalgebras coordinatized by quantum tori, Sci. China Ser. A 49 (2006), no. 11,1740-1752
[CWZ] S-J. Chen, W. Wang and R. Zhang, A Fock space approach to representation theory of osp(2|2n), Transform. Groups 23 (2007), no. 2, 209-225
[Fa] J. R. Faulkner, Barbilian planes, Geom. Dedicata 30 (1989) 125-181.
[FF] A. J. Feingold and I.B.Frenkel, Classical affine algebras, Adv. Math. 56 (1985), 117-172.
[Fr1] I. B. Frenkel, Spinor representations of affine Lie algebras, Proc. Nat. Acad. Sci. USA 77 (1980),6303-6306.
[Fr2] I. B. Frenkel, Two constructions of affine Lie algebra representations and Boson-Fermion correspondence in quantum field theory, J. Funct. Anal. 44 (1981),259-327.
[G1] Y. Gao, On the Steinberg Lie algebras st_2(R), Comm. in Alg. 21 (1993) 3691-3706.
[G2] Y. Gao, Central extensions of nonsymmetrizable Kac-Moody algebras over commutative algebras, Proc. Amer. Math. Soc. 121 (1994) 67-76.
[G3] Y. Gao, Involutive Lie algebras graded by finite root systems and compact forms of IM algebras, Math. Z. 223 (1996), no. 4,651-672
[G4] Y. Gao, Steiberg Unitary Lie Algebras and Skew-Dihedral Homology, J. Algebra, 17 (1996),261-304.
[G5] Y. Gao, Fermionic and bosonic representations of the extended affine Lie algebra (?), Canad. Math. Bull. 45 (2002),623-633.
[G6] Y. Gao, Representations of extended affine Lie algebras coordinatized by certain quantum tori, Compositio Math. 123 (2000), 1-25.
[G7] Y.Gao, Vertex operators arising from the homogeneous realization for (?)_N, Comm. Math. Phys.211(2000),745-777.
[Ga] H. Garland, The arithmetic theory of loop groups, Publ. Math. IHES 52 (1980) 5-136
[GG] M. Goto and F. D. Grosshans, Semisimple Lie algebras, Lect. Notes. Pure. Appl. Math. Vol 38, New York: M. Dekker 1978
[GS] Y. Gao and S. Shang, Universal coverings of Steinberg Lie algebras of small characteristic, J. Algebra 311 (2007), no. 1, 216-230
[GZ] Y. Gao and Z. Zeng, Hermitian representations of the extended affine Lie algebra (?)_2(C_q), Adv. math. 2007 (2006), no. 1, 244-265
[H] J. B. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts Math, Vol 9, Springer 1972
[H-KT] R. H0egh-Krohn and B. Torresani, Classification and construction of quasi simple Lie algebras, J. Functional Anal. 89 (1990), 106-136
[JK] H. P. Jakobsen and V. G. Kac, A new class of unitarizable highest weight representations of infinite-dimensional Lie algebras II, J. Funct. Anal. 82 (1989),69-90.
[K1] V. G. Kac, Infinite dimensional Lie algebras, third edition, Cambridge Univ. Press 1990
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[K3] V. G. Kac, Characters of typical representations of classical Lie superalgebras, Comm. Alg. 5(1977)889-897
[K4] V. G. Kac, Representations of classical Lie superalgebras, Lect. Notes Math 676 (1978) 597-626
[Ka] C. Kassel, Kahler differentials and coverings of complex simple Lie algebras extended over a commutative ring, J. Pure and Appl. Alg. 34 (1984) 265-275.
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[SG1] S. Shang and Y. Gao, eu_2-Lie admissible algebras and Steinberg unitary Lie algebras, Contemporary Mathematics, (to appear)
[SG2] S. Shang and Y. Gao, A class of Lie algebras graded by Klein four K_4 and their universal coverings, preprint
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[ABG1 ] B. N. Allison, G. M. Benkart, Y. Gao, Central extensions of Lie algebras graded by finite root systems, Math. Ann. 316 (2000) 499-527
[ABG2] B. N. Allison, G. M. Benkart, Y. Gao, Lie algebra graded by the root systems BC_r, r ≥ 2, Mem. Amer. Math. Soc. (751) 158 (2002)
[AF] B. N. Allison and J. R. Faulkner, Nonassociative coefficient algebras for Steinberg unitary Lie algebras, J. Algebra 161 (1993) 1-19.
[AG] B. N. Allison and Y. Gao, Central quotients and corverings of Steinberg unitary Algebras, Canad. J. Math. 17 (1996),261-304.
[BA1] G. Benkart and E. Alberto, Lie superalgebras graded by the root system A(m,n), J. Lie Theory 12 (2003), no. 2, 387-400
[BA2] G. Benkart and E. Alberto, Lie superalgebras graded by the root system B(m,n), Slecta Math. (N.S.) 9 (2003), no. 3, 313-360
[BA3] G. Benkart and E. Alberto, Lie superalgebras graded by the root systems C(n), D(m, n), D(2,1, α), F(4), C(3), Canad. Math. Bull. 45 (2002),no. 4,509-524
[BAM] G. Benkart, E. Alberto and C. Martinec, A(m,n)-graded Lie superalgebras, J. Reine. Angew Math. 573 (2004), 139-156
[BeM] G. M. Benkart and R. V. Moody, Derivations, central extensions and affine Lie algebras, Algebras, Groups and Geometries 3 (1986) 456-492.
