Lie supergroups supported over and associated to the adjoint representation

Peniche, R.; Sánchez-Valenzuela, O. A.

doi:10.1016/j.geomphys.2005.06.002

Cited by 4 publications

(4 citation statements)

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“…) of any two odd generators is symmetric, and when viewed as sections of the corresponding supermanifold sheaf, they anti-commute (see [21] or [22]).…”

Section: The Lie Algebra Versus the Lie Superalgebra Approachmentioning

confidence: 99%

The Hurwitz-Hopf map and harmonic wave functions for integer and half-integer angular momentum

2023

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Harmonic wave functions for integer and half-integer angular momentum are givenin terms of the Euler angles $(\theta,\phi,\psi)$ that definea rotation in $SO(3)$, and the Euclidean norm $r$ in ${\mathbb R}^3$,keeping the usual meaning of the spherical coordinates $(r,\theta,\phi)$.They form a Hilbert (super)-space decomposed in the form $\mathcal H=\mathcal H_0\oplus\mathcal H_1$.Following a classical work by Schwinger, $2$-dimensional harmonic oscillatorsare used to produce raising and lowering operators that changethe total angular momentum eigenvalue of the wave functions in half units.The nature of the representation space $\mathcal H$ is approached from the double covering group homomorphism $SU(2)\to SO(3)$and the topology involved is taken care of by using the Hurwitz-Hopf map $H:{\mathbb R}^4\to{\mathbb R}^3$. It is shown how to reconsider $H$ as a 2-to-1 group map, $G_0={\mathbb R}^+\times SU(2)\to {\mathbb R}^+\times SO(3)$,translating it into an assignment $(z_1,z_2)\mapsto (r,\theta,\phi,\psi)$ whose domain consists of pairs $(z_1,z_2)$ of complex variables,under the appropriate identification of ${\mathbb R}^4$ with ${\mathbb C}^2$.It is shown how the Lie algebra of $G_0$ is coupled with two HeisenbergLie algebras of $2$-dimensional (Schwinger's) harmonic oscillatorsgenerated by the operators $\{z_1,z_2,\bar{z}_1,\bar{z}_2\}$ and their adjoints. The whole set of operators gets algebraically closed either into a $13$-dimensional Lie algebra or into a $(4|8)$-dimensional Lie superalgebra.The wave functions in $\mathcal H$ can be written in terms ofpolynomials in the complex coordinates $(z_1,z_2)$and their complex conjugates $(\bar{z}_1,\bar{z}_2)$and the representations are explicitly constructed via the various highest weight (or lowest weight)vector representations of $G_0$.Finally, a new non–relativistic quantum (Schr"odinger-like) equationfor the hydrogen atom that takes into account the electron spinis introduced and expressed in terms of $(r,\theta,\phi,\psi)$and the time $t$.The equation is succeptible to be solved exactly in terms of the harmonic wave functions hereby introduced.

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“…) of any two odd generators is symmetric, and when viewed as sections of the corresponding supermanifold sheaf, they anti-commute (see [21] or [22]).…”

Section: The Lie Algebra Versus the Lie Superalgebra Approachmentioning

confidence: 99%

The Hurwitz-Hopf map and harmonic wave functions for integer and half-integer angular momentum

2023

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“…We follow the techniques of previous works (e.g., [5,10]). The computation is straightforward and we find that any left invariant super vector field can be written as a linear combination of the basis…”

Section: Heisenberg-like Super Group Structures On H := F K|k × F 1|1mentioning

confidence: 99%

“…It is easy to see that Jacobi identity holds true. Furthermore, this Lie superalgebra integrates to a unique simply connected Heisenberg-like super group (e.g., as in [4,5]). 4.…”

Section: Introductionmentioning

confidence: 99%

On Heisenberg-like Super Group Structures

Peniche

Sánchez-Valenzuela

2010

Ann. Henri Poincaré

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The ring-like structures that can be defined on the ground supermanifolds R 1|1 and C 1|1 are classified up to equivalence in the category of smooth and complex Berezin-Kostant-Leites-Manin supermanifolds. It is proved that there are three different such equivalence classes in the real case, whereas there are two for the complex field. The corresponding module structures-defined componentwise on the product of k copies of R 1|1 or C 1|1 -are also classified up to equivalence. The notions of linearity and bilinearity are reviewed and used to define Heisenberg-like super group structures. It turns out that there are three non-isomorphic real such super groups, whereas only two over the complex field. The use of the appropriate exponential maps introduces the possibility of defining Heisenberg-like super group structures on the product of k copies of the ground supermanifold, with an appropriate super circle. The corresponding classification is also obtained.

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“…Let us denote by gl 2 (C; λ, μ, ν) the C-Lie superalgebra gl 2 ⊕ gl 2 defined by the parameter values (λ, μ, ν). We now have the following statement (see [5], [6]): for any x and y in the Lie algebra gl 2 . This is the case if and only if there are nonzero constants a, b and c in the ground field C, such that,…”

Section: The Classification Problemmentioning

confidence: 99%