In abstract algebra, a representation of an associative algebra is a module for that algebra. Here an associative algebra is a (not necessarily unital)...

mathematical field of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices...

commutative Banach algebras as algebras of continuous functions; the fact that for commutative C*-algebras, this representation is an isometric isomorphism...

element of G. One may always pass from a representation of a Lie group G to a representation of its Lie algebra by taking the derivative at the identity...

the corresponding 'infinitesimal' representations of Lie algebras. A complex representation of a group is an action by a group on a finite-dimensional...

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of...

In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a...

Hopf algebra Index of a Lie algebra Leibniz algebra Lie algebra cohomology Lie algebra extension Lie algebra representation Lie bialgebra Lie coalgebra...

the fundamental or defining representation for s l 2 C {\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} } . The Lie algebra s l 2 C {\displaystyle {\mathfrak...

X)-(-1)^{AB}B\cdot (A\cdot X).} Equivalently, a representation of L is a Z2-graded representation of the universal enveloping algebra of L which respects the third equation...

abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra. That is, a representation of a Hopf algebra H...

mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero...

mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory...

{\displaystyle \pi } is unitary if and only if the associated Lie algebra representation d π : g → E n d ( H ) {\displaystyle d\pi :{\mathfrak {g}}\rightarrow...

of V. A trivial representation of an associative or Lie algebra is a (Lie) algebra representation for which all elements of the algebra act as the zero...

In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem...

In mathematics, an algebraic representation of a group G on a k-algebra A is a linear representation π : G → G L ( A ) {\displaystyle \pi :G\to GL(A)}...

Morita equivalence Progenerator Representation theory Algebra representation Group representation Lie algebra representation Maschke's theorem Schur's lemma...

a Lie algebra g {\displaystyle {\mathfrak {g}}} is solvable if its derived series terminates in the zero subalgebra. The derived Lie algebra of the Lie...

Élie Cartan in his doctoral thesis. It controls the representation theory of a semi-simple Lie algebra g {\displaystyle {\mathfrak {g}}} over a field of...

and Lie algebras. According to this connection, the different quantum states of an elementary particle give rise to an irreducible representation of the...

In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group...

that its non-trivial representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra E8 itself; it is also...

Aff(A). Similarly, an affine representation of a Lie algebra g on A is a Lie algebra homomorphism from g to the Lie algebra aff(A) of the affine group of...

\;f\cdot g)} . The Lie algebra retains its form, with indices µ and ν now taking values between 0 and d − 1. The alternative representation in terms of Ji and...

(or its Lie algebra of infinitesimal rotations), they are typically defined as elements of a vector space that carries a linear representation of the Clifford...

interest for algebra is in the process in the other direction: given a projective representation, try to 'lift' it to an ordinary linear representation. A general...

mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure...

equivalent representation: The set of traceless Hermitian n × n complex matrices with Lie bracket given by −i times the commutator. The Lie algebra s u ( n...

Lie algebra of SU(2). Since the group SU(2) is simply connected, every representation of its Lie algebra can be integrated to a group representation; we...