[BGK] S. Berman, Y. Gao and Y. S. Krylyuk, Quantum tori and structure of elliptic quasi-simple Lie algebras, J. Func. Anal. 135 (1996), no. 2, 339-389
[BK] S. Berman and Y. S. Krylyuk, Universal central extensions of twisted and untwisted Lie algebras extended over commutative rings, J. Algebra 173 (1995), no. 2, 302-347
[BM] S. Berman and R. V. Moody, Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy, Invent. Math. 108 (1992) 323-347.
[B1] S. Bloch, The dilogarithm and extensions of Lie algebras, Alg. K-theory, Evanston 1980, Springer Lecture Notes in Math 854 (1981) 1-23.
[Bou] N. Bourbaki, Groupes et algebres de Lie, Chap. IV, V, VI. Paris: Hermann 1968
[BS] S. Berman and J. Szmigielski, Principal realization for the extended affine Lie algebra of type sl_2 with coordinates in a simple quantum torus with two generators, Contemporary Mathematics 248 (1999),39-67
[BeS] G. Benkart and E. Smirnov, Lie algebras graded by the root system BC_1, J. Lie Theory 23 (2003), no. 1,91-132
[BeZ1] G. Benkart and E. Zelmanov, Lie algebras graded by the root systems, Proc. the Third Inter. Conf. on Nonassoc. alg. and appl.(S. Gonzalez, ed.), Kluwer Publ., 1994, 31-38
[BeZ2] G. Benkart and E. Zelmanov, Lie algebras graded by the finite root systems and intersection martrix algebras, Invent. Math. 126 (1996), 1-45
[C] R. W. Carter, Simple groups of Lie type, London: Wiley 1972
[CG] H. Chen and Y. Gao, BC_N-graded Lie algebras arising from fermionic representations, J. Algebra 311(2007), no. 1,216-230
[CGS] H. Chen, Y. Gao and S. Shang, BC(0, N)-graded Lie superalgebras coordinatized by quantum tori, Sci. China Ser. A 49 (2006), no. 11,1740-1752
[CWZ] S-J. Chen, W. Wang and R. Zhang, A Fock space approach to representation theory of osp(2|2n), Transform. Groups 23 (2007), no. 2, 209-225
[Fa] J. R. Faulkner, Barbilian planes, Geom. Dedicata 30 (1989) 125-181.
[FF] A. J. Feingold and I.B.Frenkel, Classical affine algebras, Adv. Math. 56 (1985), 117-172.
[Fr1] I. B. Frenkel, Spinor representations of affine Lie algebras, Proc. Nat. Acad. Sci. USA 77 (1980),6303-6306.
[Fr2] I. B. Frenkel, Two constructions of affine Lie algebra representations and Boson-Fermion correspondence in quantum field theory, J. Funct. Anal. 44 (1981),259-327.
[G1] Y. Gao, On the Steinberg Lie algebras st_2(R), Comm. in Alg. 21 (1993) 3691-3706.
[G2] Y. Gao, Central extensions of nonsymmetrizable Kac-Moody algebras over commutative algebras, Proc. Amer. Math. Soc. 121 (1994) 67-76.
[G3] Y. Gao, Involutive Lie algebras graded by finite root systems and compact forms of IM algebras, Math. Z. 223 (1996), no. 4,651-672
[G4] Y. Gao, Steiberg Unitary Lie Algebras and Skew-Dihedral Homology, J. Algebra, 17 (1996),261-304.
[G5] Y. Gao, Fermionic and bosonic representations of the extended affine Lie algebra (?), Canad. Math. Bull. 45 (2002),623-633.
[G6] Y. Gao, Representations of extended affine Lie algebras coordinatized by certain quantum tori, Compositio Math. 123 (2000), 1-25.
[G7] Y.Gao, Vertex operators arising from the homogeneous realization for (?)_N, Comm. Math. Phys.211(2000),745-777.
[Ga] H. Garland, The arithmetic theory of loop groups, Publ. Math. IHES 52 (1980) 5-136
[GG] M. Goto and F. D. Grosshans, Semisimple Lie algebras, Lect. Notes. Pure. Appl. Math. Vol 38, New York: M. Dekker 1978
[GS] Y. Gao and S. Shang, Universal coverings of Steinberg Lie algebras of small characteristic, J. Algebra 311 (2007), no. 1, 216-230
[GZ] Y. Gao and Z. Zeng, Hermitian representations of the extended affine Lie algebra (?)_2(C_q), Adv. math. 2007 (2006), no. 1, 244-265
[H] J. B. Humphreys, Introduction to Lie algebras and representation theory, Grad. Texts Math, Vol 9, Springer 1972
[H-KT] R. H0egh-Krohn and B. Torresani, Classification and construction of quasi simple Lie algebras, J. Functional Anal. 89 (1990), 106-136
[JK] H. P. Jakobsen and V. G. Kac, A new class of unitarizable highest weight representations of infinite-dimensional Lie algebras II, J. Funct. Anal. 82 (1989),69-90.
[K1] V. G. Kac, Infinite dimensional Lie algebras, third edition, Cambridge Univ. Press 1990
[K2] V. G. Kac, Lie superalgebras, Adv. Math. 26 (1977) 8-96
[K3] V. G. Kac, Characters of typical representations of classical Lie superalgebras, Comm. Alg. 5(1977)889-897
[K4] V. G. Kac, Representations of classical Lie superalgebras, Lect. Notes Math 676 (1978) 597-626
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