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Super-Algebra and Super-Geometry



A super Lie algebra is the analog of a Lie algebra in superalgebra/supergeometry.

See also supersymmetry.



A super Lie algebra over a field kk is a Lie algebra internal to the symmetric monoidal kk-linear category SVect of super vector spaces.


This means that a super Lie algebra is

  1. a super vector space 𝔤=𝔤 even𝔤 odd\mathfrak{g} = \mathfrak{g}_{even} \oplus \mathfrak{g}_{odd};

  2. equipped with a bilinear bracket

    [,]:𝔤𝔤𝔤 [-,-] : \mathfrak{g}\otimes \mathfrak{g} \to \mathfrak{g}

    that is graded skew-symmetric: it is skew symmetric on 𝔤 even\mathfrak{g}_{even} and symmetric on 𝔤 odd\mathfrak{g}_{odd}.

  3. that satisfies the 2\mathbb{Z}_2-graded Jacobi identity in that for any three element x,y,z𝕘x,y,z \in \mathbb{g} of homogeneous degree deg(x),deg(y),deg(z))deg(x),deg(y),deg(z))\in \mathbb{Z} then

    [x,[y,z]]=[[x,y],z]+(1) deg(x)deg(y)[y,[x,z]]. [x, [y, z]] = [[x,y],z] + (-1)^{deg(x)\cdot deg(y)} [y, [x,z]] \,.

Equivalently, a super Lie algebra is a “super-representable” Lie algebra internal to the cohesive (∞,1)-topos Super∞Grpd over the site of super points.

See the discussion at superalgebra for details on this.



(Kac 77a, Kac 77b) states a classification of super Lie algebras which are

  1. finite dimensional

  2. simple

  3. over a field of characteristic zero.

Such an algebra is called of classical type if the action of its even-degree part on the odd-degree part is completely reducible. Those simple finite dimensional algebras not of classical type are of Cartan type.

  1. classical type

    1. four infinite series

      1. A(m,n)A(m,n)

      2. B(m,n)=B(m,n) = osp(2m+1,2n)(2m+1,2n) m0m\geq 0, n>0n \gt 0

      3. C(n)C(n)

      4. D(m,n)=D(m,n) = osp(2m,2n)(2m,2n) m2m \geq 2, n>0n \gt 0

    2. two exceptional ones

      1. F(4)F(4)

      2. G(3)G(3)

    3. a family D(2,1;α)D(2,1;\alpha) of deformations of D(2,1)D(2,1)

    4. two “strange” series

      1. P(n)P(n)

      2. Q(n)Q(n)

  2. Cartan type


The underlying even-graded Lie algebra for type 2 is as follows

𝔤\mathfrak{g}𝔤 even\mathfrak{g}_{even}𝔤 even\mathfrak{g}_{even} rep on 𝔤 odd\mathfrak{g}_{odd}
B(m,n)B(m,n)B mC nB_m \oplus C_nvector \otimes vector
D(m,n)D(m,n)D mC nD_m \oplus C_nvector \otimes vector
D(2,1,α)D(2,1,\alpha)A 1A 1A 1A_1 \oplus A_1 \oplus A_1vector \otimes vector \otimes vector
F(4)F(4)B 3A 1B_3\otimes A_1spinor \otimes vector
G(3)G(3)G 2A 1G_2\oplus A_1spinor \otimes vector
Q(n)Q(n)A nA_nadjoint

For type 1 the /2\mathbb{Z}/2\mathbb{Z}-grading lifts to an \mathbb{Z}-grading with 𝔤=𝔤 1𝔤 0𝔤 1\mathfrak{g} = \mathfrak{g}_{-1}\oplus \mathfrak{g}_0 \oplus \mathfrak{g}_1.

𝔤\mathfrak{g}𝔤 even\mathfrak{g}_{even}𝔤 even\mathfrak{g}_{even} rep on 𝔤 1\mathfrak{g}_{{-1}}
A(m,n)A(m,n)A mA nCA_m \oplus A_n \oplus Cvector \otimes vector \otimes \mathbb{C}
A(m,m)A(m,m)A mA nA_m \oplus A_nvector \otimes vector
C(n)C(n) 1\mathbb{C}_{-1} \oplus \mathbb{C}vector \otimes \mathbb{C}

reviewed e.g. in (Farmer 84, p. 25,26, Minwalla 98, section 4.1).



According to Kac77b the definition of super Lie algebra is originally due to

  • Felix Berezin, G. I. Kac, Math. Sbornik 82, 343—351 (1970) (Russian)

The original references on the classification of super Lie algebras are

  • Victor Kac, Lie superalgebras, Advances in Math. 26 (1977), no. 1, 8–96.

  • Victor Kac, A sketch of Lie superalgebra theory, Comm. Math. Phys. Volume 53, Number 1 (1977), 31-64. (EUCLID)

See also

  • Werner Nahm, V. Rittenberg, Manfred Scheunert, The classification of graded Lie algebras , Physics Letters B Volume 61, Issue 4, 12 April 1976, Pages 383–384 (publisher)

  • M. Parker, Classification Of Real Simple Lie Superalgebras Of Classical Type, J.Math.Phys. 21 (1980) 689-697 (spire)

Further discussion of classification related specifically to classification of supersymmetry is due to

Introductions and surveys include

  • Richard Joseph Farmer, Orthosymplectic superalgebras in mathematics and science, PhD Thesis (1984) (web, pdf)

  • L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Superalgebras (arXiv:hep-th/9607161)

  • Groeger, Super Lie groups and super Lie algebras, lecture notes 2011 (pdf)

  • L. Frappat, A. Sciarrino, P. Sorba, Dictionary on Lie Superalgebras (arXiv:hep-th/9607161)

  • D. Leites, Lie superalgebras, J. Soviet Math. 30 (1985), 2481–2512 (web)

  • Manfred Scheunert, The theory of Lie superalgebras. An introduction, Lect. Notes Math. 716 (1979)

  • D. Westra, Superrings and supergroups (pdf)

  • Shiraz Minwalla, Restrictions imposed by superconformal invariance on quan tum field theories Adv. Theor. Math. Phys. 2, 781 (1998) (arXiv:hep-th/9712074).

Discussion of Lie algebra extensions for super Lie algebras includes

Revised on April 1, 2015 04:41:43 by Urs Schreiber (